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After my last question I have worked through the math quite a bit and now I'm stuck again. This time my question is less wordy.
I have two equations for $t$, one with respect to each $a_{x}$ and $a_{y}$:
\begin{align*}
t&=\frac{-2(u_{x}^{A}-v_{x}^{B})\pm\sqrt{4(u_{x}^{A}-v_{x}^{B})^{2}-8a_{x}(c_{x}^{A}-c_{x}^{B})}}{2a_... |
My notes for determining the Maximum Likelihood decision rule for classifying two fish given two features (lightness and width) have a different decision rule if the features are correlated to when they're uncorrelated.
ML Decision rule for 2 independant features:
if p(lightness | sea bass)p(width | sea bass) >= p(ligh... |
Hatcher uses the words "path-component" and "component" without giving their definition. The first occurrence of "path-component" is on p. 19, exercise 12. The first occurrence of "component" is on p. 6, example 0.2.
I think "path-component" means "path-connected component". But what is a "component" of a topological s... |
I'm doing stuck with the first half of exercise 12 on page 19 in Hatcher:
Exercise:
Show that a homotopy equivalence $f : X \rightarrow Y$ induces a bijection between the set of path-components of $X$ and the set of path-components of $Y$. and that $f$ restricts to a homotopy equivalence from each path-component of $X$... |
I'm trying to do exercise 5 on page 18 in Hatcher:
Show that if a space $X$ deformation retracts to a point $x \in X$, then for each neighborhood $U$ of $x$ in $X$ there exists a neighborhood $V \subset U$ of $x$ such that the inclusion map $V \rightarrow U$ is nullhomotopic.
My question:
Does the existence part of the... |
Are there any good online resources to learn multivariate statistics? ( With topics such as Multiple Linear Regression, Principal Component Analysis, Factor Analysis, Cluster Analysis and Discriminant Analysis)
|
Sorry for my English.
Here is the question:
$G=(v,e)$, undirected graph, $V=\{v_1,v_2,\ldots,v_n\}$. the vertices are organized in sequence from the smaller one to the biggest $v_1,\ldots,v_n$.
We will define a spread distribution $(C_1,\ldots,C_k)$ as follows:
For each $i,j$, $C_i \cap C_j = \emptyset$.
$\bigcup_{i=1... |
Define the $q$-analog $(a;q)_n = \prod_{k=0}^n \left(1 - aq^k\right)$.
I want to prove the identity $\frac{(q^2;q^2)_\infty}{(q;q)_\infty}=\frac{1}{(q;q^2)_\infty}$.
I viewed the LHS this way:
$$\frac{1 - q^2}{1 - q} \frac{1 - q^4}{1 - q^2} \frac{1 - q^6}{1 - q^3} \frac{1 - q^8}{1 - q^4} \frac{1 - q^{10}}{1 - q^5} \fra... |
Please help me prove by induction that
$\displaystyle\forall n\in {{\mathbb{N}}^{*}}$, $\displaystyle\forall {{a}_{1}},\ldots ,{{a}_{n}}\in {\mathbb{R}}^{*}_{+}$, $\displaystyle \ln \left( \prod\limits_{j=1}^{n}{{{a}_{j}}} \right)=\sum\limits_{j=1}^{n}{\ln \left( {{a}_{j}} \right)}$.
Deduce that $\displaystyle \fora... |
Take for example $f(z)=e^z$, so the inverse is $z(f) = \ln(f) + n\pi i$ for an arbitrarily chosen (but fixed) branch $n\in\mathbb N$. Now if $f$ is restricted to e.g. $0<|1-f|<1$ such that the essential singularity at $f=0$ is not part of the definition region, is $z(f)$ then analytical in that region?
|
Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why they use Jacobian matrix to illustrate this theorem.
|
I have a question about Legendre symbol.
Let $a$, $b$ be integers. Let $p$ be a prime not dividing $a$. Show that the Legendre symbol verifies:
$$\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0.$$
I know that $\displaystyle\sum_{m=0}^{p-1} \left(\frac{m}{p}\right)=0$, but how do I connect this with the previous formula... |
Let $f$ be a non-negative measurable function in $\mathbb{R}$. Suppose that $$\iint_{\mathbb{R}^2}{f(4x)f(x-3y)\,dxdy}=2\,.$$ Calculate $\int_{-\infty}^{\infty}{f(x)\,dx}$.
My first thought was to change the order of integration so that I integrate in $y$ first, since there's only a single $y$ in the integrand... but... |
For example, two Green functions:
\begin{equation}
G_1(\tau_1 - \tau_2, x) = \alpha_1(\tau_1 - \tau_2)\beta_1(x) H(\tau_1 - \tau_2) + \mu_1(\tau_1 - \tau_2) \nu_1 (x) H(\tau_2 - \tau_1)
\end{equation}
\begin{equation}
G_2(\tau_2 - \tau_3, x') = \alpha_2(\tau_2 - \tau_3)\beta_2(x') H(\tau_2 - \tau_3) + \mu_2(\tau_2 - \... |
What are the general uses of the hat and star symbol in math? Or could you please point me to a page that discusses this? Thanks.
|
How can I find the volume of the solid generated when the region enclosed by $y=0, x=0, x=1$ and $(1+e^{-2x})^{0.5}$ is rotated through $360^\circ$ about the x axis?
|
Someone asked this question about how many ways there are to prove $0.999\dots = 1$ and I posted this:
$$ 0.99999 \dots = \sum_{k = 1}^\infty \frac{9}{10^k} = 9 \sum_{k = 1}^\infty \frac{1}{10^k} = 9 \Big ( \frac{1}{1 - \frac{1}{10}} - 1\Big ) = \frac{9}{9} = 1$$
The question was a duplicate so in the end it was clos... |
Suppose $H$,$K$ are normal subgroups of a group $G$. We don't assume $G$ is finite. Furthermore assume $[G:H]$ and $[G:K]$ are both finite. Then it follows that $[G: H \cap K]$ is divisible by $[G:H]$. Why is this?
Here's what I thought. Suppose we write $G = \bigcup_{i \in I} a_{i}(H \cap K)$ as a disjoint
union of le... |
I'm reading this article by Keith Conrad, on subgroup series. I'm having trouble with a statement he does at page 6:
Any subgroup of $G$ which contains $[G,G]$ is normal in $G$.
He says this as evidence that commutator and center play dual roles, since any subgroup of $G$ contained in $Z(G)$ is normal in $G$. Now, th... |
Definition. A subspace of a vector space V is a subset H of V that has three properties
(a) The zero vector of V is in H.
(b) $\forall$ u, v $\in H$, we have u + v $\in H$.
(c) $\forall$ u $\in H$, c scalar, we have c u $\in H$.
For $n \geq$ 0, denote $\mathbb{P}_n$ to be the set of all polynomials of degree at most $... |
So, I'm given:
$$f(x) = \left\{\begin{array}{ll}
x^2\sin\left(\frac{1}{x}\right) & \mbox{if $x\neq 0$,}\\
0 &\mbox{if $x=0$.}
\end{array}\right.$$
I'm asked to show that this piecewise function is both continuous and differentiable on the real number line.
I'm not sure how to state what I know, i.e. that since $x^2$ ... |
How would I go about showing that the third coefficient of the 2nd order Newton's interpolating polynomial is :
$$a3=D^2y1= \frac{Dy2-Dy1}{x3-x1}=
\frac{\frac{y_3-y_2}{x_3-x_2}-\frac{y_2-y_1}{x_2-x_1}}{x_3-x_1}$$
I dont know how to use math equations here, but I hope putting this in word and pressing the INSERT EQUATI... |
Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is
$$\mathbb{E}[\exp(-\lambda H_a)] = \exp (-\sqrt{2\lambda} H_a)$$
by considering the martingale $$M_t = \exp \left(\theta B_t -\frac{1}{2}\theta^2 t... |
I've been reading generationgfunctionology by Herbert S. Wilf (you can find a copy of the second edition on the author's page here).
On page 33 he does something I find weird. He wants to shuffle the index forward and does so like this:
\begin{align*}
(f_{n+1})_{n\in N_0} &= \frac{(f(X)-f(0))}{X}\\
(f_{n})_{n\in N_0} &... |
What is the 2D kernel, k, that when convoluted with a 2D signal, f, that is convoluted again with a gaussian 2D kernel, g, produces a result that is closest to the original signal, f'.
Something like this, I think:
For
f' = f $\star$ k $\star$ g (where $\star$ denotes convolution)
Find k, such that |f - f'| is minimal.... |
I'm trying to rewrite Taylor's formula on $\mathbb{R}^n$.
Let $p\in \mathbb{N}$, $V \subset \mathbb{R}^n$ open and convex, $\mathbf{x, a} \in V$, and $f:V\to \mathbb{R}$ where $f \in C^p(V)$. Then
$$f(\mathbf{x}) = \sum_{k=0}^{p-1} \frac{1}{k!} D^{(k)} f(\mathbf{a}; \mathbf{x-a}) + \frac{1}{(p-1)!} \int_0^1 (1-t)^{p-1}... |
Let $\mathscr{C}$ be a strict monoidal category. I will denote the product of $\mathscr{C}$ by $\otimes$. The Drinfeld center $\mathscr{Z(C)}$ of $\mathscr{C}$ is the category with object $(X,\phi)$ where $X$ is an object of $\mathscr{C}$ and $\phi$ is a natural isomorphism from $X \otimes -$ to $ - \otimes X$. Morhphi... |
Why is the following true?
$$\sum_{j=0}^{n-1}w(2\pi j/n)\left[\frac{1}{n}\sum_{k=0}^{n-1}e^{2\pi ik(j-m)/n}\right]=w(2\pi m/n)$$
|
I was assigned a few problems in my Honors Calculus II class, and one of them was kind of interesting to do:
Suppose that $f_{n}$ are nonnegative bounded functions on $A$ and let $M_{n} = \sup f_{n}$. If $\displaystyle\sum\limits_{n=1}^\infty f_{n}$ converges uniformly on $A$, does it follow that $\displaystyle\sum\l... |
I had the following on an exam today.
Let the universe = $Z^+$, $Z = (1,2,3,....100)$
$|A| = 10$
$|B| = 10$
$|A\cap B| = 0$
(a) What is $|A^c \cup B^c|$
I thought about it like this.
Let set A be dogs
Let set B be cats.
Or let A be even numbers and B be odd numbers. Sticking with dogs and cats though.
A-compliment is... |
please help me establish(etablir):
$\forall n\in \mathbb{N}-\left\{ 0,\left. 1 \right\} \right.$ , $x\in \mathbb{R}-\left\{ \pi \mathbb{Z} \right\}$ , $\left| \sin \left( nx \right) \right|<n\left| \sin x \right|$
thx in advance...
|
Let $H^s(\mathbb{R}^d):= \{ u \in \mathcal{S}' : (1+|\xi|^2)^{s/2}\hat{u}(\xi) \in L^2(\mathbb{R}^d)\}$. It can be shown that this space is a Hilbert space and that $H^s \subset H^t$ if $t \leq s$.
Now suppose we have $t > s>0$ such that $H^s \subsetneq H^t$. Then we know that $(H^s)^* \supsetneq (H^t)^*$, where $^*$ ... |
context: i am currently learning about the Radon Nicodym Theorem.
I was wondering what the notation $du$ represent, where $u$ is a measure. I mean just $du$ by itself, not within an integral or anything.
|
In the introduction to André Weil's Number of solutions of equations in finite fields he mentions article 358 of Gauss' Disquisitiones. Can someone please show me the connection here:
How does the Gaussian sum of order $3$, for primes of the form $p = 3n+1$ determine the number of solutions for all congruence $ax^3 - b... |
Question: If $f\colon G\to H$ is a group homomorphism, and $|G:\mathrm{Ker}(f)|$ and $|H|$ are coprime, then show that $f(G)=1$
Has anyone experienced the ":" notation before? Does it have something to do with cosets? any hints would be greatly appreciated. Thanks for reading.
|
What is the most general object that defines a solution to an ordinary differential equation? (I don't know enough to know if this question is specific enough. I am hoping the answer will be something like "a function", "a continuous function", "a piecewise continuous function" ... or something like this.)
|
A steepest tangent vector to the explicit surface $z=f(x,y)$ at the fixed point $(x,y,f(x,y))$ is given by the formula:
$$ \mbox{steepTan} = (?)(?)(?) $$
You can replace the $?$s with formulas like
$$\frac{\partial}{\partial x}[f(x, y)]$$
One of the hints is that the steepest tan is the $(dx,dy,?)$. The hardest part i... |
I have the following question:
If $I,J$ are ideals in $R$, then denote $I+J = \{ r \in R | r = x+y, x \in I, y \in J \}$
It is not hard to show that $I+J$ is again an ideal in $R$
If we then assume that $I+J = R$ we have the identity $R/(I \cap J) \simeq R/I \times R/J$
I managed to show this by constructing a functio... |
Does anybody know of a good reference (preferably online) where I can find a good, rigorous description of $Hom_R(C_\bullet,D_\bullet)_\bullet$, which is a cochain complex where the module is the nth degree is given as $\Pi_{i-j=n}Hom_R(C_i,D_j)$, where $C_\bullet$ and $D_\bullet$ are chain complexes? I would give wha... |
Let $c,d>0$. I do not know how to integrate the following:
$$\int_{-d}^d \frac{z}{2\sqrt{z+c}\ \cdot \sqrt{d^2-z^2}}\ \text{d}z.$$
Sorry~~ it should be $z^2$.
I think it can be simplified to the following: for $c\ge d>0$
$$\frac{\sqrt{d}}{2} \int_{-\pi/2}^{\pi/2} \frac{\sin y}{\sqrt{\sin
y+c/d}}\ \text{d}y.$$
Thank... |
What math subjects are used in quantitative trading firms?
|
Let us assume that directional derivative of a function $f$ exists at a point $p$ (i.e.,$ D_v(f)$) for all vectors $v \in \mathbb{R}^{n}$. Does it imply that the function is differentiable?
|
If $a^{n-1} \equiv 1 \pmod{n}$, can we say that $(a,n) = 1$? If not, what conditions do we need to make this argument true? Any idea?
Thanks,
|
I am trying to prove that the set of irrational numbers $\mathbb{R} \setminus \mathbb{Q}$ is not an $F_{\sigma}$ set. Here's my attempt:
Assume that indeed $\mathbb{R} \setminus \mathbb{Q}$ is an $F_{\sigma}$ set. Then we may write it as a countable union of closed subsets $C_i$:
$$
\mathbb{R} \setminus \mathbb{Q} = \... |
I am wondering how to prove that the intersection of an infinite number of convex sets is convex.
I can prove that the intersection of two convex sets is convex, and I believe that I can simply do an induction on this result, but I've heard that it would be wrong to do this since I am working with infinity.
I guess a... |
I'm reading Fractals Everywhere by Michael Barnsley. On pp. 6-8 [1] he defines a linear space which, he says, "is also called a vector space." However, his definition of a linear space only requires closure under vector addition and scalar multiplication. There is no mention, for example, of additive inverses. (I reali... |
Today we were discussing how for an nxn orthogonal projection matrix from $\mathbb{R^{n}}$ onto a subspace W, Ker($A$)=$(Im$A$)^{\perp}$=$W^{\perp}$ and that Ker($A^{T}$) is also $W^{\perp}$. This prompted the question of what conditions are necessary for a matrix so that the kernels of it and its transpose are equal. ... |
In some integration by parts problems, such as evaluating the integral of $e^x \cos x$ or $\sec^ 3 x$, one performs integration by parts (possibly more than once, and possibly together with algebraic manipulations) and eventually the original integral appears again.
To beginning students, this may superficially appear ... |
Here is a problem from generatingfunctionology that I'm stuck on:
I'm trying to get started on part (a). I broke the string up like this. If the last digit is $0$, the number of possible strings is then $f(n-1,m,k)$. If the last digit is $1$, there are two subcases. If the $n-1^{th}$ digit is $0$, then we can cut them... |
The question is to calculate the homology groups of the chain complex:
$0 \to A \stackrel{n}{\to} A \to 0$, where $A$ is an Abelian group and $n \in \mathbb{N}$.
I don't see a nice way to get anything out of this other than $H_1 = \mathrm{ker} \ \partial_1$ and $H_0 = A / \mathrm{Im} \ \partial_1$
Some quick calculati... |
I want to move one object (dot) in circular path.
By using x and y position of that object.
Thanks.
|
I have this question that I took a shot at but I am not very familiar with Hermite or Laguerre, this my first time running across these type of polynomials and need some help please.
(a) The first four Hermite polynomials are $1, 2t,-2+4t^2,$ and $-12t+8t^3$. These polynomials arise naturally in the study of certain im... |
My attempt was:
$x^3 + 2x + 2 \equiv 0 \pmod{25}$
By inspection, we see that $x \equiv 1 \pmod{5}$. is a solution of $x^3 + 2x + 2 \equiv 0 \pmod{5}$. Let $x = 1 + 5k$, then we have:
$$(1 + 5k)^3 + 2(1 + 5k) + 2 \equiv 0 \pmod{25}$$
$$\Leftrightarrow 125k^3 + 75k^2 + 25k + 5 \equiv 0 \pmod{25}$$
$$\Leftrightarrow 5 \e... |
Let $G$ be a finite group such that for any two subgroups $H_{1}$ and $H_{2}$ of $G$ we have $H_{1} \subseteq H_{2}$ or $H_{2} \subseteq H_{1}$. Why this implies $G$ is a cyclic group?
Ah. I think this works: Suppose $g \in G$ then $G = \langle g \rangle$. Suppose otherwise, then we can find $z$ such that $z \not \in \... |
I have equation
As much I know, I have four points lets say A(x1,y1), B(x2,y2), C(x3,y3), D(x4, y4). How these 'P' are related to these set of points. And what might a Cubic Bézier curves be for the this (A, B, C, D) given set of points.
[UPDATE]
Having problem uploading image, but image somewhat looks like this
http... |
Determine if the set $Z$ of all matricies form
$ \left[ \begin{array}{cc}
a & b \\
0 & d
\end{array} \right] $
is a subspace of $M_{2 \times 2}$ (the set of all $2 \times 2$ matrices).
% This is something I came up with. Can someone look at it and let me know any useful corrections/suggestions to the question plea... |
Given a function $f:z\mapsto f(z)$ for a discrete set of points in the real interval $z\in[a,b]$ and the knowledge that $f$ is analytical along the real axis and that its Fourier transform is real valued along $\mathbb R$ (and thus $f(-z)=f^*(z)$), is it possible to reconstruct $f(z)$ for all $z\in\mathbb C$? If not, w... |
How would you be solve sin x = 1 - x, without drawing the graph and manually measuring the point of intersection?
|
It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots.
I know that for an equation $x^2 = 9$, the solution is $x = \pm 3$. But if simply given $\s... |
What are the theorems/results/widely applicable results in graph theory that everyone should know about?
|
For any module $P$, we define $\mathrm{tr}(P)=\sum \mathrm{im}(f)$, where $f$ ranges over all elements of $\mathrm{Hom}(P,R)$， and call it trace.
Why does it have such a name? Does it have any relation to the trace of matrix？
|
The definition of one-to-one was pretty straight forward. If the vectors are lin.indep the transformation would be one-to-one. But could it also be onto?
The definition of onto was a little more abstract. Would a zero-row in reduced echelon form have any effect on this? I just assumed that because it has a couple of f... |
I am trying to calculate the center of gravity of a polygon.
My problem is that I need to be able to calculate the center of gravity for both regular and irregular polygons and even self intersecting polygons.
Is that possible?
I've also read that: http://paulbourke.net/geometry/polyarea/ But this is restricted to non ... |
How should you break $\displaystyle \frac{x^n}{(1-x)(1-2x)(1-3x)\ldots(1-nx)}$ into partial fractions?
|
I'm trying to learn number theory on my own, and here's a proof I'm not quite sure I got right. It feels too simple(?), I'm thinking maybe I'm missing something. So the question is:
Prove that if $n$ is a square, then each exponent in its prime-power decomposition is even.
My proof:
Let $n=m^2$, with $m$ having prime... |
I have some trouble understanding the record value for a sequence of i.i.d. random variables of geometric distribution. Following quotation is from Univariate discrete distributions By Norman Lloyd Johnson, Adrienne W. Kemp, Samuel Kotz.
The lack-of-memory property of the
geometric distribution gives it a role
com... |
How do I express the matlab function abs(M), on a matrix $M$, in mathematical terms?
I thought about norms or just $|M|$, but these return scalars, not another matrix of the same size as $M$.
Sorry for the rough explanation, english isn't my primary language.
|
In most Descriptive Set Theory books, the rationale for working with the Baire space ($\mathbb{N}^{\mathbb{N}}$) as opposed to the real line ($\mathbb{R}$) is that the connectedness of the latter causes 'technical difficulties'.
My question is, what are these technical difficulties, and why does Descriptive Set Theory... |
Let $m > 0$ and $ a:[0,1] \rightarrow \mathbb R$ be a function with $a(\epsilon) \rightarrow 0$.
Then $ \epsilon^m a(\epsilon) \rightarrow 0 $ for $ \epsilon \rightarrow 0$.
But is $ \epsilon^m a(\epsilon) \in o(\epsilon^m) $, with O-Notation used? Put into other words, if we multiply a zero sequence which converges by... |
I have a graph with a cycle ($v_1,\ldots,v_k, v_1=v_k$).
Claim: If there is a cycle with 2 edges of the same weight, and they are the heaviest edges in this cycle, then there is more than one Minimum spanning tree.
I am trying to think about contradiction, but I can't find one. All my examples create more than one Mini... |
As we all know, the theory of class fields began at the paper by Takagi on the abelian extensions of the field of rational numbers. Then one naturally has the following
Where can one find the English version of that paper, or had this paper been translated into English yet?
Thanks.
|
Let $f(m,n,w)$ be the probability density function of $F$ variable with $m$ numerator $df$ and $n$ denominator $df$, i.e.
$$f(m,n,w)=\frac{\Gamma\left(\frac{m+n}{2}\right)(m/n)^{m/2}}{\Gamma(m/2)\Gamma(n/2)}w^{(m/2)-1}\left(1+\frac{mw}{n}\right)^{-(m+n)/2}$$
I am interested in the infimum of $\int_1^\infty f(m,n,w) dw$... |
I'm trying to find the equation of a line that passes through (1, 2, -1) and creates equivalent angles with the three positive axes. I'm at a loss, though I'd like just a small hint, not the full answer. Thanks!
|
I was trying to understand the Ergodic theory recently, but I don't really have any knowledge about linear operators, Markov semigroups etc. so I didn't even fully understand the definition.
Could someone recommend a good book, notes or similar resource to learn about Linear operators and Markov semigroups?
|
Is it possible to define a bijection $f:\mathbb{R}\setminus\mathbb{R}^{-}\rightarrow[0,1)$ such that $f$ is continuously differentiable on its entire domain?
|
The Fourier transform of a sinc function is the top hat function. So, if $\{y_k\};\ k\in\{0,1,...,n-1\}$ are samples of the sinc function, sampled $T$ apart, the discrete Fourier transform is
$$Y_m=\sum_{k=0}^{n-1}y_k e^{i 2\pi mk/n}$$
Now, if I choose $T$ to be greater than 1, it doesn't satisfy the sampling criterio... |
When one talks about the category $V_K$ of vector spaces over a field $K$ and considers the dual functor $D$ which maps a vector space $V$ to its dual $V^{*}$ I believe to have in mind something like a labeled category, the labels letting me know which object is the dual of another object. (Or can I see this by careful... |
Given the prior probability of 2 distributions N(x,y) and N(a,b), where N(μ,σ^2):
how do you make a decision rule to minimize the probability of error, if the prior probabilities are equal? Can you give an example?
What if the prior probabilities are different, such as
P(Distribution 1) = 0.70
P(Distribution 2) =... |
While playing around with programming, a problem suddenly came across my mind.
Given $n$ let's say $2^{64}$ (64-bit unsigned integer). Find a positive integer $x$, $1 \leq x \leq n$, such that $x$ has maximum divisors.
My attempt was:
Let $n = p_1^{a_1} \times p_2^{a_2} \times ... \times p_k^{a_k}.$
The number of di... |
So the cup product is not well defined over co-chain groups, but all the books claim it is well defined over co-homology groups. The only thing I am not clear on is invariance under ordering/re-ordering of simplices when we go to the co-homology level. Every book seems to gloss over this, and after doing a few examples... |
The answer is probably gonna be very straightforward but I'm missing it. I want to prove the following claim:
Suppose we play a game where we win with probability $p$ and lose with probability $q = 1 - p$, where $p \neq \frac{1}{2}$. Denote by $X_i$ the outcome of game $i$ ($X_i = 1$ if we win and $X_i = -1$ if we lo... |
Suppose a samurai wants to try out his new sword and cuts a piece of bamboo twice, randomly, so now there are $3$ lenghts of bamboo. What is the probability of these 3 pieces being able to form a triangle?
I have never came across a continuous probability problem before, but I tried doing it anyway and got a result of ... |
Note:This is more a math-recreational question
Consider the series $\exp_p(x)=\sum_{k=0}^{\infty}\frac{x^k}{(k!)^p}$ which is some systematic modification of the exponential function. It's $\exp_1(x)=\exp(x) $ if p=1. I'm interested in the behaviour for $x=0\to -\infty$ depending on change of the parameter $p$.
[e... |
I have proved the following problem, can you help me check if there is any loopholes in my proof?
Let I be an open interval in R, let $c \in I$, and let $f, g\colon I\to \mathbb{R}$ be functions. Suppose that $f(c) = g(c)$, and that $f(x) \leq g(x)$ for all $x \in I$. Prove that if f and g are differentiable at c, then... |
G=(v,e) , with weight on the edges than can be only a or b (when $a
I need to find MST of the graph in O(v+e).
I think to put all the edges in array, and than scanning the array. first only check about a, and after about b. The algorithm is like Kruskal's: check about evey edge if it doesnt form a cycle. but I'm not s... |
While studying the various algorithm implementations available on-line of the Fast Fourier Transform algorithm, I've come to a question related to the way the DFT works in theory.
Suppose you have a sequence of $N$ points $x_0, ..., x_{N-1}$. For $ k = 0, ..., N-1 $, let $ X_k = \sum_{n=0}^{N-1} x_n e^{-2ik\pi \frac{n}... |
I've seen questions like this on a lot of old qualifying exams in algebraic topology. The type of question is usually of the form:
Let $X$, $Y$ be some topological spaces. Prove that $X$ can't be embedded in $Y$.
The problem I have is that I don't see how homology or homotopy or really anything I know helps. For examp... |
Let $u=p_1^{n_1} \cdots p_t^{n_t}$ for $p_i$ prime and $n_i \in \mathbb{N}$. Let $m_i = \frac{u}{p_i^{n_i}}$. Then there exist $c_i \in \mathbb{Z}$ such that $c_1 m_1 + \cdots + c_tm_t = 1$.
This is given as a property we are allowed to use in one of our homework questions, but we're not sure why we are allowed to us... |
EDIT: I found a brief discussion of this in Husemoller's Fibre Bundles, chapter 16 section 12. Here to compute $\tilde K(\mathbb R P^{2n+1})$ he says to consider the map
$$
\mathbb R P^{2n+1} = S^{2n+1}/\pm 1\to \mathbb C P^n = S^{2n+1}/U(1).
$$
Under this map the canonical line bundle over $\mathbb C P^n$ pulls bac... |
I have been trying this notoriously difficult problem for quite some time but i haven't made any progress.
Let $\mathscr{I}(V)$ denote the set of all homomorphisms $f : V \to V$. That is $\mathscr{I}(V) = \text{Hom}(V,V)$.
Suppose $V$ is a vector space over a field $K$ and $\text{dim}_{K}(V)>1$, then prove that $\mat... |
I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions.
$\dfrac {x}{0}$ is Impossible ( If it's impossible it can't have neither infinite solutions or even one. Nevertheless, both $1.$ and $2.$ are divided by zero, but only $2. $ has infinite solutions so as $1.$ has none... |
I have a bipartite graph and I want to partition each of the two sets of nodes (I don't know the technical term for these sets I mean each part of the bipartite graph) separately.
Each node hase a multidimensional location (x1, x2, x3, xn) as well I want to consider the edges on this graph in the clustering (if two nod... |
I have a cyclotomic field $\mathbb{Q}(\zeta_8)$, and want to know how I can find a minimal polynomial of this element $\zeta_8-2*\zeta_8^3$. Can we generalize this to any number field?
|
Question 2.23 out of Boyd & Vanderberghe's Convex Optimization:
Give an example of two closed convex sets that are disjoint but cannot be strictly separated.
The obvious idea is to take something like unbounded sets which are disjoint but approach each other in the limit. For example, $f(x) = \frac1x$ and $g(x) = -\f... |
This is a question from Spivak's Calculus. Question statement:
If $\lim\limits_{x\to a}f(x)$ exists, and $\lim\limits_{x\to a}[f(x) + g(x)]$ exists, does it follow that $\lim\limits_{x\to a} g(x)$ exists?
I have not been able to find a counterexample, so it seems that it exists. Here is my attempt at proof, and I'm mo... |
An on-going effort to answer this question has brought up various other questions. Most recently, I have been wondering: given a set of $n$ distinct non-zero complex numbers, $z_1, \dots, z_j, \dots , z_n$, and given the values of the first $n+1$ the power sums, $\mathcal{Z}_0, \mathcal{Z}_1, \dots, \mathcal{Z}_n$:
$$... |
I have Borel measures $\mu$, $\nu$ on $\mathbb R^2$ with densities $g, h$ respectively with respect to the Lebesgue measure. Now I assume that $\mu(A \times \mathbb R) = \nu(A \times \mathbb R)$ for every Borel set $A$. Now I want to find a Borel measure $\gamma$ on $\mathbb R^3$ such that:
$$\mu(A \times B) = \gamma(A... |
I am looking to build a repertoire of olympiad type problems which have non-intuitive elegant solutions, If possible instead of a resource, I think problems would be the best. (i.e. select the best problem to post here).
The problem I like best that falls into this category is to prove that if a bigger rectangle has s... |
Let $f$ be a homomorphism mapping $G$ to $J$ groups. Assume that $J$ is abelian. Prove that if $H$ is a subgroup of $G$ and if $\mathrm{ker}(f)$ is a subset of $H$, then $H$ is normal in $G$.
I honestly have no idea how to start this proof. The fact that $J$ is abelian gives me that for all $a$ and $b$ in $G$, $f(ab)... |
Plotting the Bessel functions of the first kind $J_n(z)$ versus $n$ for some fixed $z\gg1$, it appears that there is a sharp cutoff just before $n=z$.
Three questions:
What is a reference describing this sharp cutoff?
What is an expression for the location of the maximum of $J_n(z)$ with respect to $n$, for fixed (la... |
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