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Can someone show me the proof that the convolution of a compactly supported real valued function on $\mathbb{R}$ with a locally integrable function is also continuous? I feel that this is a standard analysis result, but cannot remember how to prove it. Thanks!
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I am going crazy showing something that is clearly wrong, but I can't see the error in my logic.
I am calculating $H_1(D^2,S^1)$
Now the disc is contractible so $H_n(D^2)=0$ for all $n$. Also $H_n(S^1)=\mathbb{Z}$ for $n=0,1$ and is $0$ otherwise.
Now we have a long exact sequence of homology groups
$\cdots H_1(D^2)\t... |
From the wiki page Catalan number, we know the number of Young tableaux whose diagram is a 2-by-n rectangle given $2n$ distinct numbers is $C_n$. In general, given $m\times n$ distinct numbers, how many Young tableaux whose diagram is a $m\times n$ rectangle are there?
Also, what if these numbers can be repeated?
Man... |
I feel this must be an old question, but so far I have never seen it mentioned in any of the books I know. Most of modern treatment of advanced linear algebra is centered on tensor products and eigendecomposition, etc. So here is the question:
We know ordinary matrices provided a way of measuring a list of vectors $a_{... |
What are the exact meanings of the terms "operation", "function" and "map" (are they even exactly defined)? I have always believed that an "operation" was a map
$$S \times S \times \cdots \times S \rightarrow S.$$
However, according to the always-so-reliable Wikipedia, an operation can be any map on the form
$$A_1 \tim... |
I have seen the phrase "minimal number of generators of an ideal" (in a Noetherian local ring) several times. I am unable to see how this is a well defined. Explicitly, how do we show, if $x_1,...,x_m$ and $y_1,...,y_n$ are minimal generating sets of an ideal in a Noetherian local ring, then $m=n$.
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Let $Q(x,y)$ be the statement $y = 2x +1$ what are the values of the following. The universe is $Z^+$ {1,2,3,....}.
(a) $\forall x\exists y Q(x,y)$ This is true
(b) $\exists x \forall y Q(x,y)$ This is false.
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I have been looking at the birthday problem (http://en.wikipedia.org/wiki/Birthday_problem) and I am trying to figure out what the probability of 3 people sharing a birthday in a room of 30 people is. (Instead of 2).
I thought I understood the problem but I guess not since I have no idea how to do it with 3.
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Pretty easy when its only a function of the form $y = ax + b$, however I'm trying to find if this function:
$f(x,y) = 2x + y$
is onto and one-to-one. Is there a procedure to follow to check this?
Thanks!
EDIT: My reasoning.
onto:
$z = 2x + y$
$x = (z-y)/ 2$
$y = z - 2x$
Seeing as both $x$ and $y$ can be represented i... |
How to show that Rolle's theorem, the Mean Value Theorem are equivalent to the least upper bound property?
I'm thinking of starting like this: Let F be an ordered field that does not satisfy the least upper bound property, and then deduce that F does not satisfy either Rolle's or MVT. But I'm not sure how to continue, ... |
Let $\ell^2$ be the set of sequences $x = (x_n)_{n\in\mathbb{N}}$ ($x_n \in \mathbb{C}$) such that $\sum_{k\in\mathbb{N}} \left|x_k\right|^2 < \infty$, how can I prove that $\ell^2$ is a Hilbert space (with dot-product $\left(x,y\right) = \sum_{k\in\mathbb{N}} x_k\overline{y_k}$). This is a standard textbook exercise: ... |
Let $n$ be an integer.
If $3$ does not divide $n$, then $3$ divides $n^2 - 1$.
I'm trying to prove this using a "proof by cases". However, I'm lost as to how to start. I thought proof by cases started by making a logical expression, but I can't seem to find the right one.
Thanks
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How to show that the Peano Arithmetic theory is not scapegoat?
Note:
Peano Arithmetic is a consistent theory.
A theory T is scapegoat if for every formula $A$ with only one free variable there exist a closed term $s$ such that $T$ proves:
$$(\exists x \; (\neg A(x)) )\Rightarrow \neg A(s)$$
("$\neg$" means "... |
Let $f_0 = 1$, and $f_1 = 1$, and $f_n = f_{n-1}+f_{n-2}$ when $n \gt 1$ (the Fibonacci sequence)
Prove using induction that $f_n\gt 2n$, when $n \geq 6$. (note the $f_6 = 13$, $f_7=21$)
I want to rewrite $f_n \gt 2n$ as $f_n\gt f_{2n}$ is this legal?
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A machine works for an exponentially distributed time with rate μ and then fails. A repair crew checks the machine at times distributed according to a Poisson process with rate λ; if the machine is found to have failed then it is immediately replaced. Find the expected time between replacements of machines.
What I hav... |
We know that the Fourier transform of the sinc function is the rectangular function (or top hat). However, I'm at a loss as to how to prove it. Most textbooks and online sources start with the rectangular function, show that
$$\int_{-\infty}^\infty \text{rect}(x)e^{i\omega x}dx=\int_{-1/2}^{1/2}e^{i\omega x}dx=\left.\... |
I am aware of the formula.. but can someone give me a clear definition of what the total area enclosed by a polar curve represents.
Thanks
|
Homework problem I recently got:
Find the length of the curve of $y=\sqrt{a^2-x^2}$
from $x=-2$ to $x=1$. Also find the area of the surface of revolution about the $x$-axis.
Having trouble understanding this. I'm not even sure what it would look like. I assume a is supposed to be a constant. Is it just a semi-ci... |
I'm wondering if there is a function $f(x,y)$ such that
$$\frac{\partial f}{\partial x}(x, y)=2x-3y$$
and
$$\frac{\partial f}{\partial y}(x, y)=3x+4y$$
Integrating the first equation with respect to x and the second with respect to y, we get:
$$\int (2x-3y)dx = x^2-3xy+c_1$$ where $c_1$ can depend on y and:
$$\int (3... |
If $p - a \equiv -a \pmod{p}$ then what is $\frac{p-1}{2} \equiv ? \pmod{p}$? Where $p$ is an odd prime.
I read in the book, they claimed:
$$p - 1 \equiv -1 \pmod{p}$$
$$p - 2 \equiv -2 \pmod{p}$$
$$p - 3 \equiv -3 \pmod{p}$$
$$ ... $$
$$\frac{p - 1}{2} \equiv -(\frac{p - 1}{2}) \pmod{p}$$
However, I realized that the ... |
Can someone guide me in the right direction on this question?
Prove that every $n$ in $\mathbb{N}$ can be written as a product of an odd integer and a non-negative integer power of $2$.
For instance: $36 = 2^2(9)$ , $80 = 2^4(5)$ , $17 = 2^0(17)$ , $64 = 2^6(1)$ , etc...
Any hints in the right direction are appreciated... |
$\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}$
$\dots\sqrt{2+\sqrt{2+\sqrt{2}}}$
Why they are different?
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I'm having a little difficulty in proving what are probably simple induction proofs. Here is the question.
Define function $f(n)$ as follows. $f(1) = 2$ and $f(n) = n\cdot f(n-1)$ when $n > 1$.
Use induction to prove that $f(n) > 2^n$ for all $n>2$.
Note that $f(1) =2$, $f(2)=4$. I understand the basis step, so I'm no... |
Let $S$ be the set of functions $f:\mathbb{R}\to \mathbb{R}$ such that $\sqrt{f(1)+\sqrt{f(2)+\sqrt{f(3)+\dots}}}$ converges.
A function $q(x)$ dominates $p(x)$ if there exist an m such that $q(x)\gt p(x)$ for all $x\gt m$.
Take all functions $f(x)$ from $S$ and put $O(f(x))$ in $S2$.
Which function $g(x)$ in $S2$ domi... |
I have the following problem:
Suppose that $A$ is a symmetric matrix, with $A$ = $A^{T}$ . Suppose $\vec{v}$ and $\vec{w}$ are eigenvectors of $A$ associated
with distinct eigenvalues. Show that $\vec{v}$ and $\vec{w}$ must be orthogonal.
(Hint: Show that a$\vec{v}$ $\cdot$ $\vec{w}$ = $\vec{v}$ $\cdot$ b$\vec{w}$.)
I ... |
Can anyone think of sequences $\{a_n\}$, $\{b_n\}$ such that $\sum a_n$ diverges, ${b_n}\to\infty$, but $\sum a_nb_n $ converges?
Thank you.
Note that $\{a_n\}$ must have infinitely many positive terms and infinitely many negative terms.
Edit: I get the feeling that only Qiaochu Yuan could answer this one... ;)
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If I write (1 3 4 6)(2 3 4)(4 6 1) as a product of disjoint cycles, I get (164)(23), is it true that order of (1 3 4 6)(2 3 4)(4 6 1) is lcm(2,3)=6 (just the order of its disjoint cycles)?
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I need to build up the matrix of matrices. Say, for all $1\leq i,j\leq n$ I have to define a matrix $p_{ij}$ which will be a matrix $n\times n$. What I can do - it is to build up a matrix $P$ which is $n^2\times n^2$ - but for $n=20$ there is an error about memory.
Could you please tell me how to solve this problem?
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I'm sure this question has a simple answer, but I'm a very beginner to calculus...
This is the problem I have: Given a vertex point and two x-axis cutting points, how do I find a formula for parabola-like diagram?
Important part: the vertex point is not in the middle of the parabola.
So I know, there are infinite s... |
can anyone help me? I have a line $Ax+By+C=0$ (let's call it Line $1$); I want to draw a perpicidular line (Line $2$) from the center of line $1$ with a length of $10$.
what I did is get the slope and intercept of line $1$ as well as the its middle point (point $m$).
and I used the length equation $L=\sqrt{(\text{line... |
I am trying to figure out if I made a mistake in the following proof - in particular I have been trying to verify the inequality (*) below. I have attached my proof but I have been getting some error messages in Mathematica that make me think something is wrong when I try to compare the left hand side of the inequalit... |
I'm reading Hatcher's book Algebraic Topology and on page 12 he writes:
... It is less trivial to show that there are path-connected spaces that do not deformation retract onto a point...
The definition of deformation retract is given as:
A deformation retraction of a space $X$ onto a subspace $A$ is a family of map... |
$$f \left (\frac{x}{x+1} \right) = x^2 \implies f(x)=\;?$$
I encountered this exercise, and and don't know how to solve it.
In what category of math does this belong?
In what book/website I can study/exercise myself?
thank you very much.
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In the presentation here about Level set methods
http://www.cs.au.dk/~bang/smokeandwater2006/Lecture9_IntroToWaterAndLS.ppt
the author constructs a linear PDE
$\frac{\partial \phi}{\partial t} + \triangledown \phi \cdot V = 0$
In order to solve this numerically, first the author uses forward Euler time step for discre... |
For finite group $G$, Mobius function is defined on the lattice of subgroups of $G$ as: $\sum_{H\geq K} {\mu (H)}=\delta_{K,G}$. From this, $\mu (H)$ can be obtained by induction argument.
1. Can we define $\mu(H)$ explicitly (i.e. is there formula) for any $H \leq G$?
2. For cyclic groups of order $n$, does this Mob... |
There's a homework exercise I have to do for my scheduling course about which I have some questions. The exercise is this:
Consider the $1|\text{ }|\sum w_jT_j$ problem. (This means that we look at one machine which has to process a number of jobs and where the objective function is the total weighted tardiness of a ... |
For the past two weeks, I've tried to prove two different results that hold the same structure:
Suppose a property that holds for a dense subset of a metric space.
Prove that it holds for the entire metric space.
In this case, both questions were related to uniform convergence:
i) Show that if $(f_{n})$ is a sequence ... |
Let $(X, d)$ be a compact metric space and let $T:X \to X$ be a continuous mapping. Now, does there exist a probability measure $\nu$ such that $T_* \nu = \nu$ (the first thing is the image measure)?
Now I want to do this using a fixed-point theorem. So I start like this:
Define the operator $\psi_T$ from $C(X)$ into i... |
Let $a$ be a positive irrational number. Let $p_k/q_k, p_{k+1}/q_{k+1}$ be two consecutive
convergents of its simple continued fraction, where $k\ge 1$.
Is it possible that both $$|a-(p_k/q_k)|<1/(2q_k^2)$$ and $$|a-(p_{k+1}/q_{k+1})|<1/(2q_{k+1}^2)$$
are true?
I can only prove that at least one of these inequalities i... |
I know that the vector space of all real valued continuous functions on a compact Hausdorff space can be infinite dimensional. When will it be finite dimensional? And how will I identify that vector space with $\mathbb{R}^n$ for some $n$?
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Barring very elementary arithmetic, which skills from elementary school are essential for understanding the world better?
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Suppose $(X,\mathcal {M})$ is a measurable space. For function $f(r,x): \mathbb R\times X\to\mathbb R$, suppose each r-section of it is $\mathcal M$-measurable. For constant $R\in\mathbb R$, do we have $\limsup \limits_{r\to R}f(r,x)$(this is a function of $x\in X$) always $\mathcal M$-measurable?
The definition of $\l... |
I know that the functions $f$ and $g$ are Riemann-integrable on some closed set.
Also, $f$ has a min ($m>0$) on the closed set.
Let $h(x) = f(x)^{g(x)}$
I want to prove that $h$ is Riemann-integrable.
I've been trying to use the Lebesgue's Theorem but I'm not sure how to prove the set of discontinuities of $h$ is me... |
Possible Duplicate:
What is the condition for a field to make the degree of its algebraic closure over it infinite?
More generally, what is known about subfields $\mathbb{F}\subset \mathbb{C}$ such that the degree $[\mathbb{C}:\mathbb{F}]$ is finite? We all know, for example, that $[\mathbb{C}:\mathbb{C}]=1$ and $[... |
There's a question over on stack overflow about adding negative numbers to a negative number. The question surrounds why 10 - -5 is equal to 5.
I'm very happy that the result is 5, but why is it 5 and not 15. It is philosophical or is there a concrete reason for it?
The original question was asked here:
https://stacko... |
More important than knowing inequalities by hearts is knowing when and how to apply them and what they express.
Regarding Chebyshev's and Markov's inequality. What is the relation (if any) between them? Which one is more strict (and in which situation)? Is there an easy way to understand what they express (kind of like... |
Consider $\displaystyle\sum_{1}^{p-1}\frac{1}{x}$, where $p$ is an odd prime. I want to find a general formula for this sum.
I can't believe that I am having trouble figuring this out, but I can't figure out the summation in the question. I have to find a general formula $A_p/B_p$ for the summation from 1 to $p-1$ of $... |
Given two sets of finite measure in $\mathbb{R}$ say, $E$ and $F$, and their characteristic functions $\chi_E$ and $\chi_F$, can somebody show that $\chi_E\ast\chi_F(x)$ (the convolution) is a continuous function of $x$? This is a qual problem from an old qual that I'm studying, and I cannot figure it out. If we were... |
I'm reading Hatcher and I'm doing exercise 9 on page 19. Can you tell me if my answer is correct?
Exercise: Show that a retract of a contractible space is contractible.
Proof:
Let $X$ be a contractible space, i.e. $\exists f :X \rightarrow \{ * \}$,$g: \{ * \} \rightarrow X$ continuous such that $fg \simeq id_{\{ *\}}$... |
Not sure if many of you speak binary, but the question is to prove that 2n bits are sufficient to store the product of two unsigned n-bit numbers (i.e., there will be no overflow).
I've thought of a simple method by contradiction, but its way too simple and I feel theres something wrong with it. So here it is:
Suppose ... |
[Update 3]: I realized, that the series $s(h)$ below is simply a "(general) Dirichlet-series" (1), so after I know the principle how to find the Taylor-series for some function (which may be given by its Dirichlet-series) I can now make my question more precise as:
how to find the Dirichlet-series-representation o... |
We have that:
If $a \equiv 1 \pmod{n}$, then does $a^n \equiv 1 \pmod{n}$?
But I don't know if this is true:
If $a \not\equiv 1 \pmod{n}$, then does $a^n \not\equiv 1 \pmod{n}$?
I tried to find a counter example, but it always turned out to be true. Any idea?
Thanks,
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I'm really confused about these two. For example if $n = 6$, then:
Divisors: $2, 3$
Proper Divisors: $1, 2, 3, 6$
Is it right?
Update
From Elementary Number Theory and Its Application by Kenneth H. Rosen 6th edition, page 256:
Because of certain mystical beliefs, the ancient Greeks were interested in those integers th... |
Let $f \in C^1_c(\Omega)$ where $\Omega \subset \mathbb{R}^d$ is a bounded domain. Let $\phi \in C^1_c(\mathbb{R}^d)$ be an approximation of the identity (i.e. $\int_{\mathbb{R}^d} \phi=1$, $\phi \geq 0$, $\phi_\epsilon := \frac{1}{\epsilon^d} \phi(\frac{x}{\epsilon})$.
How would you prove that
$$\int_\Omega |f(x) - f... |
The given curve is rotated about the $y$-axis. Find the area of the resulting surface.
$y= (1/4 x^2) - (1/2 \ln x)$. $x$ is in between 1 and 2 (including 1 and 2). If anyone could please point me in the right direction of how to solve this I would be very grateful :)
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Potentially-related questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate.
I know there are many ways of calculating (or should I say "ending up at") the constant e. How would you explain e concisely?
It's a rather beautiful number, but when friends have ask... |
I am reading the Wikipedia article on conditional independence. There seems to be Two definitions for conditional independence of Two random variables $X$ and $Y$ given another one $Z$:
Two random variables $X$ and $Y$ are
conditionally independent given a
third random variable $Z$ if and
only if they are independent ... |
Given that $(\psi_i)_0^{|\chi|-1}$ is an orthonormal basis in $\ell^2(\pi)$ of real eigenfunctions of $K$ where $K$ is a Markov operator, then why is $\sum_0^{|\chi|-1} \psi_i^2(x)\pi(x) = 1$?
If we were summing over $x$, then it would make sense since $\psi_i$ is of unit norm. This time though, we're summing over $i$... |
In grade school we learn that a square can be equidecomposed into two congruent right isosceles triangles. Does the following three dimensional generalization hold?
Consider a trirectangular tetrahedron with vertices at $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ and volume $\frac{1}{6}$. (It has three orthogonal rig... |
Suppose I have a random variable X that is known to follow the binomial distribution B, but whose parameter $p$ is unknown.
I have observed 100 samples of X, and they all came out true.
How can I calculate a 95% confidence interval for the value of $p$ (which stands for P(true)) of the underlying distribution B?
|
Possible Duplicate:
Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?
Is it correct that $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ are isomorphic as fields?
|
If $f$ is an $L^2$ function and $\hat{f}$, its Fourier transform, also in $L^2$, can the Fourier transform and its inverse be written as
$$\hat{f}(\omega)=\int_{-\infty}^\infty f(x) e^{i\omega x}dx$$
and
$$f(x)=\int_{-\infty}^\infty \hat{f}(\omega) e^{-i\omega x}d\omega$$
respectively, disregarding the constants? Whe... |
Question:
Find the coordinates of those points on the curve given by the equation:
$\ x^2-0.25xy+y^2=16 $
at which the tangent line has slope 1.
Implicitly derived:$\ \frac{(-2x+0.25y)}{(2y-0.25x)} $
The book talked something about solving for y (from the original equation):
$\ \pm \sqrt{-x^2+0.25xy+16}$
then substit... |
I am trying to understand the concept of strong Markov property quoted from Wikipedia:
Suppose that $X=(X_t:t\geq 0)$ is a
stochastic process on a probability
space
$(\Omega,\mathcal{F},\mathbb{P})$ with
natural filtration
$\{\mathcal{F}\}_{t\geq 0}$. Then $X$
is said to have the strong Markov
property i... |
I'm new to all this large cardinal thing and I have trouble in proving the following:
If $\kappa$ is a $\gamma$-strong cardinal, for some large enough $\gamma$, then $\kappa$ is $\Sigma_2$-reflecting.
Any ideas or hints will be welcome.
Thanks in advance
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Let $X$, $Y$ be Banach spaces. Let $T \colon X \to Y$ be a bounded linear operator.
Under what circumstances is the image of $T$ closed in $Y$ (except finite-dimensional image).
In particular, I wonder under which assumptions $T \colon X \to T(X)$ is a bounded linear bijection between Banach spaces, so it is at least a... |
Let $k'$ be an field extension of an infinite field $k$. If $\{v_i\}$ is a set of vectors over $k$, and they satisfy $\sum_{i=1}^{n} c_iv_i =0$ for some nontrivial $\{c_i\} \subset k'$, then is the same set of vectors $\{v_i\}$ linearly dependent over $k$? In other words, can the $c_i$'s be chosen in $k$ only?
I see t... |
Okay, so this was labeled as a "fun problem", but I'm having trouble knowing how to approach it.
I'm given: $\lim\limits_{x \to 0^+} f(x) = A$ and $\lim\limits_{x \to 0^-} f(x) = B$.
I need to find (or at least know where to start):
a) $\lim\limits_{x \to 0^+} f(x^3 - x)$
b) $\lim\limits_{x \to 0^-} f(x^3 - x)$
Any ins... |
I'm looking at a textbook problem, which has the answer as well, but don't understand what's going on.
The problem: A part with lifetime X has a normal distribution with mean of 7 years and standard deviation of 1.2 years.
If the warranty is for 5 years, what % will fail inside the warranty?
The answer is: P(X<=5) = P(... |
Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to $\sqrt{3}/3$.
Is there a mathematical or other reason?
And does the same apply to exponential notation -- are students expected... |
I recently learned about spectral sequences (which I am still trying to understand). In class we defined a (co)homologically graded exact couple (of $A_p^n$s and $C_p^n$'s with vertical arrows in the $A$ columns) to be a certain grid of exact sequences, which given some convergence assumptions, allows us to calculate ... |
A set $\Gamma$ of probability measures on $X$ is said to be tight if for every $\epsilon > 0$ there exists a compact set $K_\epsilon$ of $X$ such that
$$\mu(K) \geq 1 - \epsilon \text{ for all $\mu \in \Gamma$}.$$
Fine, now let $\mathcal M$ be the set of measurable mappings on the probability space $(\Omega, \mathcal F... |
I have two right triangles One is a $6$-$8$-$10$ and inside is a $3$-$4$-$5$ and the space between the two triangles is a uniform amount.
I made a really awkward and weird pic of the diagram and I need to solve for $X$
How would I go about solving this? I couldn't figure out any good approaches.
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Suppose you have a triangle with vertices $(1,0),(0,1), (0,0)$, and let $P$ be the number of all integer coordinates in the triangle. So $P=3$. Then the Minkowski sum $P+P$ is the set of points found by adding all possible pairs of integer coordinates in the triangle. So $P+P$ consists of points in the original triangl... |
Let me start by saying I graduated High School in 2002 and have worked as a technician, a systems administrator, and a part time web developer up till now.
I have started taking courses at community college for my general requirements, and I am planning which major I should pursue. I have always had a passion for math... |
This post is a continuation of my previous one, but can be read as being stand-alone. I am trying to understand the concept of conditional independence of two random variables given a third one.
Following is the definition that I think is the right one:
Two random variables $X$ and $Y$ are
conditionally independent ... |
I can not figure out how to integrate
$$\int\frac{3x^2+x+4}{x^3+x} \, dx.$$
I got as far as factoring the denominator which gives us $x(x^2+1)$.
So from there I got
$$\frac{3x^2+x+4}{x(x^2+1)} = \frac{A}{x} + \frac{Bx+C}{x^2+1}$$
which would give us: $(3x^2 + x + 4) = A(x^2 + 1) + (Bx+C)x$
This is where Im stuck becau... |
There are many proofs that there is a unique division quaternion algebra over a locally compact field that is not $\mathbb{C}$. For instance this set of notes/book by John Voight contains two proofs: http://www.cems.uvm.edu/~voight/crmquat/book/quat-modforms-041310.pdf
That said, every proof that I've seen first splits... |
Let $v(x)\in L^1\cap L^2$ be a function, and $w(y)\in L^2$ be its Fourier transform. Let $\{w_k\}$ be an infinite sequence of samples from $w(y)$, sampled $T$ apart. Now,
$$v(x)=\sum_{k=-\infty}^{\infty}w_k e^{ikx}$$
in the $L^2$ norm sense. Is it possible to write $w_k=\displaystyle\int_0^{2\pi}v(x)e^{-ikx}dx$, since ... |
I'm trying to find a local minimum for $f(x,y) = x^4 + 3xy^2 + y^2$.
After finding that a critical point is at $f(x,y)$ is $(0,0)$, I am wondering if this is proper formal justification to prove that this is indeed the case?
Let $\epsilon >0$ be given. Let $|y|=-x$
(Is this proper? I just want to keep x and y to hav... |
Motivation: I'm writing a proof, and within it, I need to prove:
Conjecture: Let $p$ be an odd prime (i.e. $p \neq 2$). Let $c \geq 2$, $d \geq 1$ and $r \geq 1$. If $p^r$ divides $cd$, then $d(c-1) \geq r+1$.
This conjecture, should it prove to be true (and I suspect it will [it seems to be like "prove elephants a... |
Let n be a positive natural number , $n\ge 0$, then. $\displaystyle\sum_{i=0}^n i \cdot i!= (n+1)!-1$
Here is my attempt.I'm not going to write the base case because I understand that part.
Assuming $\displaystyle\sum_{i=0}^n i \cdot i!= (n+1)!-1$ is true. We wish to show.
$\displaystyle \sum_{i=0}^{n+1} i \cdot i!=... |
Find the length of the given curve $\displaystyle x=3y^{\frac{4}{3}}-\frac{3y^{\frac{2}{3}}}{32} $
where the bounds are given by $-8 \leq y \leq 343$.
I can solve for the positive part (0 to 343), but I am unsure of how I would go about integrating
$$\int_{-8}^{0} \sqrt{1+\left(\frac{dx}{dy}\right)^2}dy$$
Thanks agai... |
I vaguely remember seeing something like a "picture" of various groups a while back. It was as if the elements of the group were each associated with a point and many points had segments connecting them, but not all were connected.
Does anyone know what I am talking about? If so, would you care to take the time to exp... |
Reframed question:
Team A and Team B will be playing each other in a sporting event where there is one winner and one loser; a tie is impossible. A man, wanting to make money from placing a bet on the winner, consults with three omniscient seers. These seers know the outcome of the game already, being omniscient.
The... |
There seems to be a consensus that factorization of integers is hard (in some precise computational sense.) Is it known whether polynomial factorization is computationally easy or hard?
One thing I originally thought was that if we could factor polynomials easily, then we could factor $n$ by finding a polynomial $p(x)$... |
Attach a (Laurent) monomial weight $x_1^{i_1} \cdots x_n^{i_n}$ to each point $(i_1, \dots, i_n)$ of $\mathbb{Z}^{n}$ and call it $\mathbb{Z}^{n}[x_1, x_{1}^{-1}, \dots, x_n, x_n^{-1}]$. Does this formal space have a name (other than the title) or any special properties?
I understand that Newton polygons are studied us... |
Let $X_1,X_2,...$ be a sequence of integer-valued random variables that converge in distribution to some random variable $X$. Am I right in thinking that we can always pick $X$ to be integer valued?
I thought like this: Let $F_n$ be the distribution function of $X_n$ and $F$ be the distribution function of $X$. Then $X... |
I have $n$ points $(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)$ all located in the unit square $[0,1] \times [0,1]$. I am trying to compute the largest distance from a point in the unit square to the closest $(x_i,y_i)$. Is there some sort of formula for this quantity in terms of the $x_i,y_i$?
Thanks.
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Problem:
If $x^{n-1} \equiv 1 \pmod{n}$, and for all divisors $q$ of $n - 1$, $a^{q} \not\equiv 1 \pmod{n}$, then $n$ is prime. $(n > 1)$
I read the proof in the book and it was very straightforward; however, I wonder is there a way to prove it by just using congruence property?
And another related question about... |
Eisenstein's proof of the Quadratic Reciprocity (QR) (and its Jacobi symbol generalization) both rely on counting lattice points in two congruent triangles. If we take $t$-dilates of these triangles, does an extension of Gauss' Lemma still hold with some dependence on $t$? That is, can we assign a Legendre symbol $(p|q... |
Given a bipartite graph G with two partition sets $U$ and $V$ of the same size $n$. In each set, there are d vertices of degree (n-d+1), and n-d vertices of degree (n-d). Can we find a perfect matching in this graph?
So basically, the two sets are the same in terms of degree sequence, although one has no idea how verti... |
I'm working through some of the exercises in generatingfunctionology. One of the questions is to find the generating function where the $n$th term $a_n=P(n)$ for $P$ a polynomial. The answer is $P(xD)(1/(1-x))$. I thought it meant something like if $P(n)=3n+n^2$, then $P(xD)=3xD+x^2D^2$, and then you would apply that t... |
In my textbook, they said:
$$2x^{3} + 7x - 4 \equiv 0 \pmod{5}$$
The solution of this equation are the integers with $x \equiv 1 \pmod{5}$, as can be seen by testing $x = 0, 1, 2, 3,$ and $4.$
And I have no clue how do they had $x \equiv 1 \pmod{5}$. I tested as they suggested:
Let $y = 2x^{3} + 7x - 4$, we have:
$$x =... |
I want to find an $n$-vector $\hat{\theta}$ that maximizes a function $f(\theta)$ subject to the $p$-norm constraint $||\theta||_p = c$.
Is there a general parametrization of $p$-norm hyperspheres that makes this easy?
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Is there some proven statement that tells us "not all X satisfy Y", but there are currently no examples of X which do not satisfy Y?
Is it necessarily true that there are always explicit counterexamples? Or are there statements that only guarantee the existence of such counterexamples, whose construction is impossible ... |
Given $n$ independent geometric random variables $X_n$, each with probability parameter $p$ (and thus expectation $E\left(X_n\right) = \frac{1}{p}$), what is
$$E_n = E\left(\max_{i \in 1 .. n}X_n\right)$$
If we instead look at a continuous-time analogue, e.g. exponential random variables $Y_n$ with rate parameter $\la... |
I have values from two sensors stored in two vectors A and B. They both represent values of the sensors at times TA and TB which is stored in two other vectors(since it is not uniform sampling) Both A and B represent the same data but A is shifted a bit to the right because of the delay in starting the sensors.
My que... |
I was working on this problem here below, but seem to not know a precise or clean way to show the proof to this question below. I had about a few ways of doing it, but the statements/operations were pretty loosely used. The problem is as follows:
Show that ${\bf A}^{-1}$ exists if and only if the eigenvalues $ \lambda ... |
I'm reading Hatcher and I did exercise 10 on page 19. Can you tell me if my answer is correct? Many thanks for your help!
Claim: $X$ contractible $\Leftrightarrow \forall$ arbitrary maps $f:X \rightarrow Y$, $Y$ arbitrary, $f \cong const.$
Proof:
$\implies$
Given the homotopy $H: I \times X \rightarrow X$, $H(0,x) = id... |
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