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Let $p$ be an odd prime number and $T_p$ is the following set of $2 \times 2$ matrices:
$$ T_p = \left\{ A = \begin{bmatrix}
a & b \\
c & a
\end{bmatrix}\ \Biggm|a,b,c \in \{0,1,2,...(p-1)\} \right\}$$
then how to prove that the number $k$ of $A$ in $T_p$ such that the trace of A is not divisible by $p$ but the ... |
Let G(S) denote the sum of the elements of set S and F(n) be the sum of G(s) for all subsets of the set consisting of the first n natural numbers. For example, F(3) = (1) + (2) + (3) + (1 + 2) + (1 + 3) + (2 + 3) + (1 + 2 + 3) = 24. Given n, calculate F(1) + F(2) + ... + F(n).
Input:
1
2
3
Output:
1
7
31
How do i go ... |
One way to get the famous double cover $\text{SU}(2) \to \text{SO}(3)$ is to note that $\text{SU}(2)$ is isomorphic to the group of unit quaternions and to let unit quaternions $q$ act on the subspace $V$ of $\mathbb{H}$ spanned by $i, j, k$ via conjugation $t \mapsto qtq^{-1}$; this preserves the norm. (Alternately, ... |
For linear fractional transformations, it suffices to specify where three points are mapped to. I am wondering if some analogue of this holds for general conformal mappings.
|
I am attempting to self-study proof techniques and your criticism of my following proof would be greatly appreciated. Feel free to nitpick minor/trivial things; that's how I'll learn!
Edit: I have appended a revised proof with the criticisms received so that other rookies can learn from my progress too.
Edit 2: And ano... |
I have a set $S$ of $n$ elements. I select $r$ elements from the set $E_1, E_2, E_3, \ldots, E_r$ that may not be all distinct. I select again $r$ elements from the set $F_1,F_2,F_3,\ldots, F_r$ in a similar manner.
What is the probability $P$ that $E_i = F_j$ for some $i$ and $j$?
I have to prove that this probabilit... |
Question: Given is the following pipe system:
the pressure in a stationary state is at every end and cross-point (with exception of $K_i, i =\{1,...,4\}$) equal to the average pressure of the neighboring points. On $K_i$ with $i = \{1,...,4\}$ are valves with fixed pressure of $K_i = 10i$.
a) Create a system of linear... |
At the moment, I can't find a similar example in my textbook or in my class notes, and I have looked for one and contemplated this problem for the past hour or so. Can anyone give me an idea of what to do to solve this problem?
A constant force F = -3i -2j -1k acts upon an object moving along a straight line from point... |
Let A be a noetherian ring. How can I show that Spec(A) is noetherian?
Also, is there a way to show this by showing directly that the closed sets in Spec(A) satisfy the descending chain condition?
(This is exercise 6.8 from Atiyah and Macdonald.)
|
I have some questions as to some good combinatorial interpretations for the sum of elementary symmetric polynomials. I know that for example, for n =2 we have that:
$e_0 = 1$
$e_1 = x_1+x_2$
$e_2 = x_1x_2$
And each of these can clearly been seen as the coefficient of $t^k$ in $(1+x_1t)(1+x_2t)$. Now, in general, what ... |
I'm in a similar scenario as with my other question. I have tried to answer the problem correctly, and I have found similar (and even an identical) problem in my textbook, yet I still can't seem to yield the correct answer with this specific problem.
The example problem in my textbook:
A crate is hauled 8m up a ramp un... |
How can we compute the value of $\mathrm{gcd}(2^n-1,n!)$ efficiently where $n$ is very large? I couldn't think of any fast and efficient method.
|
This is an equality that I am gleaning out of some papers that I have been reading. I am not sure I am reading it right. Hopefully people will correct it.
Let $U$ be a group element and $R$ be a representation of the group. Let $sym^n (R)$ be the $n^{th}$ symmetric power of the representation and similarly $anti^n (R)... |
Given two vectors, a and b, how do I find the Orthogonal Projection?
I've already found the Scalar and Vector Projections.
\begin{align*}
\text{Scalar}&:\quad \frac{-90 + -25 + 24}{\sqrt{9^2+5^2+8^2}};\\
\text{Vector}&:\quad \left(\left(\frac{-91}{\sqrt{170}}\right)\left(\frac{-9}{\sqrt{170}}\right), \left(\frac{-91}{... |
While trying to understand artificial neural networks, I came upon an equation for finding the net input of an artificial neuron. Can someone explain this to me and what it means? Here is the original article
$$y_i=f\left(\sum_j w_{ij}y_j\right)$$
Thanks in advance, ell.
|
I am interested in alternatives to Ziegler's Lectures on Polytopes, which is the suggested textbook for a course I am attending. I find the conversational style of the book jarring.
|
I've just had to do a homework on binomial expansion for approximation:
$1.07^9$
so:
$(1+0.07)^9$
To do binomial expansion you need a calculator for the combinations button (nCr), so why would use a more complicated method, which only gives an approximation be used over just typing 1.07^9 into a calculator?
(or is this... |
Let $\phi(n)$ be Euler's totient function. For $n=4m$ what is the smallest $n$ for which
$$\phi(n) \ne \phi(k) \textrm{ for any } k<n \textrm{ ?} \quad (1)$$
When $n=4m+1$ and $n>1$ the smallest is $n=5$ and when $n=4m+3$ it's $n=3.$ However, when
$n=4m+2$ the above condition can never be satisfied since
$$\phi(4m+2)... |
If I have a grid that I know is RxCxD in size, and I have number that corresponds to an element in that grid, and the grid is mapped in Row major format, how can I find out the ordered triplet that corresponds to the index that I know? that is
(i,j,k) = index
|
I know that for a series of non-negative, continuous functions $u_{n}(x)$, a sufficient condition for uniform convergence of $\sum u_{n}(x)$ to $u(x)$ is for $u(x)$ to be continuous in $I\subset \mathbb{R}$.
But I can't think of an example where $\sum f_{n}(x)\to f(x)$ uniformly in $I\subset \mathbb{R}$ but $f(x)$ is ... |
In "The maximum number of Hamiltonian paths in tournaments" by Noga Alon, the author states the following without proof (equation 3.1):
Consider a random permutation $\pi$ of $\mathbb{Z}_n$. What is the probability that $\pi(i+1)-\pi(i) \pmod{n} <n/2$ for all $i$?
The claim is that this is $(2+o(1))^{-n}$, which makes ... |
Just wondering about this:
Is it true that there are no prime numbers in Pascal's triangle, with the exception of $\binom{n}{1}$ and $\binom{n}{n-1}$?
From the first 18 lines it appears that this is true, but I haven't looked beyond that.
Is this a coincidence or is there a reason for it?
|
I'm trying to prove that if we have the elementary symmetric polynomials that the following identity holds:(where $x = (x_1,..,x_n)$ is a vector of n variables)
$$\sum_{k=0}^n e_k(x)^2 = x_1\cdots x_n \sum_{j=0}^{\lfloor n/2 \rfloor} {2j \choose j} e_{n-2j}(x_1+1/x_1,\cdots x_n+1/x_n).$$ I'm trying to prove this comb... |
What is the best intuition for Calderon-Zygmund operators? Why are they so important in singular integrals, and complementary, which singular integrals don't they cover?
|
A student randomly asked me to compute
$$\int\sqrt{\cos(2\theta)}\, \mathrm d\theta.$$
I was unable to do so, as were several other instructors. I typed the integral into Wolfram and it says that it is an elliptic integral of type 2. I am not familiar with elliptic integrals and I don't have any idea why this student... |
My calculation was (5520*5845)*31% and I got value 10,001,964. Closest answer is 10,100,000. Is the answer right?
Thanks
|
I'm not very good at this, so hopefully I'm not making a silly mistake here...
Assuming that $\infty$ is larger than any real number, we can then assume that:
$\dfrac{1}{\infty}$ is the smallest possible positive real number.
It then stands to reason that anything less than $\infty$ is a real number. Therefore, if we t... |
I am trying to understand the proof of this fact. On page 183, Eisenbud defines a map from the formal power series ring $S[[x_1,x_2,...,x_n]]$ to $R/m^i$ where $R=S[x_1,x_2,...,x_n]$ sending $f$ to $f+m^i$. I have trouble understanding this map since, $f$ need not be an element of $R$ and so what does the coset $f+m^i$... |
Again:
$$\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$$
Also the one for $\sin$:
$$\int e^{\alpha x}\sin(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \sin(\beta x)-\beta \cos(\beta x))}{\alpha^2+\beta^2}$$
Both help avoid integration by p... |
Let P = (1, 1, -1), Q = (2, -2, -3), and R = (-1, 0, 4) be the vertices of a triangle.
Let L be a line passing through P and Q.
Find:
The point on the line x = y + 1 = z - 2 closest to the line L.
|
I remember there is some name for a special topology on a set, such that every subset is open i.e. every point forms an open set, but I cannot recall it.
Please tell me its name or say I am wrong.
Thanks!
|
I have the following question:
I've managed to do the surface integral part of stokes theorem (and got an answer of 0) however I'm having a lot of trouble doing the line integral part in order to verify the theorem.
As I understand it, I should be doing a line integral around the part of the plane $z = 2-2x-y$ that's... |
I need help on this question. Thanks in advance!
my calculation is like this:
Percentage difference: 31-27=4
4% = £50,000
100% = £1.25mil
Need to find total revenue:
Profit = Revenue - Cost
Not too sure about this part.
|
Im trying to write recursive formulas for sequences but it seems like there are different techniques depending on what type of sequence I'm dealing with. for example I want to the sequence:
$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} +.... \frac{1}{n}$
What I see when I look at this is all the next terms are $\frac{1... |
How can I rewrite this equation:
$
\hspace{2in}
V_0=\frac{E_1+B_0-B_1}{(1+r)^1}
+
\frac{E_2+B_1-B_2}{(1+r)^2}
+
\frac{E_3+B_2-B_3}{1+r)^3}
+
\dots
$
into:
$
\hspace{2in}
V_0=B_0+
\frac{E_1-rB_0}{(1+r)^1}
+
\frac{E_2-rB_1}{(1+r)^2}
+
\frac{E_3-rB_2}{(1+r)^3}
+
\dots
$
My textb... |
Problem:
Prove that if gcd( a, b ) = 1, then gcd( a - b, a + b ) is either 1 or 2.
From Bezout's Theorem, I see that am + bn = 1, and a, b are relative primes. However, I could not find a way to link this idea to a - b and a + b. I realized that in order to have gcd( a, b ) = 1, they must not be both even. I played a... |
I would like to solve the compound interest formula
$$V = Pz^n + c\left(\frac{z^{n+1}-z}{z-1}\right)$$
for $n$. Or, given what I would like the final value to be, the amount I can save a year, an interest rate, and how much money I have to invest initially, I would like to find how long it would take to reach my goal.... |
Take the differentiable vector function $\vec{v}(t)$ (a velocity vector). If its speed, $|\vec{v}(t)|=constant$, then prove that at any point which $\frac{d\vec{v}}{dt}$ is non-zero, $\frac{d\vec{v}}{dt}$ is perpendicular to $\vec{v}(t)$.
In other words, if a velocity vector has constant speed, show that whenever its ... |
For a ray from the origin with a given angle in $R^n$, I am trying to find the radius at which that ray intersects the frontier of the unit n-cube. In two dimensions, the picture is this:
Given $\theta$, find the distance r to the edge of the unit square. Or, in $n$ dimensions, given angles $\theta_1,...,\theta_{n-1}$... |
I tried to prove this theorem, but I'm really confused about its correctness.
Let's say
$$\gcd( a, b ) = 1 \Rightarrow am + bn = 1 \text{ where } m, n\in\mathbb Z$$
Then by multiplying both side by $c$, we have: $amc + bnc = c$.
From here, is it sufficient to say $c = \gcd( ac, bc )$?
I feel like this proof needs so... |
Consider a given inflation rates $I_1, I_2, \dots, I_N$ for years $1 \dots N$. What is the total (cumulative) inflation rate over the whole period of $N$ years?
I was trying to chain-link the resulting purchasing powers:
$P_1=P_0\frac{1}{1+I_1}$, $P_2=P_1\frac{1}{1+I_2}$, etc. yielding the $P_N=P_0\prod_{j=1}^{N}{\frac... |
the question asks:
Find $f '(a)$,
$f(x) = \frac{x^2 + 1}{x - 5}$
$f'(a)=$ [answer box here]
I understand that $f'(a)$ is the derivative, but i don't understand what the derivative is. Or how to solve this. I am used to finding a tangent slope with two points. This one doesn't have that. So i don't understand how i ca... |
What is the idea behind defined a Projective system of Groups/Rings. In our class an example for the Projective system was given by Taking the Ring $\mathbb{Z}/n\mathbb{Z}$ over $\mathbb{N}$. The order defined was:
$ m \leq n$ if $m \mid n$.
Moreover then we calculated the Projective Limit of this system and found i... |
I would like to know how to find the number of (complex) roots of the polynomal $f(z) = z^4+3z^2+z+1$ inside the unit disk. The usual way to solve such a problem, via Rouché's theorem does not work, at least not in an "obvious way".
Any ideas?
Thanks!
edit: here is a rough idea I had: For any $\epsilon >0$, let $f_{\ep... |
{Xn} is a sequence of independent random variables each with the same
Sample Space {0,1} and Probability {1-1/$n^2$ ,1/$n^2$}
Does this sequence converge with probability one (Almost Sure) to the constant 0?
Essentially the {Xn} will look like this (its random,but just to show that the frequency of ones drops)
010... |
I have a question concerning the formulation of an (3D) ellipsoid. The most common definition for an ellipsoid seems to be:
$$ E = \left\{ x= \left( x_1, \dots x_n \right)^T \in \Bbb R^n: \sum_{i=1}^n \left( \frac{x_i}{r_i} \right)^2 = 1 \right\} $$
based on the different radii $r_i$. However for an ellipse (2D) I also... |
I throw two coins (simultaneously). A student (very much a beginner in both math and probability theory) thought that the following 3 outcomes are equally likely: "two heads", "two tails", "a head and a tail".
However much I tried, I couldn't find a clear and obvious explanation of why it's not the case.
Of course, I ... |
Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification?
It seems that Goldbach should be true, but this is based on heuristic justification.
I am looking for conjectures that seem to be true, but where the 'why' is something other then a statistical ... |
I have a random vector $X = ( X_1 , X_2 )$ that has a bidimensional normal repartition with mean $0$ and covariation matrix :
$$
\Sigma = \left(
\begin{array}{ccc}
1 & q \\
q & 1
\end{array} \right)
$$
I need to calculate $q$, which is
$$ q = \mathrm{Corr}(X_1,X_2)$$
How can I calculate $\mathrm{Corr}(X... |
What does this set look like? $A = \{ (x, \frac{1}{x}) | x \in \mathbb{R} \backslash 0 \}$, where $A$ is a subset of $\mathbb{R}^2$ with the Euclidean topology.
I thought $A = (-\infty, 0] \cup [0, \infty)$ but I know that its projection $(x,y) \rightarrow x$ is not closed, so this must be wrong.
Thanks for your help... |
In spatial euclidean vector spaces norm is an intuitive concept: It measures the distance from the null vector and from other vectors.
The generalization to function spaces is quite a mental leap (at least for me).
My question: Is there some kind of intuition what "norm" or "euclidean norm" is even supposed to "mean" h... |
What is the closure of $\{ (x_n) | x_n = 0 $ for almost all $n \}$ in $\mathbb{R}^\mathbb{N}$ with the product topology?
I'm not sure how to think about it. A point $x$ is in the closure of a set $A$ if for every neighbourhood $N$ of $x$, $N \cap A \neq \emptyset$.
But I'm not sure what a neighbourhood of a point in ... |
To do some computations (such as finding copies of GL(2,3) in a larger of groups), I need presentation of GL(2,3). ( Because: Sylow-2 subgroups of GL(2,3) is semidihedral :$ $. It is easy to find copies of semidihedral groups in bigger groups. Therefore, to get a copy of GL(2,3) in larger group, it is sufficient to ge... |
I have to prove the following without any use of further mathematical theories except basic calculus and linear algebra:
Let $ \ell $ be a positive integer and $A$ a real, symmetric and positive definite $n \times n$-Matrix (square Matrix with $n$ columns and $n$ rows). Show that the following identity holds:
$$ \int_{... |
Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^{\theta i}=\cos(\theta)+i \sin(\theta)$ trick where you took $e^{2 i \theta}$ and $e^{-2 i \theta}$. While I am draftin... |
If I knew that by the year 2000 that there were 1 trillion humans, and that they started reproducing 2000 years before that, how would I calculate their birth rate?
Assume an initial population of 1 million, and a reproduction age of 15.
Also take into account a standard death rate, where death occurs after 75 years, a... |
I want to write down $\ln(\cos(x))$ Maclaurin polynomial of degree 6. I'm having trouble understanding what I need to do, let alone explain why it's true rigorously.
The known expansions of $\ln(1+x)$ and $\cos(x)$ gives:
$$\forall x \gt -1,\ \ln(1+x)=\sum_{n=1}^{k} (-1)^{n-1}\frac{x^n}{n} + R_{k}(x)=x-\frac{x^2}{2}+\... |
So I have a large txt file with such contents (like 20mbs long)
20 30 40 550 60 70 80 91
20 30 40 50 60 70 80 92
20 30 40 50 60 70 80 93
20 30 40 50 64 70 80 90
20 30 40 50 60 70 80 90
20 30 40 50 60 70 80 90
20 30 40 40 60 70 80 90
40 30 40 50 60 70 80 90
4 5 6 6
20 30 40 50 60 70 80 91
20 30 40 50 60 70 80 92
20 30... |
Let $k$ be an algebraically closed field. Let $X$ be the quasi-afine variety $X = A^2 - \{0,0\}$. I know from complex-analytic arguments that the ring of regular functions on $X$ is equal to $k[x,y]$. How can I prove this algebraically?
Thanks
|
https://i.stack.imgur.com/xs2AR.jpg
The first method I used was find out Profits by
Sales Revenue - Operating Cost - Payroll Cost - Waste -Shrinkage
I have Geneva the lowest and the answer is wrong.
2nd method, I used sales revenue/number of employees
London was the lowest and it's wrong again.
Could anyone help me out... |
hi
i'm looking for some historical informations about Gruenberger path.
it is a path based on conjecture that every prime could be written in the form $6k+1$ and $6k-1$ (but what is the name of this conjecture?)
i don't find a lot in the net.. the best one seems to be this site..
any other infos?
|
I'm interested in what to call the length of the sequence of ones in the Smith normal form of a matrix. Equivalently (or so it seems to me), this is the rank of the largest submodule of M, on which A acts injectively (by left-multiplication; there should be an equivalent formulation for right-multiplication). For a giv... |
As a follow-up question to What is the norm measuring in function spaces
I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal coordinate system where the two axes represent two orthogonal functions (e.g. sin and cos). Different functions, that can be r... |
Hi I am learning about Barker sequence. I have a problem, because I do not know, why for example 5, in Barker Code looks like this: +++-+. What is the base of this code. I am looking for some algorithm, because, I do not want to learn by hearth.
|
What does it mean when, in (derivation of) (partial) differential equations, a term containing an operator like:
$\delta t$
gets replaced by:
$d t$
and what are the rules pertaining to this? What is the difference between the two? It seems black magic to me :).
Thanks!
|
Two part question:
If I define a linear operator $L=\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$ and then use that operator on a variable, say $L\alpha$ then is it accurate to say that the $L$ maps $\alpha$ onto itself like this: $L\alpha=\frac{\partial^2}{\partial x^2}\alpha + \frac{\partial^2}{\... |
I have a system that is defined as follows:
$$\dot{x}(t) = Ax(t) + Bu(t),\quad\quad x(0)=x_0.$$
It has a well known solution:
$$x(t) = e^{tA}x_{0} + \int_0^t \! e^{(t-\tau)A}Bu(\tau) \, \mathrm{d}\tau $$
I know that one of the tasks during exam will be to draw the the graph of the solution given that A and B are number... |
For $x \in \mathbb{R}^n$ and $A,B \in \mathbb{R}^{m \times n}$,
$f(x) = ((Ax)^{2})^T((Bx)^2)$
where $^2$ denotes the power of 2, element-by-element of vector Ax or Bx.
(I wasn't sure how to notate this)
Is $f(x)$ convex? How can it be shown?
If the domain of $x$ is restricted to be $x$ nonnegative, then is it convex?
|
I have been working through the exercises in Tenenbaum's "Introduction to analytic and probabilistic number theory" book, and I am stumped here (Exercise I.1.6). This is not a homework assignment, but just rather for my own edification. I know Tenenbaum's exercises are usually considered quite hard, so I don't feel a... |
how to show T(S^3) isomorphic to S^3 cross R^3? so can I say it for every odd dimension?I have shown it for n=1
|
Let $\tau$ be the topology generated by half-open intervals of the form $[a,b)$ where $a$ is a rational number and $b$ is a real number. Let $C$ denote the space endowed with the previously described topology.
Prove/disprove: $C \times C$ is a Lindelof space.
How do you proceed: here $C$ is not equal to the Sorgenfrey... |
I encountered an interesting function which is called "Eulerian" by the Wolfram's MathWorld:
$$\phi(q)=\prod_{k=1}^{\infty} (1-q^{k})$$
It is interesting because it seems that roots of any polynomial can be expressed in this function and elementary functions.
I want to know more about the properties of this function, w... |
Second Update:
I see that some answers that reference my image are more closely answering my question.
Here is a second image to clarify my point.
Take this image representing a checkerboard like configuration. If I asked you to move the X negative 1 space it would be impossible. You would have to make a guess as to ... |
If A is a commutative integral domain that's not a field, and let $K$ be the quotient field of A. We know that $K$ is not finitely generated as an A-module. But can $K$ ever be finitely generated as an A-algebra?
|
It is well known that one of the first books Ramanujan studied was "Synopsis of Pure and Applied Mathematics" and that it shaped the way Ramanujan thought and wrote about mathematics. Being interested in the works of Ramanujan, I was wondering if it would be of any help if I studied this "Synopsis".
Moreover, I would a... |
I saw in my book that $2x^2 - 2x + 2$ factored became $2(x^2 - x + 1)$.
Why it does not became $2(x(x - 1) + 1)$? Is it wrong or correct as well?
|
I'm trying to solve an homework question but I got stuck.
Let A be a m x n matrix with the SVD $A = U \Sigma V^*$ and $A^+ = (A^* A)^{-1} A^*$ its pseudoinverse.
I'm trying to get $A^+ = V \Sigma^{-1} U^*$, but I'm missing something.
Can anyone help me with this please?
Thanks!
|
Is the image of the general Veronese embedding ever contained in a hyperplane of $P^{n}$? I'm guessing no, but I can't prove it.
|
I'm learning about $\log_2$ for an algorithms class and theres a problem in the book that is confusing me.
It asks:
Find a formula for $\log_2(n!)$ using Stirling's approximation for $n!$, for large $n$.
Stirling's approximation for $n!$ is $\sqrt{2\pi n}(\frac{n}{e})^n$
Does anyone have guidance on how to go about cre... |
What do you think is the best way to rank universities participating in a competition if we know for each university the number of recognitions that it's students won. Each university can participate with 3 students, and approximately half of them get a recognition - which can be first (5%), second (15%) or third place... |
In the chapter on completions in Atiyah Macdonald, they define a topological abelian group. Let $G$ be such a group. Denote $H$ to be the set of intersection of neighbourhoods of $0$. Then it is shown that $H$ is a subgroup. Then they say $G/H$ is Hausdorff because points are closed. I don't quite understand this.
|
http://en.wikipedia.org/wiki/Covariance_matrix
Cov(Xi,Xj) = E((Xi-Mi)(Xj-Mj))
Is the above equivalent to:
(Xi-Mi)(Xj-Mj)
I don't understand why the expectancy of (Xi-Mi)(Xj-Mj) would be different than just (Xi-Mi)(Xj-Mj)
Addendum:
Let's say I have two sets of data:
Set 1: 1,2,3 avg: 2
Set 2: 4,5 avg: 4.5
Is the follo... |
Let $$\varphi(x) = e^{-x} + \frac{x}{1!} \cdot e^{-2x} + \frac{3x^{2}}{2!}\cdot e^{-3x} + \frac{4^{2} \cdot x^{3}}{3!} \cdot e^{-4x} + \cdots$$
Then what is the value of: $$ \displaystyle\lim_{t \to 0} \frac{\varphi(1+t) - \varphi(t)}{t}$$
I am not getting any idea as to how to proceed for this problem. I tried summing... |
In this example problem in my textbook:
"Find a power series representation for ln(1-x) and its radius of convergence."
They integrate both sides:
-ln(1-x) = integral (1/1-x)dx which comes out to be SUM x^n/n + C.
They solve for C, C=0. This is where I get stuck. They proceed to show what the series looks like when C=0... |
Suppose you have a deck of cards. Each card has n images on it. Any two cards will have exactly one matching image. What would be the formula for the maximum number of unique images and the maximum number of unique cards?
At first I thought the maximum number of images would be:
$\sum\limits_{i=1}^n i$
But that doesn't... |
Let $M$ and $N$ be graded $R$-modules (with $R$ a graded ring). $\varphi:M\rightarrow N$ is a homogeneous homomorphism of degree $i$ if $\varphi(M_n)\subset N_{n+i}$. Denote by $\mathrm{Hom}_i(M,N)$ the group of homogeneous homomorphisms of degree $i$. We define $^*\mathrm{Hom}_R(M,N)=\bigoplus_{i\in\mathbb{Z}}\mathrm{... |
This is a homework question, so I'd appreciate hints (or perhaps explanations of concepts I've not properly digested)
Anyhow: This is exercise 1.3.6 in Weibel's book on homological algebra. Let $0 \to A \to B \to C \to 0$ be an exact sequence of double complexes of modules. Show that there is a short exact sequence of ... |
I think this is false, a counter example could be:
$c = 100,$
$b = 10,$
$a = 5$
But the book answer is true :( ! Did I misunderstand the problem or the book's answer was wrong?
Thanks,
Chan
|
In reading "Real Analysis: Modern Techniques and Their Applications" (Folland), I've come across a few different notions of "distribution" or "distribution functions."
The distribution function of a finite Borel measure $\mu$ on $\mathbb{R}$ is defined by $F(x) = \mu((-\infty, x])$.
The distribution function of a meas... |
Suppose f(z) is a polynomial such that its derivatives are non-zero for all $|z| <1$. Is the restriction of $f$ to $|z|<1$ 1 to 1?
I know that $f$ must be locally 1 to 1. It is obvious that $f$ is 1-1 for polynomials of order 1. The case of order 2 follows from Gauss-Lucas Thm.
I am stuck on how to prove the gener... |
I've thought about trying to prove this by induction, and by contradiction, but I can't seem to phrase this in a formal way. Any help would be appreciated.
|
Suppose $z,w \in \mathbb{C}$ s.t. $|z|,|w| < 1$. How would you show that $\frac{|z-w|}{|1-\bar{w}z|} \leq |z| + |w|$?
I calculated that
\begin{align*}
&\frac{|z-w|}{|1-\bar{w}z|} \leq |z| + |w| \\
&\Rightarrow \frac{|z|^2-2Re(\bar{w}z)-|w|^2}{1-2Re(\bar{w}z) + |w|^2|z|^2} \leq (|z|+|w|)^2 \\
&\Rightarrow -2Re(\bar{w}z... |
In 1556, Tartaglia claimed that the sums
1 + 2 + 4
1 + 2 + 4 + 8
1 + 2 + 4 + 8 + 16
are alternative prime and composite. Show that his conjecture is false.
With a simple counter example, $1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255$, apparently it's false. However, I want to prove it in general case instead of using ... |
Here is a nice proof of the isoperimetric inequality (equality part ommited):
Isoperimetric Inequality
If $\gamma$ is any simple closed piecewise $C^1$ curve of length $l$, with it's interior having area $A$, then $4\pi A \le l^2$. Furthermore, if equality holds then $\gamma$ is a circle.
Proof.
Take two parallel stra... |
Prove:
$[ (A \cap B) \cup C ]\cap (A^c\cup C) = C$
My attempt:
$(A\cup C)\cap(B\cup C)\cap(A^c\cup C) = C$ I used the distributive property,
but this is where I'm lost at. I think that I need to get A and $A^c$ together. So I should use the commutative property.
$(A\cup C)\cap(A^c\cup C)\cap(B\cup C) = C$ Should I di... |
I wrote a program for myself in Mathematica to generate the answer for the first 300, which was really easy, but I can't find a pattern. The results are here. This is a problem in Underwood Dudley's Elementary Number Theory, but for the LIFE of me I can't figure it out! My initial thought was this: Let m be the smalles... |
I recently learned about summation methods when dealing with divergent series to give them a finite value. An example of this isusing Cesàro summation on Grandi's series to get 1/2. However every method I know of is unable to sum the harmonic series. Are there any summation methods that work on the harmonic series or i... |
Implicit diff on $\frac{2x}{y} - \frac{3y}{x} = 8$
I'm doing this:
$ \frac{2x^2 - 3y^2}{xy} = 8 $
$ 2x^2 - 3y^2 = 8xy $
$ 2x^2 - 3y^2 - 8xy = 0 $
Derive
$ 4x dx - 6y dy - 8( ydx + xdy ) = 0 $
$ dx( 4x - 8y ) - dy ( 6y + 8x ) = 0 $
$ \frac{dy}{dx} = \frac{4x - 8y}{ 8x + 6y } $
But if you use the quotient rule from the... |
A laser is suspended 1 metre above an infinite plane. The laser swivels on its aperture, which remains at point A. The aperture is directly above point B. The laser shines a infinitely small dot onto point C. We'll call the angle BAC θ, and the distance BC d. Therefore, d = tan θ.
As θ approaches 90°, d approaches infi... |
If we take $S^1$ acting on $S^2$ by rotation, then the height function $h: S^2\to R$ is an example of an invariant map (or an equivariant map where the action on $R$ is the trivial one).
I'm looking for an easy example of an equivariant map which is as simple as this one (and which can be visualized).
|
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