All Questions
0
votes
0answers
4 views
Fast way to calculate Eigen of 2x2 matrix using a formula
I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html
Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
0
votes
0answers
3 views
Question on a third-order boundary value problems
This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems"
if $$u'''=\lambda (\sum_{i=1}^{m} c_i(t)u^{\mu_i}-d(t))+e(t), t\in (0,1)$$
and
...
0
votes
2answers
6 views
quadratic equation precalculus
from Stewart, Precalculus, 5th, p56, Q. 79
Find all real solutions of the equation
$$\dfrac{x+5}{x-2}=\dfrac{5}{x+2}+\dfrac{28}{x^2-4}$$
my solution
...
0
votes
0answers
8 views
Is a Relationship Quadratic?
I have a relationship $y=f(x)$ for which I can obtain data through simulation.
I have good reason to suspect that this relationship is quadratic (rather than, say, exponential), and would like to ...
0
votes
3answers
15 views
Finding the Fourier Series of $\sin(x)^2\cos(x)^3$
I'm currently struggling at calculation the Fourier series of the given function
$$\sin(x)^2 \cos(x)^3$$
Given Euler's identity, I thought that using the exponential approach would be the easiest ...
0
votes
0answers
24 views
question on summation?
Please, I need to know the proof that
$$\left(\sum_{k=0}^{\infty }\frac{n^{k+1}}{k+1}\frac{x^k}{k!}\right)\left(\sum_{\ell=0}^{\infty }B_\ell\frac{x^\ell}{\ell!}\right)=\sum_{k=0}^{\infty ...
3
votes
1answer
36 views
Last non zero digit of $n!$
What is the last non zero digit of $100!$?
Is there a method to do the same for $n!$?
All I know is that we can find the number of zeroes at the end using a certain formula.However I guess that's of ...
2
votes
1answer
17 views
Euler lagrange equation is a constant
I'm working through exercises which require me to find the Euler-Lagrange equation for different functionals.
I've just come across a case where the Euler Lagrange equation simplifies to
$$1=0.$$
...
0
votes
0answers
3 views
Characteristic equation/expression for addtion of n lognormal distributions
I have to find the expression for addition of n lognormal distributions (lognorm1+lognorm2+lognorm3+.....+lognorm nth) with mean values and error factors known for each lognormal distribution.Please ...
0
votes
3answers
16 views
Finding sequence in a set $A$ that tends to $\sup A$
I have been reading the book at http://www.neunhaeuserer.de/short.pdf, and have noticed that in the proof of the intermediate value theorem (Theorem 5.8 in the book), it seems to be quietly assumed ...
0
votes
0answers
9 views
Linearizing a series expansion
In the context of statistical mechanics the "classical trace" is defined as $Tr(A e^{-\beta H}) = \int dr^N dp^N A e^{-\beta H}$ where $r^N$ and $p^N$ are phase space variables. So if $\Delta H$ is a ...
0
votes
0answers
11 views
Topology of the Segre product vs. the product topology
In general, the product topology on two (quasiprojective) varieties is not the same as the topology of the product variety given by the Segre embedding. This is something I've often seen asserted is ...
1
vote
0answers
11 views
A special random subset of uniformly distributed numbers is still uniformly distributed?
Assume that I have a value range [1,1000].
Goal: I want to have 10 numbers randomly sampled from [1,1000].
case1:
...
0
votes
1answer
21 views
Combinations, Expected Values and Random Variables
A community consists of $100$ married couples ($200$ people). If during a given year, $50$ of the members of the community die, what is the expected number of marriages that remain intact?
Assume ...
-4
votes
0answers
34 views
What Hilbert-Style actually do? What is deduction? Why do we need it? Amazing stuff!
So, the problem is here:
We can use the following axioms:
$$\begin{align}
&A\to(B\to A)&\tag{A1}\\
&[A\to(B\to C)]\to[(A\to B)\to(A\to C)]&\tag{A2}\\
&(\lnot A\to\lnot B)\to(B\to ...
0
votes
0answers
11 views
Showing it is a joint probability density function
I have two random variables $X,Y$ with a joint density function $f_{X,Y}(x,y)=x+y$ if $(x,y)\in[0,1]\times [0,1]$ and otherwise $f_{X,Y}(x,y)=0$
I want to analyze this case in different cases, first ...
2
votes
1answer
24 views
calculate kernels of matrices with angles
So my professor gave me this question:
I have to find the basis of the eigenvalues of this matrix
\begin{pmatrix}
\cos(q) & \sin(q)\\
\sin(q) & -\cos(q)\\
\end{pmatrix}
so I calculate ...
1
vote
3answers
39 views
If $x \leq g(x) \leq x^2-x+1$ where $x \in [0,2]$, can we say that $g(x)$ is continuous at $x=1$?
If $x \leq g(x) \leq x^2-x+1$ where $x \in [0,2]$, can we say that $g(x)$ continuous at $x=1$ ?
Is $g(x)$ continuous in $[0,2]$?
3
votes
2answers
38 views
For every prime of the form $2^{4n}+1$, 7 is a primitive root.
What I want to show is the following statement.
For every prime of the form $2^{4n}+1$, 7 is a primitive root.
What I get is that
$$7^{2^{k}}\equiv1\pmod{p}$$
...
1
vote
0answers
44 views
For what $p$ is $x^p$ Lebesgue Integrable?
Revising for an exam on Monday any help with the following question would be greatly appreciated;
If $f$ is a function on $(0, \infty)$ taking values in $\mathbb R$, defined $f(x)=x^p$ ($p$ is a real ...
1
vote
0answers
6 views
Polynomials, integrals convergence
Let $P_n(x)= \frac{x^n(bx -a)^n}{n!}, \ \ \ a,b,n \in \mathbb{N}$.
Prove that $\int_0 ^{\pi}P_n(x) \sin xdx \rightarrow 0 \ \ \ \ $ and $ \ \ \ \ \int_0 ^r P_n(x)e^xdx \rightarrow 0$ $ \ \ \ \ \ \ (n ...
2
votes
0answers
21 views
The smallest nontrivial conjugacy class in $S_n$
Find the smallest nontrivial conjugacy class in $S_n$.
For small $n$, the answer is not hard to find:
$$\begin{array}{cc}
n & \text{smallest nontrivial class(es)} \\
1 & \text{none} \\
2 ...
0
votes
0answers
13 views
Coons patches using Matlab
I have studied Bilinearly Blended Coons patches, practically they are Coons Patches with blending funcions which are linear. I want to represent the following situation using matlab: 2 patches given ...
1
vote
1answer
25 views
Way to have inutition about the shape of a curve?
Past the usual memorized curves like $y=\sin(x), y=|x|, y=1/x, y=\ln(x), \ldots,$ is there a way to have an intuition about the shape of a curve from looking at an arbitrary function/term? (that is, ...
2
votes
0answers
13 views
Abelian Elliptic Surfaces
By abelian surface we mean a 2-dimensional algebraic complex torus. Thus
$$ S=\Bbb{C}^2/\Gamma$$
where $\Gamma$ is a rank $4$ lattice in $\Bbb{C}^2$ and such that $S$ is algebraic. It has trivial ...
1
vote
1answer
15 views
Sequence of continuous functions, integral, series convergence
Let $f_k$ be a sequence of continuous functions on $[0,1]$ such that $\int _0 ^1 f_k(x)x^ndx = \int _0^1 x^{n+k} dx$ for all $n \in \mathbb{N}$.
Is $\sum _{k=1} ^{\infty}f_k(x)$ convergent?
Could ...
0
votes
2answers
22 views
what is the diffrence between a term , constant and variable in first order logic languages ?
in the text , the author says that the language contains parenthises , sentintial connectives and n-place functions , n-place predicates , equality sign = , terms , constans and variables
i have two ...
-1
votes
0answers
42 views
I want help with $4\times 4$ symmetric matrix
I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to ...
1
vote
0answers
13 views
Generalized eigenspaces of a compact operator are finite dimensional
Let $T : H\rightarrow H$ be a compact operator on a Hilbert space $H$. Say that $\lambda \in \mathbb C$ is a generalized eigenvalue of $T$ if there is some $n \geq 1$ such that $(\lambda - T)^n$ is ...
0
votes
0answers
23 views
Right Triangles and Lagrange Multipliers
Suppose that you have a right triangle $a^2+b^2=c^2$ with integral sides. Given a perimeter $p=a+b+c$, how can you use Lagrange multipliers to determine the maximum length of $a$?
0
votes
1answer
19 views
Is the the number of generators of a group the number of different generators that one finds if one counts over every generating set of the group?
Consider the additive group of integers as an example as mentioned at the bottom of the Wikipedia article. There are two generating sets that are mentioned; The set consisting of the number 1, {1}, ...
4
votes
5answers
68 views
Limit as $x$ approaches $1$ from the right of $\frac{1}{\ln x}-\frac{1}{x-1}$
$$
\lim_{x\rightarrow 1^+}\;\frac{1}{\ln x}-\frac{1}{x-1}
$$
So I would just like to know how to begin to solve this limit, or what topic does this problem fall under so that I can search for ...
5
votes
0answers
27 views
What is the limit of the multidimensional integral?
What is the limit of the integral $$\int_{[0,1]^n}\frac{x_1^5+x_2^5 + \cdots +x_n^5}{x_1^4+x_2^4 + \cdots +x_n^4} \, dx_1 \, dx_2 \cdots dx_n$$ as $n \to \infty ?$
0
votes
0answers
13 views
Information about a particular polynomial
This question is related to this post Consider the following polynomial
$$
\alpha^3 xyzw + \alpha^2(1-\alpha) yzw + \alpha(1-\alpha) zw + (1-\alpha)w.
$$
One can of course generalize this to $n$ ...
1
vote
0answers
16 views
Is the Cauchy principal value “invariant” under change of variables?
Let $f \in C^{\gamma}_c(\mathbb{R}) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties:
1) K smooth everywhere except ...
1
vote
0answers
18 views
What is the broader name for fibonacci and lucas sequences?
Fibonacci and Lucas sequences are very similar in their definition. However, I could just as easily make another series with a similar definition; an example would be:
$$x_0 = 53$$
$$x_1 = 62$$
$$x_n ...
5
votes
0answers
59 views
High school contest question
Some work on it reveals the possibility of using gamma function. Is there any easy way to compute it?
$$\lim_{n\to\infty}\left(\frac{1}{n!} \int_0^e \log^n x \ dx\right)^n$$
1
vote
2answers
30 views
Expected number of pieces of a chessboard
If n squares are randomly removed from a $p \ \cdot \ q$ chessboard, what will be the expected number of pieces the chessboard is divided into?
Can anybody please provide how can I approach the ...
1
vote
1answer
35 views
A binary quadratic form: $nx^2-y^2=2$
For which $n\in\mathbb{N}$ are there $(x, y)\in\mathbb{N}^2$ such that $nx^2-y^2=2$ ?
0
votes
0answers
12 views
count the number of connected induced subgraphs in a graph with bounded degree
Let $G=(V,E)$ be a graph where the maximum degree of a vertex is 4. I've been asked to find an efficient algorithm for counting how many connected induced subgraphs are in $G$.
What I have so far is a ...
1
vote
1answer
20 views
Differentiation problem of power to infinity by using log property
Problem:
Find $\frac{dy}{dx}$ if $y =\left(\sqrt{x}\right)^{x^{x^{x^{\dots}}}}$
Let ${x^{x^{x^{\dots}}}} =t. (i)$ Taking $\log$ on both sides $ \implies {x^{x^{x^{\dots}}}}\log x = \log t$
This ...
2
votes
0answers
26 views
Use Fourier Series Techniques..
$x'' + 3x= 7$
given conditions $x'(0)=x'(5)=0$.
I checked the list and I went through three books. I am doing intro to differential equations. I just don't know how to get the extensions... I was ...
0
votes
2answers
26 views
Is there any specific terminology to refer to an initial sequence of a sequence?
Lets say you have a sequence $S = (0, 1, 2, 3, 4, 5, 6, 7, 8)$
And another sequence $T = (0, 1, 2, 3)$
Is there any specific mathematical term that defines the relationship between $S$ and $T$, ...
4
votes
2answers
49 views
How does a calculator calculate the sine, cosine ,tangent using just a number?
Sine Θ = oposite/hypotenuse
Cosine Θ = adjacent/hypotenuse
Tangent Θ = oposite/adjacent
So in order to calculate the Sine or the cosine or the tangent I need to ...
0
votes
2answers
38 views
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$.
Here is what I have here:
Aut$(G)$ consists of 6 bijective functions, which maps $G$ to ...
0
votes
1answer
29 views
Flow of $rot \overrightarrow{F}$
We've got vector field: $\overrightarrow{F} = \begin{bmatrix} yz\\x^3z\\e^z\end{bmatrix}$. I want to compute flow of $rot\overrightarrow{F} $($=curl \overrightarrow{F}$) through the area of the side ...
0
votes
0answers
8 views
Asymptotic recurrences?
$$T(n) = 2T(n/2) + \Theta(n), n > 1$$
$$T(n) = \Theta (1), n \le 1$$
$$G(n) = G(\lfloor n/2 \rfloor) + G (\lceil n/2 \rceil) + \Theta(n), n > 1$$
$$G(n) = \Theta (1), n \le 1$$
Prove $T(n)$ ...
2
votes
0answers
15 views
Bernstein type inequalities. Is there a standard list?
Suppose I have a sequence of iid random variables $X_i\geq 0$ with mean $\mu$ and $\mathbb E \left(e^{tX_i}\right) = G(t)$. Set $$S_n = \sum_{i=1} X_n.$$
For the purpose of this question the ...
1
vote
1answer
26 views
Proving integrability in integration by parts in Rudin's text
Integration by parts, as stated in W. Rudin's Principles of Mathematical Analysis, Theorem 6.22, goes as follows:
Suppose F and G are differentiable functions in $[a,b]$, $F'=f\in \mathcal{R}$, ...
4
votes
3answers
67 views
Definition(s) of rational numbers
The definitions of rational numbers are somewhat confusing for me. The definition of rational numbers on wikipedia and most other sites is:
In mathematics, a rational number is any number that ...
