# All Questions

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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $$1=0.$$ ...
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 ...
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 ...
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 ...
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 ...
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: ...
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 ...
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 ...
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 ...
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 ...
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 $[0,2]$?
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 $$7^{2^{k}}\equiv1\pmod{p}$$ ...
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $a$?
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}, ...
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 ...
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 $n \to \infty ?$
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 $n$ ...
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 ...
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 ...
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? $$\lim_{n\to\infty}\left(\frac{1}{n!} \int_0^e \log^n x \ dx\right)^n$$
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 ...
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 $nx^2-y^2=2$ ?
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 ...
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 ...
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 ...
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 $T$, ...
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 ...
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 ...
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 ...
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 $T(n)$ ...
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 ...
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 $[a,b]$, $F'=f\in \mathcal{R}$, ...
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 ...

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