1
vote
0answers
50 views

Why is the covariance a measure of how much two random variables change together?

In descriptive statistics, the empirical covariance of a point cloud $\left\{(x_i,y_i) : 1\le i\le n \right\}$ is defined as ...
1
vote
1answer
127 views

Functions of 2 variables and applications to economics

Given the production function $Q := \sqrt K + L^2$, determine the optimal level of production and the relative demand of the two inputs capital $K$ and work $L$. The cost of a unit of capital ...
0
votes
2answers
74 views

How do I count all values that satisfy X mod N=1 in the range [A,B]

I want to count how many values of x in range [A,B] give remainder of 1 when divided by N. Is there any formula I can apply?
1
vote
2answers
34 views

Property of a function $f$ satisfying $|f(x) - f(y)| \leq (x - y)^2$

I am given a relation about differentiable function that $$|f(x)-f(y)| \le (x-y)^2$$ and if $f(0)=0$ then possible value of $f(1)=?$ I saw the slope form and rewrote it as $$ \frac{|f(x)-f(y)|}{x-y} ...
3
votes
4answers
408 views

Theorems in number theory whose first proofs were long and difficult

What are the examples of important theorems of number theory that has been shown to have surprisingly simple proofs though their first demonstration wasn't at all simple enough. Now simple proof is an ...
2
votes
1answer
293 views

Union of Sequences

My Analysis professor mentioned the following theorem: If a sequence $(a_n)$ is the union of finitely many disjoint subsequences, and if all the subsequences converge to $l$,then sequence $(a_n)$ ...
0
votes
2answers
41 views

Calculate an edge of a cube

The question is: How far the edge of the cube is increased knowing that if the $2\text{ cm}$ edge of the volume is $2402 \text{ cm}$ ? I already found that an edge is $(x+2)^3$ but I can't find the ...
1
vote
1answer
31 views

Equality of subgroups in groups locally finite

Let $G$ be a locally finite group and $X = \{[a,b] ; (\mid a \mid, \mid b \mid)=1, a,b\in G\}.$ Let $K = \left< X \right>.$ Consider $L = \displaystyle{\bigcap_{N \unlhd G} N}$ such that $N ...
1
vote
2answers
102 views

Questions on dihedral group and orthogonal group

I am learning dihedral groups in my abstract algebra course. My teacher leaves us the following exercise. I can work out the first three parts only and I cannot even understand the question in part 5. ...
1
vote
0answers
47 views

Trigonometric polynomials are dense

Is the set of all trigonometric polynomials in the space of continuous functions on [$-\pi,\pi]$ which are $2\pi$-periodic dense?(with sup-norm topology)Please give hints on how to find a sequence of ...
0
votes
1answer
48 views

Problem with a proof of theorem about diagonalization for selfadjoint operators

Suppose that $F: X \rightarrow X$ is a self-adjoint operator on a $n$-dimensional unitary vector space $X$ with $Spec T=\{a_1,..., a_r\}$. Let $E_i$, for $i=1,...,r$, be orthogonal projections on ...
1
vote
1answer
71 views

Minimization over two lines

This is a minimization question where the minimizing points can be chosen freely on two lines: $$\mbox{minimize}\, \prod_{i=1}^K {y_i}\quad \mbox{such that}\quad \prod_{i=1}^K ...
2
votes
2answers
61 views

Negation of a statement

So I am trying to prove a proposition. It goes like this Let there be $\emptyset\neq X\subset\mathbb{R}$ which is bounded from above. The next two statements are equivalent about $s\in\mathbb{R} $ ...
2
votes
1answer
236 views

What makes a line “straight”?

In Euclidean space, there can be several definitions that makes a straght line: Line of shortest distance between two points Line that is linear, i.e with the points satisfying a linear equation ...
4
votes
3answers
76 views

If A represents area of ellipse $3x^2+4xy +3y^2=1$ then the value of $\frac{3\sqrt{3}}{\pi}A = ?$

If A represents area of ellipse $3x^2+4xy +3y^2=1$ then the value of $\frac{3\sqrt{3}}{\pi}A = ?$ My approach : Since area of ellipse is $\pi ab$ where a is semi major axis and b is semi minor ...
2
votes
1answer
66 views

Any suggestions for a Math book to revive my long lost math skills and knowledge?

Since I got my bachelor degree in computer science in 2005, I have rarely been in touch with or even need to do any significant mathematical calculations. I have consequently lost most of my math ...
2
votes
2answers
154 views

CR Equations using Polar Form

I have a question to check whether following function is analytic or not using CR Equations. The question is f(z) = 1/(z-z^5) I just don't know how to start and ...
2
votes
2answers
43 views

Ellipse 1 : E$_1 = \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ( a > b)$ another ellipse 2 : $E_2$ which passes thru…

Problem : Ellipse 1 : E$_1 = \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ( a > b)$ another ellipse 2 : $E_2$ which passes thru extremities of major axis of $E_1$ and has its foci at ends of its minor axis. ...
2
votes
1answer
44 views

effective way to solve isomorphism of groups

Hello I am studying group isomorphisms but I fail to check in a few cases whether two groups are isomorphic or not.for example in $S_3 \times Z_4 $ and $ S_4$ I have checked with ...
5
votes
1answer
59 views

Solving $2\arcsin\frac{x}{2}+\arcsin(x\sqrt{2})=\frac{\pi}{2}$

How do I solve this equation: $$ 2\arcsin\frac{x}{2}+\arcsin(x\sqrt{2})=\frac{\pi}{2} $$ We know that: $$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$$ So letting $\alpha = ...
1
vote
0answers
55 views

Rotation Number of Polynomial

I conjectured that Maximum Rotation Number of $n$-th degree polynomial image of unit circle (in the complex plane) is $n$. (for example, if $f(z)=z^n$, then rotation number is $n$) Is it right?
0
votes
2answers
40 views

Terms in Fourier Series

Can any one explain why? $$\int_0^\pi \sin(nx)\sin(mx)\,dx=\begin{cases}0,&n\not=m,\\ {\pi\over 2},&n=m,\end{cases}$$ and $$\int_0^\pi \cos(nx)\cos(mx)\,dx=\begin{cases} 0, &n\not=m,\\ ...
1
vote
0answers
13 views

Stable flag for a self adjoint operator in a symplectic vector space

I'm trying to learn some facts concerning symplectic spaces and I have found this affirmation I cannot prove: If $x$ is self adjoint in a symplectic vector space then there's an isotropic flag ...
1
vote
1answer
36 views

Surjectivity of a map $D^{2n} \to \mathbb{CP}^n$

I'm solving an exercise about the complex projective space, and during a step of the solution I'm asked to find a surjective map $D^{2n} \to \mathbb{CP}^n$. I defined the map in this way $$ ...
0
votes
0answers
51 views

Primality of Stirling numbers of second kind

Apart from the Mersenne primes $M_p=2^p-1=\begin{Bmatrix}p+1\\2\end{Bmatrix}$, and the four primes $\begin{Bmatrix}n\\4\end{Bmatrix}$ where $n$ is given in http://oeis.org/A100958, are there other ...
0
votes
0answers
28 views

Which boundary condition dominates in elliptic boundary value problem?

I am working on a solution to a boundary value problem (which is too complicated for me to reproduce here) but have a question about the boundary. In many dimensions, my function is infinite along ...
0
votes
1answer
1k views

How mean, median and mode changes with change in one element of set.

I have N numbers. Let mean of these numbers be A. Mode of these numbers be B.Median be C. Now if I change one element in these, how will my mean, median and mode change ? Can I calculate it directly? ...
7
votes
0answers
358 views

A question about the article 'You can't hear the shape of a drum'

I have read the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
1
vote
0answers
27 views

LINEAR MAPS and their representation in ordered tuple

If $L:F^n\to F^m$ is linear map show that it can be written as like. $L(x)=(f_1(x),\cdots,f_m(x))$ Where $f_j\in (F^n)' and x\in F^n$ I tried to do something involving their basis but ...
2
votes
0answers
40 views

quotient by contractible space and homotopy equivalent? [duplicate]

Let $X$ be a space and $L$ be a subspace of $X$ that is contractible. Then is $X/L$ homotopy-equivalent to $X$ itself? it doesn't seem trivial to me at al...If so, how to show it rigorously? Could ...
1
vote
2answers
72 views

Use the formal definition of the derivative to find the derivative of $f(x) = \frac1x$

Working through some exercises which I have been set in a Stats module. I'm stuck on this problem. I can get to $$\lim_{h\to0}\frac{\frac1{x+h}-\frac1x}h.$$ Then I'm unsure as to where to go from ...
0
votes
1answer
23 views

Topic for attribute exploration

I'm writing my bachelor theses about the interactive algorithm 'attribute exploration'. For this I want to add an example. In the literature I found many such examples, like exploration of finite ...
2
votes
0answers
55 views

Finding volume using triple integral

Find the volume of solid bounded by the $x^2+y^2=a^2$ , $y^2 + z^2 =a^2$ , $x^2 + z^2 = a^2$ I can see that shadow in $x$y region is given by $x^2 + y^2 =a^2$ . but when I draw ray from shadow to up ...
0
votes
1answer
140 views

Smallest $\sigma$-algebra and $\sigma$-algebra generated by a function

I'm reading through the following theorem: Let $X=\{X_t,t\in T\}$ be a stochastic process. Then $\sigma (X)=\sigma ( \cup_{t\in T} \sigma (X_t))$ From my basic knowledge of measure theory, I ...
0
votes
1answer
79 views

If $\{X_n\}$ is a martingale, then $E[X_n-X_{n-1}]=0$

Apparently this should be quite simple, but I have been trying for a while and can't seem to get this. Let $\{X_n\}$ be a martingale, then we have: $$E[X_n-X_{n-1}]=0$$ According to some notes I found ...
0
votes
1answer
56 views

Bigger and Smaller for numbers - Works in both directions?

I wanted to know how to use the right term when explaining the difference between numbers. For example, I have two lenses: Lens 1 = 10x zoom Lens 2 = 5x zoom I know I can say that the 1 has 2x ...
6
votes
2answers
98 views

Limit of 2 variables function

$$ \lim_{(x,y) \to (0,0)} \frac{\sin^2(xy)}{3x^2+2y^2} $$ If I pick $ x = 0$ I get: $$ \lim_{(x,y) \to (0,0)} \frac{0}{2y^2} = 0$$ So if the limit exists it must be $0$ Now for ${(x,y) \to (0,0)}$ ...
1
vote
3answers
117 views

Boundary points of a domain bounded by a continuous curve

Suppose $F:\mathbb{R^2}\to \mathbb{R}$, which is given by $F(y_1,y_2)=\frac{y_1^2}{4}+\frac{y_2^2}{9}-1$. $S=\{(y_1,y_2) |F(y_1,y_2)=0\}$, and $D=\{(y_1,y_2) |F(y_1,y_2)<0\}$. I want to show ...
0
votes
0answers
29 views

Notation for bounds on derivative

I am working on a problem where the assumptions are that some derivatives are bounded. I want to refer to the individual bounds in the proof but there are about 7 of them in total. I am wondering if ...
1
vote
0answers
32 views

How do I know that min-term can't be combined any further?

I'm trying to learn (and implement) Quine-McCluskey algorithm for boolean function minimalisation. I'm learning the algorithm from wikipedia example. From that I understood the following: Take all ...
3
votes
0answers
60 views

Unipotent representations of SL(2,R) by quantization

I'm a PhD student in mathematical physics and I happen to need some elements of Kirillov's "orbit method" for producing representations of Lie groups. I'm familiar with symplectic geometry, geometric ...
0
votes
3answers
38 views

Proving $x^2 < y^2$ by means of the Ordering Axioms [closed]

How do I prove $x^2 < y^2$, if $0 \le x < y$ with the ordering axioms? thanks!
1
vote
1answer
51 views

Check the convergence (& absolutely) of parametric integral [closed]

$$\int\limits_{-1}^{1} \left(\frac{1+x}{1-x}\right)^{\alpha} \ln(2+x)dx$$ Don't know where to start..
3
votes
1answer
73 views

Finding the oblique asymptote of:

Given $$f(x)=\frac{x^2+1}{(x+1)^{\frac{1}{2}}}$$ how would you find the oblique asymptote of that?
0
votes
2answers
99 views

Is $\sum i^{1/i}$ bounded?

I'm trying to find the limit $$ \lim_{n\to\infty}\sum_{i=1}^n \frac{i^{1/i}}n\,. $$ I was going to say that $\lim_{n\to\infty} \frac1n=0$ and $\sum i^{1/i}$ is bounded but I can't prove it.
1
vote
0answers
37 views

Huygens formula

Let AB - arc of the circle , and the scale degree of the arc, and as the radius of the circle are unknown . For approximate calculation of the length of the arc used as follows. Notes on the arc of ...
0
votes
1answer
54 views

Simplifying the quotient $\frac{4x^4+2x^2+x+1}{x^2+1}$

I got stuck simplifying the following quotient. How to divide it? $$\frac{4x^4+2x^2+x+1}{x^2+1}$$ Thanks a lot!
2
votes
1answer
75 views

Varieties over a field $K$ are also varieties over any subfield of $K$.

Suppose that $f:X\longrightarrow\text{Spec} K$ is a variety over $K$, namely $X$ is an integral, separated $K$-scheme of finite type. Now if $L$ is a subfield of $K$, it is clear that there exists a ...
5
votes
2answers
146 views

Show that there do not exist 3 $\times$ 3 matrices $A$ over $\mathbb{Q}$ such that $A^8 = I $and $A^4 \neq I.$.

Show that there do not exist 3 $\times$ 3 matrices $A$ over $\mathbb{Q}$ such that $A^8 = I $and $A^4 \neq I.$. I am aware that the minimal polynomial of $A$ divides $(x^8−1)=(x^4−1)(x^4+1)$.If the ...
2
votes
1answer
50 views

Can we calculate the order of $\hom (G,G')$ in terms of $|G| $ and $ |G'|$ , when $G,G'$ are finite abelian groups?

Let $G,G'$ be abelian groups and let $\hom (G,G')$ be the set of all homomorphisms from $G$ to $G'$. We define an operation $\ast$ on $\hom (G,G')$ as: for $f,g \in \hom(G,G') \space , (f\ast ...

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