0
votes
1answer
47 views

Find quotient space on $\mathbb{N} $

On $\mathbb{N}$ is given equivalence relation R with $nRm \iff 4|n-m$. Topology on $\mathbb{N}$ is defined with $\tau=\{\emptyset\}\cup\{U\subseteq\mathbb{N}|n\in U \wedge m|n \implies m\in U\}$. I ...
3
votes
1answer
61 views

Automorphism of $A[t]/(t^m)$

Let $A$ be a commutative ring and $t$ an indeterminate over $A$. If $f$ is an automorphism of the ring $A[t]/(t^m)$ satisfying $f(x)\equiv x\pmod{(t)}$ for each $x\in A[t]/(t^m)$ with $m$ a positive ...
2
votes
0answers
165 views

Show that the projection map is Orientation preserving iff n is even

My question is that Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere $U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ...
1
vote
1answer
63 views

Second-order ODE with substitution

I’m struggling with this question: Use the substitution $y(t) = z(t)\,e^{-t}$ to transform the ordinary differential equation $$\frac{d^2 y}{dt^2} + 2\,\frac{dy}{dt} + y = t^2 e^{-t}$$ into an ...
6
votes
2answers
81 views

Embeddings of $GL(n-1,q)$ into $GL(n,q)$

Let $n>2$ and $q$ be a prime power. I want to find all the embeddings of $GL(n-1,q)$ into $GL(n,q)$. An embedding I first thought of is as following. Let $V$ be an $n$-dimensional vector space over ...
3
votes
1answer
68 views

On the primality of integers of the form $p^2+k$

I am not able to find an answer to the following question: For which positive even integers $k$ is the integer $$p^2+k$$ prime, where $p$ is a prime number $\gt5$?
1
vote
0answers
204 views

Econometrics Simultaneous equation Indirect Least Squares and Two Stage Least Squares

I still can't figure out this problem. PLEASE HELP! (1) $F_t = a_1 + a_2.C_t + a_3.P_t + e_t$ (2) $P_t = b_1 + b_2.F_t + b_3.S_t + b_4.I_t + u_t$
0
votes
2answers
305 views

Proving that $L^p \subset L^q$ when $1 \le q \le p$ [duplicate]

Let $(E,\mathcal{F},\mu)$ be a measure space such that $\mu(E)=1$ and let $L^p=L^p(E, \mathcal{F},\mu)$. Prove that $L^p \subset L^q\text{ if } 1 \le q \le p$. I let $f \in L^p$. Then $(\int_E ...
0
votes
0answers
51 views

Help me prove the supremum property.

Let $A$ and $B$ be nonempty sets and $f$ be a function from any nonempty set $S$ to subset of real number. Prove that $$\sup_{x \in A} \{ \min \{ \sup_{y \in B} \{ \min \{ f(y) \} \}, f(x) \} \}= ...
1
vote
0answers
48 views

Howuse this $R_{l}=\frac{1}{n}\left(\frac{(-1)^l}{2n}+\sum\limits_{m=1}^{n-1}\frac{1}{m}\cos{\frac{ml\pi}{n}}\right)$ and MATLAB get this four fig?

we consider Tikhonov's regularization method for $\delta =0.1, 0.01,0.001,$ and $\delta =0$ The Tikhonov's regularization method you can see:http://en.wikipedia.org/wiki/Tikhonov_regularization and ...
3
votes
1answer
58 views

For polynomial prove that $r(x)=0$ or $\deg [r_k(x)]<\deg [b(x)]$

Suppose $a(x),b(x)\in \mathbb R[x]$, $\deg(b(x)) \geq 1$. Show that there exists $m=0,1,\dots$ and $r_0(x),\dots,r_m(x)\in\mathbb R[x]$ such that $$ a(x)=r_0(x)+r_1(x)b(x) + \cdots + r_m(x)b(x)^m$$ ...
1
vote
1answer
282 views

Definite integration of a high order exponential function mixed with rational function

I would like to solve the integral $$\int_{x>0}xe^{ax^m+bx^n}~dx,\qquad m>n>0$$
1
vote
1answer
79 views

How can I prepare effectively for this test?

I live in Australia and I will be doing my final high school mathematics test in 5 months from today. The test is about the following topics: Differentiation and Integration and their applications ...
1
vote
1answer
57 views

This proof that Folner sequences imply amenability

During various travels, I encountered the following page on Wikipedia: http://en.wikipedia.org/wiki/Folner_sequence The proof in the "Proof of amenability" section is kind of neat and I'd like to ...
3
votes
1answer
184 views

Calculate an integral in a measurable space

Let $(X,\mathcal{M})$ a measurable set with measure $\mu$. Let $f$ be an integrable non negative function, such that $K:=\int_{E}f \mathrm d\mu<\infty$, where $E\in(X,\mathcal M)$. Let ...
9
votes
2answers
488 views

A hard 'if and only if' trigonometric identity proof

Prove $$ \frac{-2+2\tan A+2\cos B\cdot\sin B+\cot^2 A\cdot({\sec^4A-\operatorname{cosec}^2A-2)}}{2+\tan^2A-2\sin^2A} =(\sin A+\cos A)^2 $$ if and only if B is the double angle of A, or ...
0
votes
0answers
85 views

Find best k maximum spanning trees of graph

Is there any way to calculate best k maximum spanning trees of graph without finding all spanning trees?
1
vote
2answers
75 views

Positive defnite matrices

Hei guys, I would like to check with you a prove I have Let the matrix $X_1 = \begin{bmatrix} A_1 & B_1\\ B_1 ^ T & A_1\\ \end{bmatrix}$ be positive definite. Based on this I would ...
0
votes
2answers
280 views

Shading A Venn Diagram Using A Specific Equation

The expression is: $A\triangle(B\cap C')$. The $\triangle$ refers to $-$ and the $\cap$ refers to an intersection, whilst the $\;{}'$ refers to the prime of $C$. There are no numbers or items ...
1
vote
1answer
108 views

Implication: F $\implies$ T

Why is F $\implies$ T taken as true? Why is this the "convention"?
4
votes
0answers
53 views

“Disjoint” elements of a lattice - what's the correct terminology?

Given a set $X$ and a pair of subsets thereof, call them $A$ and $B$, we say that $A$ and $B$ are disjoint iff $A \cap B = \emptyset$. This generalizes to lattices with a least element. Given such a ...
1
vote
2answers
477 views

Relationship between second order derivatives and cross derivative of smooth surfaces

Probably a silly question, but I wonder if $z=f(x,y)$ is a smooth surface, and the values of its two second order derivatives $\frac{\partial^2f}{\partial x^2}$ and $\frac{\partial^2f}{\partial y^2}$ ...
1
vote
1answer
33 views

The definition of Pisot number

A Pisot number is an algebraic integer $>1 $ and all of whose conjugates have modulus $<1$. and I want to ask that is the polynomial is unique? I mean that the degree of the polynomial is ...
0
votes
3answers
74 views

What wrong with pointwise convergent

It is true that the uniform limit of continuous functions is continuous as it has proved as a theorem, but what is wrong with pointwise limit? I mean why this theorem it doesn't work for if the ...
1
vote
2answers
62 views

Proving that $\frac{\sigma_{n-1}}{\omega_n} = n$ in $\mathbb{R}^n$

If $\sigma_{n-1}$ was the surface area of the unit sphere in $\mathbb{R}^n$ and $w_{n}$ was the area of the unit ball in $\mathbb{R}^n$, my lecture notes prove that $$\frac{\sigma_{n-1}}{\omega_n} = ...
0
votes
1answer
54 views

Fourth moment of Z - why does this not work

I thought I could find $E(Z^4) = Var(Z^2)+E(Z^2)^2$ from the variance formula using $X$ as $Z^2$ (std normal square). i got $Var(Z^2) = 2$ because it's a 1 degree of freedom chi square and $E(Z^2)^2 ...
1
vote
3answers
60 views

Consider $x^4 \pmod {pq}$, with $p = q = 3 \pmod4$.

Consider $x^4 \pmod {pq}$, with $p = q = 3 \pmod 4$. Would someone explain to me why exactly one of the four square roots of $x^4 \pmod {pq}$ is also a square? This result was given without proof ...
1
vote
1answer
48 views

question about sets

I have this as a beggining to a question: $A\subseteq Z^2$ $$ A = \left \langle \left ( 1,7 \right );\left ( 7,2 \right );(2,3) \right \rangle = \left \{ ...
3
votes
1answer
120 views

How to find the discontinuity set?

What is the discontinuity set of the function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$, $ f(x,y) := \sup \{ \sin (tx) + \sin (ty) : t \in \mathbb{R} \} ?$
4
votes
1answer
1k views

Graph theory: Finding the number of different paths through a vertex on a complete graph

If G is a complete graph on n vertices and u,v,w are three distinct vertices in the vertex set of G, then how many different paths are there from u to v passing through w? For 3 vertices it is ...
1
vote
1answer
64 views

explain $df(tx).x = \sum_{i=1}^n {\partial f\over \partial x_i}(tx)x_i \hspace{1cm} x\in \mathbb R^n$

the question is : let $U$ be a Neighbourhood of the origine of $R^n$ and : $x\in U \Rightarrow tx \in U , \forall t\in U $ let f be a numeric function defined in U , and $f(0)= 0$ if we have ...
1
vote
1answer
60 views

is this function injective?

Is this function injective? $f:\mathbb{N}\to \mathbb{N\times N}$ defined as $f:n\to (n, n+1)$ $f(n_{1})=f(n_{2})\Rightarrow (n_{1},n_{1}+1)=(n_{2},n_{2}+1)\Rightarrow$ $n_{1}=n_{2} \wedge ...
3
votes
2answers
74 views

Integral of $ \int_{-1}^{1} \frac{x^4}{x^2+1}\,dx $

Any suggestions how to solve it? by parts? $$ \int_{-1}^{1} \frac{x^4}{x^2+1}dx$$ Thanks!
2
votes
2answers
61 views

Bernoulli differential equation help?

We have the equation $$3xy' -2y=\frac{x^3}{y^2}$$ It is a type of Bernouli differential equation. So, since B. diff equation type is $$y'+P(x)y=Q(x)y^n$$ I modify it a little to: $$y'- \frac{2y}{3x} ...
1
vote
1answer
126 views

The weight of the extended ternary Golay code

I need to show that the extended ternary Golay code with parameters $[12,6]$ is a self dual code and has minimum weight $6$ so i will obtain a $[12,6,6]$-code. I showed the self duality but how can I ...
2
votes
0answers
90 views

3D rotation group

It is known that the group $\text{SO}(3)$ of rotation-matrices (matrices $A$ with det(A)=1) are generated from three parameters. This can be expressed by the fact, that any rotation matrix is a ...
0
votes
1answer
400 views

How do you parametrize the paraboloid to find the flux of $F$ across $S$?

Evaluate the surface integral double integral of $S$ of $F * dS$ for the given vector field $F$ and the oriented surface $S$. In other words, find the flux of $F$ across $S$. For closed surfaces, use ...
4
votes
1answer
53 views

About parallel time computation

I am studying a paper where it is mentioned that Newton iteration may be used to compute the inverse of $n \times n$, well- conditioned matrix in parallel time $o(\log^2n)$ and that this computation ...
1
vote
1answer
41 views

Consider a Group $G$ of order $20$ such that $G=Aut$($Z_{25},+) \cong (U_{25}, \cdot)$. Analyze the Sylow subgroups in G.

$G=Aut$($Z_{25},+) \cong (U_{25}, \cdot)$ I know that there is one 5-Sylow subgroup and number of $2-Sylow$ subgroups is either $1$ or $5$. (a) How do I decide whether the number of distinct 2-Sylow ...
2
votes
2answers
64 views

$(x-a)^2(x-a-1) + p \equiv 0 \bmod{p^3}$

Let p be a prime and a belong to Z. Find all solutions to the equation $$(x-a)^2(x-a-1) + p \equiv 0 \bmod{p^3}$$ I'm having a hard time working with this as such few variables are given. We know p ...
2
votes
2answers
68 views

Can flux be proportional to $r^2$ in divergence theorem?

The motivation for the divergence being interpreted as the flux of stuff used the following: $$\text{div} F(a) = \lim_{r\to0}\frac{3}{4\pi r^3}\int_{|x-a|=r} F\cdot n dA$$ Without the $r^3$ in the ...
2
votes
2answers
83 views

Units on a ring

Let $A$ be a ring such that there exists units $u_1, \dots , u_p : u_1 + \dots + u_p=0$ and $p$ a prime. Did $A$ have a subring isomorphic to $\mathbb{Z}/p \mathbb{Z}$ in general Thanks in advance.
0
votes
1answer
283 views

Proving an Integral domain is a field. [duplicate]

Let $R$ be an integral domain containing a field $F$ as a subring. Show that if $R$ is a finite dimensional vector space over $F$, then $R$ is a field. This is a Ph.D. entrance question, I recently ...
9
votes
4answers
278 views

What will be a circle look like considering this distance function?

I am working on some exercises in the book Geometry: A Metric Approach with Models by R.S. Millman. He defines the following map: $$d_S(P,Q):\mathbb R^2\times\mathbb R^2\to\mathbb R\\\ ...
3
votes
4answers
111 views

Closed form of a recurrence relation using generating functions

It's been awhile since I have done this. The sequence is $\displaystyle a_n = a_{n-1} + 5~a_{n-2}$ with $a_{0}=0$ and $a_{1}=1$. I found the generating function to be $\displaystyle G(x) = ...
6
votes
1answer
185 views

Prove that if $Q^tQ = I$ and $A = QR$, then $\|Ax - b\| = \|Rx - Q^tb\|$

I have a linear algebra final tomorrow and was practicing a few proofs. I want to make sure this proof is correct. Prove that: If $Q^tQ = I$ and $A = QR$, then $\|Ax - b\| = \|Rx - Q^tb\|$ ...
-2
votes
2answers
91 views

Question about derivation in Jordan algebra

Let $(G,\circ)$ be a Jordan algebra, then $\sigma:G\to G$ given by $$c\mapsto a\circ(b\circ c)-b\circ(a\circ c),\quad\forall c\in G,$$ is a derivation, where $a$ and $b$ are two fixed elements of $G$. ...
33
votes
4answers
2k views

Find $\lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$

Find$$\lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}.$$ This is a recent exam question, which I couldn't figure out in the exam. My guess is it doesn't ...
3
votes
2answers
274 views

Is it an abuse of notation to omit the leading zero in a decimal less than 1?

Is it acceptable to write $.001$ rather than $0.001$ when using decimal notation? Are there contexts in which omitting the leading zero is acceptable, and other situations in which it is not?
1
vote
1answer
114 views

Finding Partial Derivative ($n$-dimensional) using implicit differentiation vs explicitly solving

This is a book example (not a homework question) about implicit differentiation on a composite of functions in $n$-dimensional space. But my book explains this example in a very unclear manner. So I ...

15 30 50 per page