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 ...
6
votes
1answer
113 views

Niceness of the projection of a closed subscheme of affine space?

Let $k$ be an algebraically closed field, and suppose $C\subseteq \mathbb{A}^{n+m}_k$ is a closed subscheme. What can we say about the image under the projection $\pi: \mathbb{A}^{n+m}_k\rightarrow ...
3
votes
0answers
68 views

How to show that $\exists~f\in C(X,\mathbb R)$ such that $f$ is unbounded? [duplicate]

Let $X$ be a non-compact metric space. How to show that $\exists~f\in C(X,\mathbb R)$ such that $f$ is unbounded?
1
vote
2answers
145 views

Trees with vertex set

I am having hard time understanding and solving the following question: There are exactly three trees with vertex set {1,2,3}. Note that all these trees are paths; the only difference is which ...
2
votes
1answer
125 views

Total no. of ordered pairs $(x,y)$ in $x^2-y^2=2013$

Total no. of ordered pairs $(x,y)$ which satisfy $x^2-y^2=2013$ My try:: $(x-y).(x+y) = 3 \times 11 \times 61$ If we Calculate for positive integers Then $(x-y).(x+y)=1.2013 = 3 .671=11.183=61.33$ ...
2
votes
1answer
107 views

Block matrix notation

Given that $A$ is a real, rectangular matrix of dimension $m \times n$ and $\begin{align} A = \left[\begin{array}{c} I \\ e^{\intercal} \\ -e^{\intercal}\end{array}\right] \end{align}$ is represented ...
1
vote
3answers
412 views

Relationship between matrix equations and determinants?

What is the relationship between a matrix equation and the determinants of the matrix? Say if you have the matrix equation $A^3 - 4A = 0$, what can I learn about the matrix $A$'s determinants? Or ...
2
votes
1answer
158 views

About measure theoretic interior and boundary

Reference:- Evans-Gariepy, Federer, other books and notes of geometric measure thoery. I just want to clarify whether these definitions of measure theoretic interior and boundary are correct. Given ...
5
votes
2answers
120 views

Number Theory: $x^2+y^2=a^2$

Is there a coprime triple $(x,y,z)$ such that $x^2+y^2=a^2, x^2+z^2=b^2, y^2+z^2=c^2$, where $a,b,c$ are integers P.S. such solution doesn't exist for $a,b,c<1000$, as the computer says P.P.S. ...
10
votes
1answer
534 views

Has this phenomenon been discovered and named?

If $$x-\frac{x}{2}=\frac{x}{2},$$ and $$\frac{x}{\sqrt{x}}=\sqrt{x},$$ and $$x-\uparrow(x-\uparrow^22)=x-\uparrow^22$$ when $(x\uparrow^n-[A])\uparrow^nA=x$, where $A$ is some constant, and one uses ...
5
votes
2answers
401 views

What are some reasonable things to prove about the Collatz Conjecture?

I am writing an undergraduate paper on the $3n+1$ problem, and I am looking for some theorems related to the problem that would be reasonable for someone with my mathematical background to prove. I'm ...
1
vote
1answer
101 views

Conditional Random Variable and Iterated Expectation

First, I can't seem to find a definition of a "conditional random variable"; I understand the concepts of conditional expectation and conditional probability, but is there a notion for the random ...
7
votes
2answers
149 views

Find a lattice with exactly three congruence relations

A lattice $L$ is a partially ordered set such that any two elements $a$ and $b$ have a least upper bound $a\lor b$ and a greatest lower bound $a\land b$. A congruence relation on $L$ is an ...
4
votes
4answers
33k views

Definite Integral of square root of polynomial

I need to learn how to find the definite integral of the square root of a polynomial such as: $$\sqrt{36x + 1}$$ or $$\sqrt{2x^2 + 3x + 7} $$ EDIT: It's not guaranteed to be of the same form. ...
0
votes
0answers
112 views

Free basis and free group

How can I prove the following result? Let $G$ be a group, $X\subseteq G$ and let $F_a(X)$ be the free group on $X$. Then the subgroup of $G$ generated by $X$ is isomorphic with $F_a(X)$ if and only ...
1
vote
1answer
266 views

area of square between tangent(externally) circles

Two externally tangent circles,have a square between them,standing on the same base as the two circles.The circles have a radius of 1 unit each.The top two vertices of the square are touching one ...
0
votes
2answers
127 views

Find the flux of the vector field out of the region $T$ bounded by $z = x^2 + y^2$ and $z = 4$. Please help

Find the flux of the vector field $$\vec{F}(x,y,z)= <x,y,3>$$ out of the region $T$ bounded by $z = x^2 + y^2$ and $z = 4$. It says the unit vector $n$ is $<0,0,1>$, but how do you find ...
3
votes
1answer
61 views

Trying to define an abstract notion of a function that turns sets of sentences into the set of models satisfying those sentences.

Let $\mathsf{Sen}$ denote a Boolean algebra, thought of as a collection of sentences, and let $\mathsf{Mod}$ denote a set without any additional structure, thought of as a collection of models. I'm ...
8
votes
1answer
76 views

If $A$ is orthogonal, for any vector $x$ such that $Ax = b$, $\Vert x \Vert = \Vert b \Vert$

Is this statment true: For any vector $x$ such that $Ax = b$, $\Vert x \Vert = \Vert b \Vert$, if $A$ is orthogonal. I was working on a proof for my linear algebra class, when I noticed that the ...
2
votes
1answer
49 views

Simplex with edges of length at least s having smallest circumradius

Is it true that of all $k$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? Please supply a proof or ...
1
vote
2answers
46 views

Complex Solutions to Polynomials

I'm trying to use topology to prove that: $z^{n} + a_{n-1}z^{n-1} + ... + a_{1}z + a_{0} = 0$ has a solution in $\mathbb{C}$ if and only if, for each positive real number $c$, the equation $z^n + ...
3
votes
2answers
175 views

$A^3+A=0$ We need to show $\mathrm{rank}(A)=2$

Let $A\ne 0$ be a $3\times 3$ matrix with real entries such that $A^3+A=0$. We need to show $\mathrm{rank}(A)=2$. $\det A(A^2+I)=0\Rightarrow\det A=0\Rightarrow \mathrm{rank}(A)<3$, Suppose ...
2
votes
1answer
141 views

Map Quadrant Conformally onto the Unit Disc and find $|g'(1+i)|$.

If $w = g(z)$ maps the quadrant $\{z = x + iy \; : \; x,y >0\}$ conformally onto $|w|<1$ with $g(1) = 1$, $g(i) = -1$, and $g(0) = -i$, find $|g'(1+i)|$. My initial reaction was to directly ...
0
votes
0answers
184 views

Evaluating a polynomial at a root of unity?

Let $R = \mathbb{Z}[x]/(x^n+1)$ be the $2n$th cyclotomic ring (for $n$ a power of $2$ in which case $\Phi_{2n}(x) = x^n+1$). Let $g$ be an $n$-dimensional vector chosen at random from $\mathbb{Z}^n$ ...
0
votes
1answer
184 views

How can I prove that the solutions of this differential equation is monotone?

I'm trying to proof if $x:I\to \mathbb R$ a maximal regular solution of $x'=f(x)$, such that the image $x(I)\subset \mathbb R$ is bounded and $f:\mathbb R\to \mathbb R$ is $C^1$, then $x$ is strictly ...
2
votes
2answers
83 views

Improper Integral $\int_0^1\frac{dx}{x^p}$

Is this integral convergent only for $p<1$? $$\int_0^1\frac{dx}{x^p}$$
2
votes
1answer
152 views

How to calculate smallest pattern size based on set of ratios

So I have x number of items, each one has its own appearance ratio for making a pattern, of which I'm trying to determine the smallest possible pattern. The appearance ratios for all items total up to ...
5
votes
1answer
113 views

For any $11$-vertex graph $G$, show that $G$ and $\overline{G}$ cannot both be planar

Let $G$ be a graph with 11 vertices. Prove that $G$ or $\overline{G}$ must be nonplanar. This question was given as extra study material but a little stuck. Any intuitive explanation would be great!
1
vote
2answers
47 views

$P\left( n,\left( {{\lambda }_{1}}+{{\lambda }_{2}} \right)T \right)$ Disaggregating Tail of Poisson

I have a Poisson tail $P\left( x,\left( {{\lambda }_{1}}+{{\lambda }_{2}} \right)T \right)$ which is sum of two independent Poisson distribution with rate $\lambda_1$ and $\lambda_2$. I am trying to ...
3
votes
2answers
135 views

Geometric solution to classic committee problem

Most people know the classic committe style problems. I read this solution to one of the basic version of the committe problem and was impressed, but not sure why it works. I was hoping someone ...
2
votes
2answers
1k views

Let G be a graph in which every vertex has degree 2.

Is G necessarily a cycle? I suspect not but I'm having hard time showing this. Also, Let be a tree. Prove that the average degree of a vertex in T is less than 2. I know that the sum of degrees of ...
0
votes
0answers
71 views

an upper semi-continuous semi-strictly quasi-concave function

May you help me? I was confused that why "an upper semi-continuous semi-strictly quasi-concave function is quasi-concave". If there is a picture,it may be better. I really need your help ~ Thank ...
4
votes
2answers
135 views

existence continuous function $f:X\to [0,1]$ such that $f(x_1)=0 , f(x_2)=\frac{1}{2}, f(x_3)=1$

Let $X$ be a Hausdorff normal topological space and $x_1, x_2, x_3$ are three distinct points.prove that there exist continuous function $f:X\to [0,1]$ such that $f(x_1)=0 , f(x_2)=\frac{1}{2}, ...
1
vote
1answer
127 views

Bounded and invertible operator on dense subspace

Who can give me an operator like this or show it doesn't exist: Operator T: X-->Y, is a bijection from normed linear space X to normed linear space Y. X, Y are equipped with the same norm, and X is a ...
0
votes
1answer
540 views

Tricky Expectation

Let $X$ denote a Poisson distributed random variable and let $Z = \max(X-\min(X,c),0)$, where $c$ is a constant. How do I compute $\mathbb E[Z]$? Any help is greatly appreciated!
3
votes
2answers
707 views

Non-homogeneous linear recurrence relation

I have this recurrence relation to solve: $$a_{n+1}=3a_n+2^{n-1}-1.$$ The homogeneous part's solution is obviously $a_n=k3^n.$ Now I don't know how to solve the original equation, but I do know what ...
3
votes
2answers
145 views

Rotman's Introduction to to the theory of groups. Exercise 3.45.

Can you give me a hint on the first part of the exercise? Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
2
votes
2answers
53 views

What's the meaning of oscI(fn-f)?

I haven't seen the form of osc$_I(f_n-f)$.I expect your explanation.

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