3
votes
2answers
172 views

a simple tensor product question

I have just started to learn about this today, and though i understand this is probably a very simple question, i'm still quite not sure about it: is ...
2
votes
1answer
96 views

Does the category of graded rings have limits?

Let $\mathfrak{C}$ be the category of ($\mathbb{Z}$)-graded-commutative rings. Does this category have limits in it? I am particulary interested in power series rings over a field. Is there a ...
4
votes
4answers
1k views

Prove sequence $a_n=n^{1/n}$ is convergent

How to prove that the sequence $a_n=n^{1/n}$ is convergent using definition of convergence?
1
vote
2answers
185 views

Theorem on Iterative Method Convergence

Theorem on Iterative Method Convergence If $\vert \vert I - Q^{-1}A \vert \vert <1$ for some subordinate matrix norm, then the sequence produced by $Qx^{(k)} = (Q-A)x^{(k-1)} + b$ converges to ...
4
votes
2answers
179 views

Projective modules over $k[X,Y]/(X^3,Y^5)$

I'm searching for an example of a module, that is not projective for $k[X,Y]/(X^3,Y^5)$, but projective for the two subalgebras $k[X]/(X^3)$ and $k[Y]/(Y^5)$. (I don't think it is relevant, but in ...
2
votes
1answer
135 views

Basis of a submodule of $\mathbb{Q}[X]^3$ over $\mathbb{Q}[X]$

I want to find the basis of a submodule of $\mathbb{Q}[X]^3$ generated by $(2X-1, X, X^2+3)$, $(X,X,X^2)$ and $(X+1,2X,2X^2-3)$. I determine that $$ \begin{pmatrix} 2X-1 & X & X^2+3 \\ X ...
1
vote
1answer
132 views

Let $y_{1},…,y_{k}$ be in $\mathbb{Z}$. Show that $\exists y \in \mathbb{Z}$ so that $y\equiv y_{1} \pmod {m_1},\dots,y \equiv y_{k} \pmod {m_k}$

Let $k\ge 2 $ and $m_{1},\ldots,m_{k}$ in $\mathbb{N}$ with $\gcd(m_{i},m_{j})= 1 \ (i \ne j)$. We show that $f(x) = x,\ldots,x)$ ist a ring isomorphism $f: \mathbb{Z}/ m\mathbb{Z} \rightarrow ...
4
votes
1answer
161 views

How do you find the base of a module given a set of generators?

Doing a little self-study, and I'm given a submodule of $\mathbb{Z}^3$ generated by $f_1=(1,0,-1)$, $f_2=(2,-3,1)$, $f_3=(0,3,1)$ and $f_4=(3,1,5)$. How do I find a base of the submodule? I put them ...
4
votes
2answers
606 views

first countable $\Leftrightarrow$ compact and Hausdorff?

Can someone give me a short sketch of a proof or a space that serves as a counterexample to the fact that every first countable space is characterized by being compact and Hausdorff (or, stronger than ...
3
votes
2answers
171 views

Example of matrix $M\in GL_3(\mathbb{Z}/7\mathbb{Z})$ such that $\langle M; +, \cdot \rangle \simeq GF(7^3)$ and the multiplicative order of $M$ is 3

I would want to make an example of a matrix $M \in GL_3(\mathbb{Z}_7)$ such that $\langle M; +, \cdot \rangle \simeq GF(7^3)$ and the multiplicative order of $M$ is $3$. Any hints how to do that ...
2
votes
2answers
96 views

What are equivalence classes that would not have (1+1 and 2 ) or (2 x 3 and 6) in same classes?

Are there any ordered equivalence classes that can be used to distinguish 1+1 and 2 , On one side there is an operation and on the other side just a single number. This becomes more obvious when ...
2
votes
2answers
149 views

basis of a vector space

1) what is an explicit basis for $\mathbb R$ as a $\mathbb Q$-vector space? 2) what is a basis for $\mathbb C$ as a $\mathbb C$ vector space? i think you will say $\{1\}$ is a basis since $\forall z ...
1
vote
2answers
73 views

Solving $2x=\frac{x}{y}-\frac{1}{1-y}$

This is a bit embarrassing, but I can't seem to solve for $x$ in $$2x=\frac{x}{y}-\frac{1}{1-y}.$$ Could someone please give me a hand!
3
votes
1answer
403 views

Question about order statistics

I saw a paper which says that: Let $Z_i$ be i.i.d. exponential random variables with mean $1$, and let $S_n = Z_1 + \dots + Z_n$ for all $n$. For a fixed $n$, let $U_j = S_j/S_{n+1}$, then ...
8
votes
3answers
789 views

an important property of an ellipse

Good morning everybody. I would like to know the proof of the following observation on the ellipse. A circle is drawn with the right latus rectum as diameter. Another circle is drawn with its ...
1
vote
1answer
141 views

Is there a unique solution for this equation..?

I need to solve the below mentioned equation and try to find a unique solution for $\epsilon$ for the range between (-1,1) in terms of $n$. $$\begin{equation} \sum_{j=1}^{n-1} \frac{(2{\alpha_j} + ...
3
votes
2answers
687 views

How the center of a non-cyclic free Group is trivial?

How can I show that "A center of a free group that is non-cyclic is trivial" ?
2
votes
0answers
93 views

Likelihood Function of Random Process

Given the following data: $$ x(t) = A + \omega(t) $$ where $ \omega(t) $ is an AWGN with zero mean, what would be likelihood function $p(x(t);A)$? I know it could be proven to be: $$ p(x;A) = C ...
9
votes
7answers
7k views

a circle graph is not a function?

I'm a little confused by the rule: If you draw a vertical line that intersects the graph at more than 1 point then it is not a function. Because then a circle like $y^2 + x^2 = 1$ is not a function? ...
2
votes
1answer
516 views

Lie bracket and covariant derivatives

I came across the following equality $[\text{grad} f, X] = \nabla_{\text{grad} f} X + \nabla_X \text{grad} f$ Is this true, and how can I prove this (without coordinates)?
3
votes
2answers
102 views

Is there a discrete probability distribution with some specific property?

The property I want is that I can change the variance with the mean fixed. I know it is easy to come up one in continuous case, i.e normal distribution with mean $\mu\ $ and variance $\sigma^2$. In ...
2
votes
1answer
114 views

Differentiability and the Chain Rule

I am not sure how to proceed with this question: Construct counterexamples for the following statements. (a) If a function $g(x)$ is differentiable at $x=a$ and a function $f(x)$ is not ...
4
votes
2answers
220 views

conditional expectation of a solution to the SDE

Suppose we have an SDE, which is the Wiener process with drift $dr_t=c dt+\sigma dB_t$, where $B_t$ is brownian I want to find $\mathbb{E}[e^{-\int_0^t r_s ds} |r_t=r]$ so my approach is this : ...
11
votes
3answers
3k views

Euclid / Hilbert: “Two lines parallel to a third line are parallel to each other.”

Background Many geometry books used to teach high-schoolers these days try to transfer Hilbert's reworking of Euclid's axioms into a (somewhat) palatable form for students. They don't usually seem to ...
1
vote
2answers
940 views

Show every subgroup of D4 can be regarded as an isotropy group for a suitable action of D4

Show every subgroup of D4 can be regarded as an isotropy group for a suitable action of D4 I know that D4={1,R,R2,R3,D1,D2,M1,M2} and the subgroups are {1,R,R2,R3} {1,D1} {1,D2} {1,M1} {1,M2} ...
1
vote
2answers
102 views

Interscholastic Mathematics League Senior B #12

Compute the product of the nonreal roots of the equation $x^4+4x^3+6x^2+1004x+1001=0$. So here is what I have done so far. I got two of the roots to be zero and 4 since ...
4
votes
3answers
870 views

Another Question in Hatcher

First of all, I apologize for asking yet another question about the hypotheses of a problem in Hatcher, but the statement of one of his problems has stumped me again. The problem is 1.3.15. It reads ...
1
vote
1answer
174 views

What does unique “minimal” partition mean (Context: Partitioning of Vertex-Sets)?

I am studying R. Diestel's Book Graph Theory and I encountered a formulation which I don't quiet understand. Mr. Diestel speaks in this proof on page 180 (Google Books Link) in the second last line of ...
1
vote
1answer
217 views

This is the most difficult question I could get without using mass point geometry

In triangle ABC, points D and E are on sides BC and CA respectively, and points F and G are on side AB with G between F and B. BE intersects CF at point O_1 and BE intersects DG at point O_2. If FG ...
1
vote
1answer
198 views

Pi approximation

If $d(a,b)=$ largest $n$ such that $a$ and $b$ agree on all digits upto $n$. Eg. $d(\pi,3.14)=3$, $d(0.1234667,0.1234669)=7$. What is the asymptotics of $d(\pi/4,1-1/3+1/5-1/7+\cdots(\pm)1/m)$ as ...
2
votes
1answer
118 views

Why can't $\mathbb{Z}/(p^k)$ for $k>1$ be the direct sum of two submodules?

If you mod out $\mathbb{Z}$ be a nontrivial prime power $p^k$, $k>1$, then why can't $\mathbb{Z}/(p^k)=\mathbb{Z}/(n)\oplus\mathbb{Z}/(m)$ for some such submodules? If that where the case, then ...
3
votes
2answers
358 views

Real Analysis: Continuity Proof

This problem has me stumped. I'm not sure how to proceed. Let $A = (0,\infty)$ and let $k: A \to \mathbb{R}$ be defined as follows: $$ k(x) = \begin{cases} 0 & \text{for } x \in ...
3
votes
1answer
92 views

Prove inequality using optional sampling

I proved the inequality below using Wald's identity and some tricky but easy manipulation, but I cannot do it using the suggestion from the source: "Hint: optional sampling!" Here is the problem: ...
-1
votes
1answer
71 views

Interscholastic Mathematic League Senior B Division [closed]

The number 2011 has the property that one of its digits is the sum of its other digits, i.e., 0+1+1=2. Compute the sum of the two largest integers less than 2011 with this property.
3
votes
3answers
496 views

What Does the Associative Property Mean Intuitively Across All Notational Schemes?

You can find descriptions of associativity as intuitively meaning that the order of operations performed does not matter, e. g. such as that of Wikipedia. However, if you write what associativity ...
0
votes
2answers
1k views

Simpler mathematic formula to find latitude coordinate mapping to lines “equally sized” on mercator projection?

I'm implementing a map visualization atop a mercator projected map (e.g google maps) where each circle appears to be the same size on the map: . At the equator, a circle plotted with a one degree ...
1
vote
2answers
120 views

Interscholastic Mathematic League Senior B Division #10

In traingle ABC, Angle A=45 degrees, Angle B is 60 degrees, and AC= radical 15. D is also a point on AB so that AB is perpendicular to CD. The circle with diameter AB intersects CD at point E. Compute ...
1
vote
1answer
78 views

Interscholastic Mathematic League Senior B Division #11

The roots of the equation 3x^3-38x^2+cx-192=0 form a geometric progression. Compute c.
3
votes
2answers
133 views

Infinitely many $n$ such that $p(n)$ is odd/even?

We denote by $p(n)$ the number of partitions of $n$. There are infinitely many integers $m$ such that $p(m)$ is even, and infinitely many integers $n$ such that $p(n)$ is odd. It might be proved ...
0
votes
1answer
103 views

Interscholastic Mathematics League Senior B Division #2

Points P,Q,R, and S are chosen on the sides of parallelogram ABCD, so that P is on line AB, Q is on line BC, R is on line CD, S is on line DA, and AP=BQ=CR=DS=1/3 AB. Compute the ratio of the area of ...
5
votes
2answers
256 views

Is $\mathbb R$ terminal among Archimedean fields?

I was wondering why metrics and norms are always defined to be real, rather than generalized to some other fields (or whatever). The best guess I have so far is: Because every Archimedean ordered ...
0
votes
1answer
86 views

Interscholastic Mathematic League Senior B Division #1

Let n be a positive integer less than 1000. If n^3 has 10 factors, compute the largest value of n.
4
votes
1answer
192 views

Limits of Functions

I'm self studying real analysis and currently reading about the limits of functions. Naturally everything in the chapter is about determining if a limit exists at a single point. But what about ...
7
votes
5answers
668 views

How many arrangements of $\{a,2b,3c,4d, 5e\}$ have no identical consecutive letters?

How many arrangements of $\{a,2b,3c,4d, 5e\}$ have no identical consecutive letters? I find it very tough... Could anyone have some good ways?
1
vote
3answers
108 views

How do I solve this matrix equation?

How do I solve this matrix equation and what is the answer: $$\begin{bmatrix} -122.366667 \\ 37.61666667 \end{bmatrix} = \begin{bmatrix} 0.000046 & 0.000032 & -122.413307 \\ ...
1
vote
0answers
354 views

how to find correlation between 2 arrays of 1's and 0's?

For my case, I have 2 arrays or sets of data, 100 elements, and the values are only 0 and 1. What test or procedure would measure the correlation or independence of the 2 sets? To give an example of ...
11
votes
1answer
389 views

How to “explain” Szemerédi's Regularity Lemma so that classmates may understand its value?

I am a student, preparing myself for a talk in which I want to present and prove Szemerédi's Regularity Lemma. I understand the proof and I am able to reproduce it - that is no problem. But I am ...
3
votes
3answers
800 views

functions $f=g$ $\lambda$-a.e. for continuous real-valued functions are then $f=g$ everywhere

I am supposed to show that if $f$ and $g$ are continuous, real-valued functions on $\mathbb{R}$, then if $f=g\;\;$, $\lambda$-a.e., then $f=g$ everywhere. So I have been reading and I think that this ...
7
votes
1answer
357 views

Is this a well known determinant identity? Are there any generalizations?

Let $A$ be a $3\times3$ matrix and for any $i,j\subseteq\{1,2,3\}$, let $A^{i,j}$ denote the $2\times2$ matrix resulting from removing row $i$ and column $j$ from $A$. Then: ...
2
votes
1answer
202 views

Interpretation of a question: “group of all p-power roots of unity”

I have a homework problem I'm trying to do, but I'm not sure what it's asking. The problem is as follows: Recall that $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the group of all roots of unity in ...

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