# All Questions

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 ...
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 ...
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?
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 ...
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 ...
135 views

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" ?
93 views

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: ...
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.
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 ...
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 ...
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 ...
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.
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 ...
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 ...
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 ...
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.
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 ...
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?
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 \\ ...
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 ...
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 ...
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 ...
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: ...
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 ...