0
votes
1answer
9 views

$\lim f(x) = L, \lim g(x) = M\implies \lim \max\{f(x),g(x)\} = \max\{L,M\}$

I need to prove the following: $$\lim_{x\to a} f(x) = L, \lim_{x\to a} g(x) = M\implies \lim_{x\to a} \max\{f(x),g(x)\} = \max\{L,M\}$$ I know I need to, somehow, come up with this: given any $\...
0
votes
0answers
11 views

Theory of Games - What distinction am I missing between these two sets?

In outlining what specifically a Game looks like, Neumman introduces two sets: Definition of B(k) (http://i.imgur.com/zswNVFH.png) - In other words, BK(k) is all the possible combinations of moves up ...
0
votes
0answers
3 views

Can this heuristic about Sorli's conjecture and odd perfect numbers be made rigorous?

Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$. That is, we have $q \equiv k \equiv 1 \pmod 4$. Sorli (page 89) conjectured that $k=1$ always holds. Suppose we rewrite $N$ as $$N = ...
0
votes
0answers
9 views

forward path transfer function (draw root locus)

I have a question regarding drawing the root locus of a forward path transfer function of a unity feedback control system. I can find the zeros and poles on the s-plane however I have trouble ...
0
votes
0answers
13 views

Lower bound for a series involving complex numbers

If $z_k,\;\;1\leq k\leq n$ are any complex numbers such that $|z_k|\geq 1 \;\;1\leq k\leq n,$ then on $|z|=1$ what is the best lower bound (expression interms of |z_k|) for $$\left|\frac{n-\sum_{k=1}^...
-1
votes
0answers
11 views

Statistics Problem - Comparing Two Populations

http://imgur.com/XYGBryy (problem is too long to type, I've attached the link to the screenshot of the problem) Solved.
0
votes
0answers
11 views

A discrepancy in a vector identity involving gradient of product of 2 vectors.

We know that: $\nabla (Α\cdotΒ)=Α\times (\nabla \times Β)+Β\times (\nabla \times Α)+(Α\cdot\nabla )Β+(Β\cdot\nabla )Α$ Where $A$ and $B$ are two vectors. Now, suppose that the curl and divergence ...
0
votes
1answer
15 views

Why is $P(X\in[a,b])=P(X\in[a,b))=P(X\in(a,b])=P(X\in(a,b))$

I saw, for any continuous random variable $X$, $P(X\in[a,b])=P(X\in[a,b))=P(X\in(a,b])=P(X\in(a,b))$, where $a,b\in\mathbb{R}$, in my textbook. I don't quite understand why the openness/closeness of ...
-1
votes
1answer
18 views

Let A and B be n by n matrices . Prove that if A is symmetric and B be skew-symmetric , then {A,B} is a linearly independent set.

can anybody help me plz? Let $A$ and $B$ be $n \times n$ matrices ($A$ and $B$ are not $0$) . Prove that if $A$ is symmetric and $B$ be skew-symmetric , then $\{A,B\}$ is a linearly independent set.
0
votes
2answers
19 views

Find $\lambda$ and $\theta$ such that it validates this matricial equation

Find $\lambda$ and $\theta$ such that it validates the matricial equatial $$ \left( \begin{array}{cc} 1 & 2 \\ 2 & 3 \end{array} \right) % \left( \begin{array}{cc} \cos \theta \\ \sin \theta ...
0
votes
2answers
33 views

Uniqueness of Inverse

I am having trouble understanding the logic for a few steps in the following. I'll point the steps out at the end. If B and C are inverses of a square matrix A, then B = C. Proof: Since B is an ...
0
votes
1answer
24 views

Where to draw the line in scientific papers in respect to definitions

Now, a science paper about i.e. "the decrease of sunken pirateships as a function" would look (despite the title) very silly if it would define addition, multiplication, heck even exponentiation or ...
0
votes
1answer
23 views

How to find largest coefficient in matrix?

$$\begin{bmatrix} 1&2&6\\ 7&8&3\\ 0&4&7 \end{bmatrix} $$ I want know the algorithm to find largest value in matrix .
-1
votes
0answers
18 views

Prove that there exists such $A_1, \ A_1A_2, \ A_1A_2A_3, \cdots, A_1A_2A_3\cdots A_n$ having distinct remainders

Let A1 ... An be a permutation of 1 to n. Prove that there exists such A1, A1A2, A1A2A3, ..., A1A2A3...An having distinct remainders when divided by a prime n.
0
votes
0answers
22 views

Solutions to recurrence relations

Consider functions $s_{m},c_{m},d_{m}$ defined by the following recurrence relations $$s_{1}=n$$ $$c_{1}=s$$ $$d_{1}=0$$ $$s_{2}=n$$ $$c_{2}=s-n$$ $$d_{2}=d$$ $s,n, d$ are integers. If $c_{m}>...
-1
votes
0answers
28 views

prove that there exists $n $ such that $(x+1)^n(x^2 +bx +1) $ has all positive coefficients

Let $ n$ be a positive integer, prove that there exists $n$ such that $(x+1)^n(x^2 +bx +1)$ has all positive coefficients for$-2<b<0$.
-2
votes
0answers
21 views

Prove angle ABD + angle ACD = 60

D is inside triangle ABC such that AB=ab then BC=bc, CA=ca, DA=da, DB=db, DC=dc. Prove that angle ABD + angle ACD = 60 degrees
0
votes
1answer
16 views

Find the Moment Generating Function $Y$. What is the distribution of $Y$?

Let $X_1$ and $X_2$ be independent normal variables with means 2 and 5 and variances 9 and 1. Let $Y = 3X_1 + 6X_2 - 8$. Find MGF. What is the distribution of $Y$. attempt: Im not sure about ...
0
votes
2answers
33 views

What kind of operation is this?

$3\circ 7=37$ $7\circ 4=74$ $a\circ b=10*a+b$ This question came me from a mathpuzzle here. Basicly we multiply the first number by 10 and add the second number, but would this be a operation by ...
0
votes
1answer
26 views

How is the following derivative equal to the limit on the right side of the equation?

I am puzzled on how the following derivative is equal to the limit on the right side of the equation. I have tried to use the limit definition of a derivative to explain it, but I believe I am making ...
2
votes
1answer
19 views

Hyperplane Problem

Given $M$ points in the $\mathbb{R}^{N}$, I was wondering if there is an approximation algorithm to find a hyperplane which goes through the origin and also intersects as many points as possible. I ...
4
votes
1answer
49 views

Solutions to $a_1+2a_2+\cdots+ka_k = 1979$

For $k = 1,2,\ldots$ consider the $k$-tuples $(a_1,a_2,\ldots,a_k)$ of positive integers such that $$a_1+2a_2+\cdots+ka_k = 1979.$$ Show that there are as many such $k$-tuples with odd $k$ as there ...
1
vote
0answers
18 views

$R$ be an infinite commutative ring with unity , also a PIR , such that for every non-zero ideal $I$ of $R$ , $|I|=|R|$ ; is $R$ an Integral Domain?

Let $R$ be an infinite commutative ring with unity , which is also a PIR , such that for every non-zero ideal $I$ of $R$ , $|I|=|R|$ i.e. the sets $I$ and $R$ are bijective ; then is it true that $R$ ...
1
vote
0answers
12 views

The range of a continuous function on the order topology is convex

Let $(W, \leq)$ be a linear order, and let $f : [0, 1] \rightarrow W$ be a continuous function (where [0, 1] has the usual topology and W has its order topology). Show that the range of f is convex. ...
1
vote
0answers
16 views

Does distance characterize open sets in $\Re ^n$

I was wondering if given the standard topology for $\Re ^n$ that I might be able use the concept of distance in open sets to prove certain properties of the topological space. For example if I have ...
0
votes
3answers
38 views

How to find an exact limit of a function which is subtracting square roots of quadratic equations?

Find exact value of limit: $$ \lim \limits_{x \to \infty} \sqrt{( 3x^2 + 8x + 6)}-\sqrt{( 3x^2+3x+4)} $$ Here is what I've got so far: $$\lim \limits_{x \to \infty} \frac{(\sqrt{( 3x^2 + 8x + 6)}-\...
0
votes
0answers
5 views

Is this solution to a pairwise Procrustes Problem correct?

I have two sets of points, let's say, $y$ and $x$. Each set has exactly $N$ elements, and each element $element$ is such that $element \in I\!R^{d}$. That is, I have two shapes with $N$ $r-\text{...
0
votes
1answer
18 views

When is minimizing the sum of function equivalent to minimizing sum of independent variables

I have to admit I am not good at math, but this is a problem I am having trouble with. What kind of function $f$ can guarantee $min\sum_{i=1}^Kf(x_i)$ is equivalent to $min\sum_{i=1}^Kx_i$. Thank you. ...
1
vote
4answers
157 views

How does 5.9-7.66=-1.76?

Sorry for such an easy question but this has been confusing me like crazy. I have borrowed like normal and subtracted like every other problem, and the answer I get is -2.24. How is this wrong?
1
vote
1answer
29 views

Riemann Zeta Function, Stirling's Numbers, and Infinite Series

A while back I was able to prove the following identity, $$\sum_{k=1}^{\infty}\frac{\Gamma(k+r)}{\Gamma(k)(k+r)^s}=\sum_{k=0}^{r}s(r+1,r+1-k)\zeta(s-r+k)$$ where $s(k,n)$ are the Sterling numbers of ...
0
votes
0answers
23 views

Theorem 2 of perfect powers

I understood theorem 1 in Perfect Powers With All equal Digits but one where a is equal to $0$,{thanks to piquito and peter for helping me}. But theorem 2 seems so complicated when a is not equal to $...
0
votes
2answers
16 views

Conditional probability and tree diagram

This is the question I am working on: For two events, M and N, P(M) = 0.4, P(N|M) = 0.6, and P(N|M') = 0.4. Find P(M|N) I am stuck on this part of the question: Which branch of the tree diagram ...
1
vote
1answer
29 views

How to correctly write this in set notation?

I have defined the following set $$\{\mathrm{skill}_i^j\mid 1\leq i\leq 100\text{ and }1\leq j\leq n\}$$ I'm unsure if this notation is correct. My intention is to say that the set contains $n$ ...
0
votes
1answer
23 views

Integration by substitution in two dimensions.

I want to integrate over the unit square $||(x,y)||_\infty<1$, using the substitution: $u=\frac{1}{1-xy}$ and $v=\frac{1-x}{1-xy}$. What is the correct expression for $dxdy$ in terms of $u,v,du,...
1
vote
1answer
17 views

expected value of game involving uniform variable and its square

I am trying to determine the expected value of the following game: Let $u$ be drawn from a uniform distribution on $[0,1]$. We write down $u$ on one side of a piece of paper and $u^2$ on the other ...
1
vote
1answer
29 views

Can someone offer a way to simplify $x_1(y_1 - x^Ty) + x_1(w_1 - x^Tw)^2 - x_1^2(w_1 - x^Tw)^2 + x_1x_2(w_1 - x^Tw)^2$

Let $x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, $y = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$, $w =\begin{bmatrix} w_1 \\ w_2 \end{bmatrix}$ I have the following vector: $V = \begin{bmatrix} ...
1
vote
1answer
15 views

Proportionality between two quantities

Its known that if one variable is proportional to two others than it is also proportional to their product. $$\forall a,b,c\in ℝ:a\propto b\wedge a\propto c\Rightarrow a\propto b\cdot c$$ I think i`ve ...
0
votes
0answers
38 views

$i^i$ is real and also infinitely many sets in $\mathbb C - \mathbb R$

$i^i$ is real and also infinitely many sets in $\mathbb C - \mathbb R$ where operations in proper superset/field maps to a proper subfield. Is this of mapping between superfields to subfields of any ...
1
vote
1answer
40 views

Interesting Trigonometry Identity

Prelude While studying trigonometry, I came across this very interesting problem. It wasn't very difficult to solve, however it's result was quite interesting. I have given the solution below. Try to ...
0
votes
1answer
14 views

Total Probability Theorem / Partition

In the Total Probability Theorem we assume that the sample space is partitioned into subsets. If we consider $B$ to be the sample space and $A_1$, $A_2$ to be the partition then the theorem says: $$\...
0
votes
4answers
35 views

Helping understanding finding a limit without L'hopital's rule?

$\lim \limits_{x \to 0}x\sin\left(\frac{1}{x}\right)$ I need to find and prove this limit. I can easily plug it into wolfram alpha, but I want to make sure I learn something. It's been 3 years ...
1
vote
0answers
34 views

Concavity when 2nd derivative is zero

I was self-studying for a CLEP calculus exam when the following problem came up. It basically asks you to sketch a graph of a function based on the information given. My question is about the part ...
4
votes
3answers
72 views

Are there ways to solve equations with multiple variables?

I am not at a high level in math, so I have a simple question a simple Google search cannot answer, and the other Stack Exchange questions does not either. I thought about this question after reading ...
0
votes
2answers
42 views

Real Numbers Raised to Imaginary Powers?

What is a real number to the power of an imaginary or complex number? e.g. 3i. I have searched through sites about imaginary numbers, but none seem to say anything about imaginary indices. Examples ...
-4
votes
0answers
26 views

What some good lecture notes for self learning that are complete (nothing but the most trivial is left as an exercise)?

Subject that are bit difficult to self learn such as Galois theory, algebraic geometry, etc. I have two months of free time and I'd like a quick entry to these subjects and wish to understand more ...
1
vote
2answers
30 views

Two variable definition of derivative

Let $f:(0,1)\rightarrow \mathbb R$ be a real valued map from the unit interval. Let $$A:=\left \{a\in (0,1):\exists f^*(a)=\lim_{(x,y)\to(a,a)} \frac{f(x)-f(y)}{x-y}\right \}$$ It is true that if ...
0
votes
1answer
17 views

Two different remainders for same expression

$$\frac{n! + 1}{n} = (n-1)! + \frac{1}{n}$$ The remainder is $\frac{1}{n}$ $$n! + 1 \equiv 1 \cdot 2 \cdot 3 \cdot \dotsc (n-1) \cdot 0 + 1 \equiv 1 \mod n$$ The remainder is $1$ What is going on? ...
0
votes
1answer
13 views

Relationship of $L_1$ distance between CDFs and PDFs?

Let $F:(-\infty,\infty)\rightarrow[0,1]$ and $G:(-\infty,\infty)\rightarrow[0,1]$ two CDFs with PDFs $f$ and $g$, respectively. Is there a connection/inequality between: $$d_1 = \int_{-\infty}^{\...
1
vote
0answers
14 views

What is the relationship between the function $\mathbb{E}(Y \mid X = x)$ and linear regression?

Consider the function $$ r(x) = \mathbb{E}(Y \mid X = x) $$ This has been called the regression function in a textbook I'm using. I'm trying to figure out the relationship between this function ...
2
votes
4answers
65 views

Compute $ \lim_{x\to\infty}\frac{1}{x}\int_1^x\cos\frac{1}{t}\,dt $

I have a limit I can't solve. I'm studying for a test in Calculus II. I'm asked to compute the limit: $$ \lim_{x\to\infty}\frac{1}{x}\int_1^x\cos\frac{1}{t}\,dt $$ It's hinted that there is no need ...

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