0
votes
1answer
8 views

Proof of Equation by Well Ordering Principle

I have an assignment question I am able to solve this question using basic Induction, but not able to figure out how to do it by using Well Ordering Principle. Any Solutions or hints would be very ...
1
vote
3answers
8 views

Finding the expectation value of a random variable counting the occurrences of certain events

Let there be $M$ (distinguishable) boxes and $N$ balls, which we uniformly distribute among the boxes. For $k \leq N$, let $g_k: \Omega \rightarrow \mathbb{Z}$ be the function counting the number of ...
0
votes
0answers
9 views

References about algebraic geometry

My question is very simple. I'm studying a course telling about algebraic surfaces but i think that i need some knowledge about basic algebraic geometry. Do you have some suggestions?
0
votes
0answers
4 views

Primitive subgroups with different order

I have been studying the permutation group $S_{n}$ in combination with transitivity and blocks. Which leads to primitive groups. Now I was wondering if for $n=6$, so $S_{6}$ could have two primitive ...
6
votes
4answers
53 views

Square root of 10=0?

I found a question that asked to find the limiting value of $$10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$$If you make the substitution $x=10\sqrt{10\sqrt{10\sqrt{10\sqrt{10\sqrt{...}}}}}$ it ...
1
vote
0answers
13 views

showing a local minimum of a function

I have a problem that is a little bit of a struggle for me. I'm pretty sure what I'm trying to prove is a minimum at the point a, but how I go about proving it is eluding me. Any help would be greatly ...
0
votes
1answer
11 views

Arrange the following growth rates in increasing order: $O (n (\log n)^2), O (35^n), O(35n^2 + 11), O(1), O(n \log n)$

I want to Arrange the following growth rates in increasing order This order are following : $O (n (\log n)^2), O ((35)^n), O(35n^2 + 11), O(1), O(n \log n)$ Please give me idea how to arrange growth ...
0
votes
1answer
13 views

State the value of x after the statement

State the value of x after the statement if P(x) then x := 1 is executed, where P(x) is the statement “x > 1,” if the value of x when this statement is reached is x = 0. x = 1. x = 2. this answer ...
0
votes
1answer
17 views

Introduction to Real Analysis Continuity Proof

Suppose $f,g: D\to R$ are both continuous on D. Define $h: D \to R$ by $h(x) =$ max{$f(x),g(x)$}. Show that $h$ is continuous on $D$. Here is what I have so far: Since $f,g$ are continuous, for ...
1
vote
2answers
18 views

Does an operator of x commute with the differential operator with respect to x?

While solving a problem in Quantum Mechanics I got an expression $ \frac{d}{dx}V(x)-V(x)\frac{d}{dx} $. The first term is just the derivative of the potential but the second one seems a bit weird. Is ...
0
votes
0answers
8 views

calculating the density on balance scale

If i had 1531 grams of silver on one end of a balance scale, how much of 24ct gold would i have to put on the other end of the scale to balance out? Keep in mind that the density of both metals are ...
0
votes
0answers
17 views

How to prove the eigenvalues of tridiagonal matrix?

Assume the tridiagonal matrix $T$ is in this form: $$ T = \begin{bmatrix} a & c & & & \\ b & a & c & ...
2
votes
1answer
23 views

The definition of an elliptic curve?

I've seen two different definitions of an elliptic curve. The first one being that it is a cubic curve of the form $y^2=x^3+ax^2+bx+c$, where all the (complex) roots are different. The other ...
0
votes
3answers
13 views

no. of monomials in variables $w,x,y,\ldots,z$ of degree $m$

The formula for no. of monomials in variables $w,x,y,\ldots,z$ of degree $m$,$\,\,\,\,\,$ (where e.g. $x^iy^jz^k$ degree $m=i+j+k$) is: $\,\,\,\,\,$$\binom{m+n-1}{n-1}$ where $m$ is degree of ...
4
votes
2answers
20 views

Can we find an $x, y : x < y$ and $x, y > 0$ and $\lfloor \frac{n}{x}\rfloor$ < $\lfloor \frac{n}{y}\rfloor$ for some integer $n > 0$?

I know there are no solutions when we have just the fraction without the floor, but how do we consider solutions when the floor is there?
0
votes
1answer
14 views

'Product rule' when $f$ differentiable at $c,$ and $g$ not differentiable at $c$, but $f(c)=0$

Take $h(x)=f(x)g(x)$ mapping some subset of $\mathbf{R}^n$ to $\mathbf{R}.$ If $f(c)=0$ and $f$ is differentiable at $c,$ $g$ continuous at $c$, can we say that the derivative is still ...
0
votes
0answers
10 views

Extending a theorem true over the integers to reals and complex numbers

How does one generally extend a theorem proved over the integers to the real numbers and beyond e.g. induction proofs, De Moivre's Theorem? I am aware that to extend a theorem proved over ...
0
votes
1answer
9 views

Proving property for all predicates in first order logic

Let's consider language with predicate $P$ and following derivation $${{{[P[a/x]]^1} \over {P[a/x] \rightarrow P[a/x]}}\rightarrow I^1 \over {\forall_x (P(x) \rightarrow P(x))}}\forall I$$ Doesn't ...
1
vote
2answers
15 views

Probability problem involving waiting time at pharmacy

I'm working on the following problem: The case of Safeway is very easy as the answer is simply the mean of $f_T$, namely $1$ minute. However, I'm having problems solving this problem for Target. ...
0
votes
0answers
18 views

finding presentation of a group

I was working through presentation of the quaternion group (with element 8), and I let $a = i$ and $b = j$. I immediately said $a^4 = b^4 = 1$, and $ab^2 a = 1$. Since I have a relation for each ...
2
votes
1answer
13 views

Probability of random assignment to form pairs

So the question goes: I have 100 individuals and 100 different buses, and I randomly assigned each individual to sit on a bus (each bus has equal probability of being selected). How many buses are ...
0
votes
0answers
8 views

Bordered minor and rank of a matrix

Let $M\in\mathbf{R}^{n\times n}$ be a matrix. Suppose that there is a $k\times k$ minor $M_k$ of rank k. Now this reference (Algebra For Iit Jee 7.65) here states that if all the $k+1$th minors ...
0
votes
0answers
13 views

Morphism between two $\mathbb Q-$ modules

I am trying to do the following exercise: Let $V$ and $W$ be two $\mathbb Q-$ modules and $f: V \to W$ a function. Prove that $f$ is a $\mathbb Q-$module morphism if and only if it is a group ...
0
votes
0answers
8 views

Questions on Strongly Differentiability.

Definition: Let $U\subseteq \Bbb R^m$ be an open set. Let $f: U \to \Bbb R^n$ be a function and $T: \Bbb R^m \to \Bbb R^n$ be a linear transformation. We say that $f$ is strongly differentiable ...
0
votes
3answers
28 views

Use the law of logarithms to expand an expression

$$\log\sqrt [ 3 ]{ \frac { x+2 }{ x^{ 4 }(x^{ 2 }+4) } } $$ How is this answer incorrect? $$\frac { 1 }{ 3 } [\log(x+2)-(4\log x+\log(x^ 2+4))]$$
0
votes
1answer
9 views

Conditional Probability Problem for a Joint Distribution

We have a joint probability distribution $f_{X,Y}(x,y)=\frac{1}{10}$, defined over the domain $(x,y)\in[-1,1]\times[-2,2]\cup[1,2]\times[-1,1]$. From this, I need to find the conditional PDFs ...
1
vote
0answers
18 views

Find the characteristic polynomial and the eigenvalues $\lambda_1, \lambda_2,\dots,\lambda_n$ of the matrix $A$

Let $\lambda_1, \lambda_2, \dots, \lambda_n$ be eigenvalues of the matrix $A=(a_{ij})_{n\times n}$. Is it true that $|A|=\lambda_a\lambda_2\dots\lambda_n$, and $tr(A)=\displaystyle\sum_i^n\lambda_i$. ...
0
votes
0answers
3 views

Can the power of a test of an inconsistent estimator still go to 1 as N goes to infinity?

This question is related with this one.«Can the power of a test of an inconsistent estimator still go to 1 as N goes to infinity?» However, the accepted answer doesn't explain one thing that I would ...
2
votes
3answers
30 views

4 equations 3 unknowns

If I have 4 equations and 3 unknowns, I could solve for the 3 unknowns using the first 3. How does it ensure that the 4th equation is also satisfied? In this case, what should be the usual strategy to ...
0
votes
0answers
5 views

The effect on k(G) of removing a vertex or an edge

Consider an (a, b) graph G (ie a graph with a vertices and b edges). Let k (G), the minimum amount of vertices that can be removed to disconnect the graph, be n>=1. What would be the possible effects ...
3
votes
5answers
34 views

Linear independence of the functions $1,\cos(x),\cos(2x)$

I want to show that the functions $1,\cos(x),\cos(2x)$ are linearly independent in $C[-\pi,\pi]$. I computed the Wronskian determinat of these functions but at the points $x=0,-\pi,\pi$ the obtained ...
0
votes
1answer
17 views

a problem in exact sequence

First, sorry for this ugly diagram Suppose $\require{AMScd}$ \begin{CD} ...@.\acute M @>f>> M @>g>>\check M @>>> O\\ @. @V \alpha V V\ @VV \beta V @VV ...
1
vote
0answers
14 views

how to prove the eigenvalues of tridiagonal matrix

Assume the tridiagonal matrix $T$ in this form: $$ T = \begin{bmatrix} a & c & & & \\ b & a & c & ...
0
votes
0answers
10 views

What's the easiest way to approximate an answer to n places after the decimal?

Say you wanted a decimal answer. How do you get one? I know evalf() can be used but it counts ALL digits, not just after the decimal. For example evalf(19/4, 2) gives 4.8 but I'm looking for a ...
0
votes
2answers
20 views

Xlosure subsets of a metric space X

If A and B are subsets of a metric space X, then cl(A ∩ B)= cl(A)∩ cl(B) Why is it false?? but it is true when it is AuB.
1
vote
1answer
13 views

Talyor's Theorem in Nocedal's Numerical Optimization

Please kindly refer to the figure below. I understand that (2.4) is just another formulation of Mean Value Theorem and I understand its geometrical meaning in 1-D case. However, I do not know what ...
1
vote
0answers
11 views

How to approach a minmax problem?

Starting with a certain geometric problem, I have reached this function: $$R(s,t,u,v)=\max(s-u,s+u,t-v,t+v,sX+tY+u, tX-sY+v)$$ where $X\geq0$ and $Y\geq0$ are parameters. I have to find the minimum ...
2
votes
2answers
12 views

Big-O Function for f(x)

I'm currently taking a Discrete Mathematics course which just started the chapter on The Growth of Functions. A (very) brief overview was given in lecture that covered the Big-O definition. Let ...
0
votes
0answers
3 views

Practical convolution of poisson and log-normal distribution

Hi guys, Im trying to make a Loss Distribution (in Excel), following the "Loss Distribution Approach". I do understand that the main idea is that we have a severity distribution and a frequency ...
0
votes
0answers
12 views

Problem with constructing a uniform probability measure over $\mathbb{N_0}$ using rationals on the unit interval

I've been toying with the possibility of constructing a uniform probability measure over $\mathbb{N^0}$. Obviously, one cannot just assign each non-negative integer a probability of 0 and call it a ...
0
votes
0answers
8 views

3D extension of Euclidean algorithm jigsaw method - help!

Recently I've been learning about how the Euclidean algorithm = jigsaw method (filling a rectangle with squares) = forming continued fractions. And today I'm wondering how a 3D version of the jigsaw ...
0
votes
0answers
14 views

Example of a special kind of infinite dimensional vector space and a linear map on it

Give example of an infinite dimensional vector-space $V$ and a linear transform $T$ on $V$ such that $T \circ S=S\circ T , \forall S \in \mathscr L(V) $ , but $V$ has a non-zero vector which is not ...
1
vote
0answers
23 views

Finishing a problem using equalities

This is my problem: Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that $$\frac{a^{n+2}}{a^n + (n-1)\,b^n} + \frac{b^{n+2}}{b^n + (n-1)\,c^n} + \frac{c^{n+2}}{c^n + ...
0
votes
1answer
30 views

Countable and uncountable sets.

a) Show that $\left \{ n^{2}+m^{2}:n,m\in \mathbb{N} \right \}$ is countable. b) Show that $\left \{ x\in \mathbb{R}:x(x-2)<0 \right \}$ is uncountable. My answers: a) Is it possible to define ...
0
votes
0answers
22 views

statistics problem, where did I mistake?

I searched interesting problem about statistic from http://www.mast.queensu.ca/~stat353/resources/pastfinals/final12sol.pdf $$ $$ But at the question No.2, I have some problem the red box $$ $$ ...
0
votes
3answers
28 views

Prove by induction for every integer$\; n\ge 5$, $2^n\gt n^2$.

Prove by induction for every integer$ \;n\ge 5$, $2^n\gt n^2$. My try: $$p(n):\;2^n>n^2$$ verify $P(5)$ $$ p(5):\;2^5>5^2 = 32 > 25 $$ Of course the trick is in the induction step and ...
0
votes
0answers
10 views

Bruns-Herzog, Cohen-Macaulay Rings, Exercise 10.1.16

This question is from the Bruns-Herzog, Cohen-Macaulay Rings, Exercise 10.1.16(a). Let $x_1,..., x_n, y, z$ be elements of $R$ such that ideals $(x_1,..., x_n,y)$ and $(x_1,..., x_n,z)$ are ...
0
votes
2answers
21 views

Find the maximum points of $f(x)=e^{-x}\sin^2(\pi x) \hspace{0.4cm},0<x<10$

Find the maximum points of $$f(x)=e^{-x}\sin^2(\pi x) \hspace{0.4cm},0<x<10$$ My calculations:I have calculated $f'(x)=\pi e^{-x}\sin(2\pi x)-e^{-x}\sin^2(\pi x)$ $f''(x)=e^{-x}\sin^2(\pi ...
0
votes
1answer
17 views

Expand and simplify

I've removed the brackets from the first equation in where it is 5a^2 + 2a - 5 and then multiplied 3 to the numbers inside the bracket. But I'm not sure what steps to take after that. (5a^2 + 2a - 5) ...
0
votes
1answer
22 views

Gaps between primes: bounds

Legendre's conjecture implies that the prime gap above any natural number $n$ is bounded by the product of a constant and the $\sqrt n$. Actually, I have strong reason to think that this factor by ...

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