0
votes
0answers
6 views

Linear map - how to show this?

Assuming that I have a map $A: \mathbb{R}^2 \rightarrow \mathbb{R}$ and we have $A(-x,x) = -A(x,x)$ and $A(x+y,x) = A(x,x)+ A(y,x)$. Is this sufficient to conclude that $A( \lambda x+y ,x ) = \lambda ...
1
vote
1answer
28 views

L'Hospital Rule: Requirement that the limit exists

Could someone define what it means for a limit to "exist"? Must the limit after using L'Hospital Rule approach a specific value? What if the limit after using L'Hospital Rule approaches infinity? ...
0
votes
0answers
3 views

$0$ is an stable equilibrium of $x' = Ax$ iff $A$ is semisimple, given that all of its eigenvalues have real part 0.

$0$ is an stable equilibrium of $x' = Ax$ iff $A$ is semisimple, given that all of its eigenvalues have real part 0. I'm kind of confused here: I had understood that if all of the eigenvalues of $A$ ...
0
votes
1answer
43 views

prove conjunction of consecutive implications

$n\ge 2,p_1,p_2,p_3,...,p_n,p_{n+1}$ are statements. Prove $(p_1\rightarrow p_2)\wedge (p_2\rightarrow p_3)\wedge ...\wedge (p_n\rightarrow p_{n+1})$ $\Rightarrow (p_1\wedge p_2\wedge ...
0
votes
0answers
20 views

Are there functions for which the cyclic integration-by-parts technique does not work?

There are a lot of functions where you can use what my teacher has described as the 'cyclic' method of integration. An example is $$\int e^x\sin x\,dx$$ where you designate $u=\sin x$ and ...
0
votes
1answer
8 views

Solve ${z_1/\overline{z_2}} = z^3$

Let $z_1,z_2$ be complex numbers such that: $$z_1= 4\sqrt{2}-48\sqrt{2}$$ $$z_2= \cos{135^\circ} +i\sin{135^\circ}$$ Find all the complex numbers $z$ that fulfill the following equation: ...
1
vote
1answer
121 views
+50

can any identity involving integers be proved by mathematical induction

Hello mathematics community, Today I was studying mathematical induction which is an axiom. I was wondering Can "ANY" identity or inequality involving integers which is already proven can also be ...
0
votes
3answers
36 views

A tangent circle element between 2 intersecting vectors

2 Vectors, which are originating from one point I. I want to the replace the sharp corner (I) with an arc (circle element) with a radius of r. The arc touches the vectors at T1 & T2. What is the ...
-1
votes
0answers
8 views

Showing the product of two normal subgroups is normal

Prove that if $H$ or $K$ are normal subgroups then $HK=\{hk\mid h\in H,k\in K\}$ is a subgroup. Then if both are normal subgroups, prove that HK is normal.
1
vote
1answer
5 views

Spans and Dot Product: Findin the linear combination

Suppose $(v_1, v_2, v_3)$ is a set of vectors mutually perpendicular. Assume that $\|v_1\|= \sqrt{27}\quad \|v_2\| = \sqrt{14}\quad \|v_3\|= \sqrt{ 4}\ $ Let $w$ be a vector in ...
2
votes
3answers
26 views

General solution of recurrence relation if two equal roots

Consider the recurrence relation $$ ax_{n+1}+bx_n+cx_{n-1}=0 $$ If the characteristic equation $$ a\lambda^2+b\lambda+c=0 $$ has two equal roots, then the general solution is given by $$ ...
0
votes
2answers
26 views

Parametrization of $ax^2+bxy+c=0$

Can I just fix $y=t$ and use quadratic formula to get the rational points of the diophantine $$ax^2+bxy+c=0?$$ or is there another method? I feel like I am turning in circles with the quadratic ...
0
votes
1answer
28 views

Is this feature of the product topology still true if we take product to infinity?

We have been asked to show "Let $X_1, \ldots, X_n$ be topological spaces. Show that the product topology is the unique topology on $X_1 \times \cdots \times X_n$ with the property that, for any ...
0
votes
1answer
11 views

Additive categories.

The question is: Prove that an equivalence between two additive categories is an additive functor. Additive categorie, i mean a categorie with zero object, that every Hom set of morphisms is a ...
0
votes
0answers
9 views

Deleting 0's from a random mod 2 matrix

I am fairly new to optimization problems, so please forgive my lack of knowledge. That said, I'm trying to write a program that takes an NxM matrix randomly filled with 0's and 1's, then reduces this ...
-1
votes
2answers
12 views

Rules of i ($\sqrt -1$) to a power

$i^{2014}$ power =? A. $i^{13}$ B. $ i ^{203}$ C. $i^{726}$ D. $i^{1993}$ E. $i^{2100}$ I don't understand the concept that powers of i repeat in fours and that "two powers of i are equal if ...
0
votes
1answer
7 views

Probability of Drawing a Card from a Deck (Part 2)

This is a continuation on a question I asked a few years back: Say you have a 60 card deck containing 12 red cards and 48 black cards. After drawing 7 cards, what is the probability you will have 2 ...
0
votes
0answers
4 views

Let P1 = (x1, y1). Describe the set of all points P = (x,y) in R2 such that ||P-P1|| = 9 by identifying the type of conic and finding its equation.

Let P1 = (x1, y1). Describe the set of all points P = (x,y) in R2 such that ||P-P1|| = 9 by identifying the type of conic and finding its equation. I'm sorry, but this question throws me off in many ...
1
vote
2answers
19 views

If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3$ [on hold]

This Question May Be Deleted Is the following True or False: If $\sum_{n=0}^\infty c_n x^n$ is convergent for $x=-3 \implies:$ a) $\sum_{n=0}^\infty c_n 2^n$ converges. b) $\sum_{n=0}^\infty c_n ...
0
votes
1answer
6 views

Proving $f(z)$ has a zero on the unit disc using the Maximum Modulus Principle.

If $f$ is non-constant, continuous function on $\bar{D}_1(0)$, which is analytic on $D_1(0)$ and $|f(z)|=1$ on all $z$ on the unit circle, then $f$ must have a zero in $D_1(0)$. I know that ...
1
vote
2answers
16 views

Clarification of the notation $f: \mathbb{R} \setminus\{3\} \to\mathbb{R}\setminus\{2\}$

I have a question that uses the following notation: the function $f: \mathbb{R} \setminus\{3\} \to\mathbb{R}\setminus\{2\}$ is defined by $$f(x)=\frac{2x-3}{x-3}.$$ I understand that the left side ...
0
votes
0answers
5 views

Constructing a Borel set A on R such that 0<m(A intersect I) < m(I) for all intervals I.

I need help constructing a Borel set A on R with the following property: For every open interval I, 0 A obviously needs to be dense in R and it also must have empty interior, but honestly I don't ...
0
votes
1answer
18 views

Uniqueness of Thin QR Factorization.

Let $A \in \mathbb C^{m x n}$, have linearly independent columns. Show: If $A=QR$, where $Q \in \mathbb C^{m x n}$ satisfies $Q^*Q=I_n$ and $R$ is upper triangular with positive diagonal elements, ...
1
vote
1answer
22 views

Multiplying and adding fractions

Don't be angry with me. Just comment to delete if you think this is really a bad question. Why multiplying fractions is equal to multiply the tops, multiply the bottoms? $$\frac{a}{b}\times ...
0
votes
0answers
3 views

Increasing annuity problem

I learned increasing/decreasing annuity and am tackling the following problem for hours now without success. Amy deposits $Z$ into a bank account that has effective annual interest rate of $5\%$ ...
0
votes
1answer
30 views

if each $(i,j)\space:\space g_ig_j=g_jg_i$, then $G$ is abelian

Let $G$ be finite group. say that $a,b\in G$ hold that $(a,b)\in R\subseteq G\times G$ iff $\exists g\in G \space:\space gag^{-1}=b$ note that $R$ is an equivalence relation. let $g_1,...,g_n$ be the ...
0
votes
1answer
10 views

Splitting of Primes in a Given Field

Find how $p=2,3,5,7$ splits in $\mathbb{Q}(\sqrt{-5})$ (i.e. find those $e_i,f_i$ for $1 \leq i \leq r$). Can somebody please explain how this is done? My attempt is the following: Let K = ...
63
votes
21answers
12k views
1
vote
1answer
18 views

How to prove that an M-matrix is inverse-positive?

Wikipedia says that The inverse of any non-singular M-matrix is a non-negative matrix." To be more precise, if $A$ is an M-matrix, then the entries of the inverse of $A$ are all non-negative, ...
0
votes
1answer
21 views

Simplify |2x^2+3x-2| so we can obtain it and control it in terms of |x-1|

First example I worked, I had $|2x^2 + x - 3|$ after some manipulations and simplifications I obtain: $|x-1|(2|x-1|+5)$. The final answer is in terms of $|x-1|$ with multiplication between two ...
0
votes
0answers
9 views

Probability Question

A casino game has two dice, each with faces numbered 1 to 6. One of the dice is fair. The other die is biased such that a 6 is twice as likely to appear on top as any one of the other faces. ...
0
votes
0answers
6 views

Conditional expectation, indication function

I am given that $X,Y$ are independent Bernoulli RVs with parameter $p\in (0,1)$. I am also told $Z=1_{(X+Y=0)}$. I am asked to find $E[X\mid Z]$ and $E[Y\mid Z]$. I can see that the expected values ...
5
votes
0answers
39 views

Characterizing $\text{PGL}_2(\mathbb F_p)$

Where can I find a description and proof of the basic structure of $\text{PGL}_2(\mathbb{F}_p)$ (Number of elements with each order, conjugacy classes, etc.) which is understandable by an ...
0
votes
1answer
17 views

Factorials with fractions

I don't understand how $$ \frac {n(n-1)!+(n+1)n(n-1)!}{n(n-1)!-(n-1)!} $$ becomes $$ \frac {(n-1)![n+(n+1)n]}{(n-1)!(n-1)} $$ and then how it becomes $$ \frac {n+n^2+n}{n-1} $$ I've tried applying ...
1
vote
2answers
29 views

How to prove $D^2\setminus\{0\}$ is not homeomorphic to $\mathbb{R}^2\setminus\{0\}$?

Here $D^2$ denotes the closed unit disk in $\mathbb{R}^2$. I know that $D^2$ is not homeomorphic to $\mathbb{R}^2$ as $D^2$ is compact. Intuitively I believe that $D^2\setminus\{0\}$ is not ...
1
vote
0answers
4 views

Submagmas of natural numbers

What is known about submagmas of natural numbers under addition/multiplication? For example, all submagmas of integers under addition are of the form $~n \mathbb{Z}~$. Are there similar results for ...
1
vote
2answers
20 views

proving that for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ (verify)

for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ I wasn't sure how to go about this correctly so what I did was set $v+w$ to $v$, yielding $w = v-v = 0$, since it ...
0
votes
0answers
9 views

Probability of one device among 6 failing

A certain component of an electronic device has a probability of 0.1 of failing. If there are 6 such components in a circuit. What is the probability that at least one fails? This is not a duplicate ...
3
votes
1answer
171 views

Generators of the intersection of prime monomial ideals

Let $ [n] = \lbrace 1,2,\dots,n \rbrace $, and $ F \subset [n] $. We denote by $ P_F \subset K[X_1,\dots,X_n] $ the monomial ideal generated by the variables $ X_i $ with $ i \in F $. Given an ...
1
vote
0answers
38 views

Statement about $(I-A)^{-1}$ matrices

Let $A \in \mathbb{R}^{n \times n}$ and let denote $I$ the $n \times n$ identitiy matrix. Theorem. If $(I-A)$ is invertible and $(I-A)^{-1}$ is a nonnegative matrix and there is a diagonal element in ...
0
votes
1answer
14 views

Checking if transformation T(p(x)) is diagonalizable?

Say you have a transformation of $P_3$ to $P_3$ defined by, say, $T(p(x)) = p'(x) + p''(x) + p'''(x)$. How would you determine if this is diagonalizable? Do I sub in a standard basis of ...
0
votes
1answer
12 views

Testing numerical solvers with analytic solution to Ornstein-Uhlenbeck SDE?

I have an SDE I want to solve numerically that is fairly close to the Ornstein-Uhlenbeck process: $$ dx_t=θ(μ−x_t)dt+σdW_t $$ which has analytic solution $$ ...
3
votes
2answers
17 views

Primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field

I am looking for the primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field. I am not looking for a solution here, rather a hint or two. Is there a general strategy for approaching ...
1
vote
6answers
55 views

Prove that $gcd(a, b) = gcd(b, a-b)$

I can understand the concept that $\gcd(a, b) = \gcd(b, r)$, where $a = bq + r$, which is grounded from the fact that $\gcd(a, b) = \gcd(b, a-b)$, but I have no intuition for the latter.
0
votes
1answer
42 views

If the inner product of Ax with x is 0 for all x, then A=0.

Given matrix $A\in M_{n}(\mathbb{C})$, if $\left<Ax,x\right>=0$ for all $x\in \mathbb{C^n}$, then $A=0_{n}$. (Here $\left<a,b\right> = b^{\ast}a$ where "*" is the conjugate transpose.) ...
0
votes
0answers
6 views

How do I calculate the aspect ratio of a width and height?

I am creating an image upload form. I want to restrict users from uploading images that are not of the aspect ratio of 16:9. What is the formula for calculating an aspect ratio such as this? If w = ...
0
votes
2answers
33 views

For every prime of the form 6x-1 are there comparable number of primes of the form 6x+1

All primes except $2$ and $3$ are of the form $6x-1$ and $6x+1$. For every prime of the form $6x-1$ are there comparable number of primes of the form $6x+1$ in the first $10000$ primes or is there an ...
3
votes
1answer
23 views

Is there a simpler approach to this application of Dominated Convergence?

For a measure theory class, I'm trying to evaluate: $$\lim_{n\to\infty}\int^\infty_1\frac 1 {nx} e^{-x/n}\ \text d\lambda$$ Obviously I want to try and move the limit through the integral and ...
0
votes
0answers
4 views

Analysis of Data given mean, median, sd, quartiles

The statistics below provides a summary of IQ scores of 100 children Mean: 100 Median: 102 Standard Deviation: 10 First Quartile: 84 Third Quartile: 110 About 50 of the children in this sample have ...
0
votes
1answer
12 views

Working of selections

There are eight finalists in the 400 m athletics at the world championships. Three of the finalists are from the USA, and the others are from five different countries. The rules for allocating a lane ...

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