1
vote
1answer
70 views

Triangle Inequalities

Anybody have a hint on how to begin to Prove that $\lvert x-y \rvert \lvert z-w \rvert \leq \lvert x-z \rvert \lvert y-w \rvert + \lvert x-w \rvert \lvert y-z \rvert$ for any $w,x,y,z \in \Bbb R$?
3
votes
0answers
204 views

How do I group unique pairs of sequential numbers in a grid?

Not sure if this SE site the best place to help find a solution for this problem. I am open to suggestions! It basically boils down to this: Given a grid of numbers where the numbers are ordered ...
0
votes
1answer
29 views

integration limit

How the integration, \begin{align}I= \Im\int_{-\infty}^{\infty} {\rm e}^{-\left(r - {\rm i}kR/2\right)^{2}} \,r\,{\rm d}r \\[3mm]\end{align} can be written as \begin{align}I= \Im\int_{-\infty - {\rm ...
1
vote
2answers
752 views

Finding a non-recursive formula for a recursively defined sequence

So I have a recursive definition for a sequence, which goes as follows: $$s_0 = 1$$ $$s_1 = 2$$ $$s_n = 2s_{n-1} - s_{n-2} + 1$$ and I have to prove the following proposition: The $n$th term of the ...
0
votes
1answer
117 views

Existence of a twice differentiable function

Let $U\subseteq \mathbb{R}^{m}$ open and simply connected, $B:U \rightarrow \mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{n})$ differentiable. If $$(B^{\prime}(x). v).w= (B^{\prime}(x). w).v \ \ \ \ \ ...
0
votes
1answer
61 views

$E[X_1+X_2+\cdots+X_n]=E[X_1]+E[X_2]+\cdots+E[X_n]$ Proof

I am trying to proof (from myself I have the case in my book for continuous random variable but want to find the proof for discrete random variables) that: ...
0
votes
2answers
319 views

null space and Image of V are finite dimensional, prove that V is finite dimensional

This seems so easy, but I don't see what I'm supposed to do. IF V IS FINITE DIMENSIONAL, then we know the dim(V) = dim(nullspace(V)) + dim(Image(V)). But this doesn't help me. So what should my ...
1
vote
1answer
72 views

Matrices over a ring: does $PAQ=A'$ imply $\mathrm{Coker}A\cong\mathrm{Coker}A'$?

In A Singular Introduction to Commutative Algebra by Greuel & Pfister, there is written on p. 127: Let $R$ be a commutative unital ring and $A\in R^{n\times k}$, $P\in R^{n\times n}$, $Q\in ...
2
votes
1answer
101 views

a logical proposition

Here I have a proposition: ((¬p ∨ x) ∧ (p ∨ y)) → (x ∨ y) To prove whether a tautology or contradiction or neither. ≡ ¬[((¬p ∨ x) ∧ (p ∨ y))] ∨ (x ∨ y) implication equivalence ≡ (¬(¬p ∨ x) ∨ ...
1
vote
1answer
59 views

How to solve a number which is raised to a large number?

How can I easily evaluate a number which is raised to a very huge number? For example; Is there an easy way to evaluate $2^{100}$ ($2$ raise to the power $100$)? It will get very messy and difficult ...
2
votes
2answers
131 views

Showing that there exists a surjective homomorphism from the integers to a cyclic group

How do I show that if $G$ is a cyclic group, say $G=\langle g \rangle$, then there exists a surjective homomorphism from the set of integers to $G$? Do I start by listing the elements of $G=\langle g ...
0
votes
1answer
69 views

Can we use $\land$, $\lor$ and $\lnot$ outside logic?

In most parts of mathematics I've seen logical symbols like $\implies, \exists, \forall$, etc. But I haven't seen $\land, \lor, \lnot$. Like " $\forall x\in\mathbb{R} \land y>0$ " instead of " ...
1
vote
1answer
38 views

Understanding the vertices in Cauchy's theorem

How to guess the contour vertices in Cauchy's integral. What would be the vertices for this integral and why? :$$\oint_C dx \, x \, e^{-x^2}$$ Is the vertices means that they limit the contour so we ...
2
votes
3answers
86 views

Evaluating $\lim_{t \to 0} \left(\int_0^1[bx+a(1-x)]^t \mathrm dx \right)^{1/t}$

$0 \lt a \lt b,$ find $$\lim_{t \to 0} \left(\int_0^1[bx+a(1-x)]^t \mathrm dx \right)^{1/t}$$ I think substituting $(b-a)x+a=u$ will make $$\begin{align*} \lim_{t \to ...
3
votes
2answers
1k views

Financial Linear Programming Problem

I'm very new at linear programming and I'm trying to figure out a way to approach this problem below: ...
2
votes
2answers
566 views

Help Getting a Zero out of the Denominator of a Limit

Why is the following limit equal to $1/2$. I get undefined. :-( $$\lim_{x\to 0}\frac{(x+1)^{1/2}+1}x$$ this should $= 1/2$. When I multiply the top and bottom by $(x+1)^{1/2} - 1$, I end up with ...
6
votes
1answer
60 views

Automorphisms of free groups

Suppose $U$ is a subgroup of finite index in the free group on $k$ generators $F_k$. Suppose $\sigma$ is an automorphism of $F_k$ such that $\sigma|_U = \text{id}$, then must $\sigma = \text{id}$?
1
vote
1answer
55 views

let $B=(f:\mathbb{C} \rightarrow \mathbb{C}/f(x)=x^ke^{cx};\mbox{ }k\in \mathbb{Z},\mbox{ }c\in \mathbb{C})$

Let $\mathbb{Z}^*=\mathbb{Z}^+\bigcup \left \{ 0\right \}$ and let $B=(f:\mathbb{C} \rightarrow \mathbb{C}/f(x)=x^ke^{cx};\mbox{ }k\in \mathbb{Z^*},\mbox{ }c\in \mathbb{C})$ Prove that $B$ is a ...
4
votes
2answers
181 views

What is the most motivating way to introduce modular arithmetic?

What the best way to introduce congruences in a number theory course? I am looking for something which will have an impact. What are the really interesting applications of congruent mathematics?
1
vote
2answers
224 views

Can the limit of a product exist even if the limit of one of the factors doesn't?

Show an example where $\lim_{x\to c} f(x)$ does not exist and $\lim_{x\to c} g(x)$ exists but $\lim_{x\to c} f(x)g(x)$ exists.
1
vote
4answers
107 views

What are the details behind structures such as $F[X]$?

I see the notation $F[X]$ often, where $F$ is an algebraic structure (usually a field). The notation $\Bbb R[X]$ has never been explained to us in class, other than that it refers to the polynomials ...
3
votes
0answers
512 views

What is the principal branch of $f(z)=\sqrt{1-z}$, $z\in\mathbb{C}$?

My question is, essencially, about the definition of the principal branch of a function that is not a Logarithm. In this case, $f(z)=\sqrt{1-z}$.
2
votes
1answer
144 views

Constructing a rational map from a divisor

This problem arose in an algebraic geometry course I'm taking, and my understanding of it comes from Shavarevich's "Basic Algebraic Geometry." The question is this: Given a projective variety $$X = ...
1
vote
1answer
53 views

If $X_n$ is binomial(n,p), then for any $b>0, P(X_n\le b)\rightarrow 0$.

If $X_n$ is binomial(n,p), then for any $b>0, P(X_n\le b)\rightarrow 0$. I used Hoeffding's inequality here: Since $EX_n=np$ and $X_n\in \{0,1\}\forall n\ge 1$, then $$\displaystyle P(\bar{X}_n ...
1
vote
1answer
178 views

books on fundamentals on statistics

I have mathematical backgroud but I am new to Statistics. So, could you please advice any books on Statistics and/or Biostatistics for beginners?
1
vote
1answer
53 views

Expressing elementary matrices in terms of each another

How can I express an elementary matrix of type 2 in terms of the product of elementary matrices of types 1 and 3? Just for clarity, here are the types: Type 1: \begin{bmatrix}1&a\\0&1\\ ...
0
votes
1answer
35 views

Are ideals of an sup semilattice always non-empty?

I am trying to do an exercise from the book A Compendium of Continuous Lattices. Exercise: Let $L$ be a set with a transitive relation, and let $A,B$ be ideals of $L$. (i) $A\cap B$ is an ideal of ...
1
vote
1answer
65 views

Is there a triangle like this?

This is my question that I posted at http://mathematica.stackexchange.com/questions/32338/is-there-a-triangle-like-this "I want to find the numbers $a$, $b$, $c$, $d$ of the function $y = \dfrac{a x + ...
1
vote
2answers
112 views

Intuition behind throwing two dice and conditional probability in general

I realize this is a rather trivial question when it comes to some that are asked on this exchange, but I was hoping for an intuitive (AND mathematical - I can't make sense of it if I don't understand ...
2
votes
4answers
626 views

Proper subsets of $\{a,b,c,d\}$.

List the members of $\mathcal P\left(\{a, b, c, d\}\right)$ which are proper subsets of $\{a, b, c, d\}$. Sorry, I know this is basic, but I'm knew to this. I think the answer is just $\{a\}, \{b\}, ...
2
votes
1answer
129 views

Using Janet Basis to solve a nonlinear polynomial system

I am trying to solve a nonlinear polynomial equation system using Janet basis, when they have finite many solutions. For example the solution of the system: $$xy^2-y^3-3x^2=0,x^2+y^2+xy=0.$$ There ...
2
votes
0answers
69 views

If X is Geometric(p), show that $\displaystyle E\left[\frac{1}{1+X}\right]=\log\left( (1-p)^{\frac{p}{p-1}}\right)$

If X is Geometric(p), show that $\displaystyle E\left[\frac{1}{1+X}\right]=\log\left( (1-p)^{\frac{p}{p-1}}\right)$ I found that $E\left[\frac{1}{1+X}\right]=\log p^{\frac{p}{p-1}}$ instead using ...
5
votes
3answers
740 views

Finite ring has zero divisors

”If the nonzero element $a$ of the ring $\mathbb{D}$ does not have a multiplicative inverse, then $a$ must be a zero divisor”. If $\mathbb{D}$ has finitely many elements, then the statement is true. ...
0
votes
2answers
305 views

Epsilon-Delta questions

Prove the following limits using only the epsilon delta definition: Q1: $$\lim_{x\to2^-} \sqrt{4-x^2}= 0$$ and Q2: $$\lim_{x\to\infty}\dfrac{x^2+2x}{x^2+1} = 1$$ For 1, I got stuck at the ...
2
votes
3answers
279 views

Why does the Method of Successive Approximations for a Differential Equation work?

Time dependent perturbation theory in quantum mechanics is often derived using the Method of Successive Approximations for a Differential Equation. I have not seen an explanation or a more rigorous ...
1
vote
0answers
63 views

Describing multilinear maps as linear operators

Let a finite dimensional complex vector space $V$ be given. Let $T^k(V)$ denote the vector space of multilinear maps $V^k\to\mathbb C$. My original question was going to be as follows: Does there ...
2
votes
1answer
95 views

Is open induction as strong as bounded induction without free bounds?

As was established in my question here, one reason that $Q$ + induction on formulas with bounded quantifiers is stronger than $Q$ + induction on quantifier-free formulas is that the variable that ...
1
vote
3answers
61 views

How to find the following Integral

I am unable to get the following integral. I know the basics of integration. I have tried looking it up but to no avail. $\int_0^\infty x^{-\frac{1}{2}}e^{-\frac{x}{2}}\,dx$ Thanks for the help.
1
vote
1answer
149 views

Problem involving trigonometric equality

Hello I am having some problems trying to solve the following trigonometric equation when is $$ -\tan(x) +3\sin(x) = \cos(x) $$ on the interval $0$ to $2\pi$.
1
vote
2answers
110 views

Does $xe^{1/\log(x)}$ have an oblique asymptote when x tends to infinity?

Does $xe^{1/\log(x)}$ have an oblique asymptote when $x$ tends to infinity? If so, what is the equation of this asymptote and how can we find it?
6
votes
2answers
184 views

Olympiad inequality: is this reasoning sound?

I am trying to show that for $a,b,c>0,\;abc=1:$ $$\underbrace{\frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)}}_{X}\geq \frac{3}{2}$$ This problem is from the Zhautykov Olympiad of 2008. ...
1
vote
1answer
48 views

I need help in limits

Find the values of real constants $a$ and $b$ such that $$\lim_{x\to \infty} \sqrt{ax^2 + bx} - \sqrt{x^2 + x + 1} = 1$$ If I obtained $a = 1$ and $b=0$, is that the correct answer? Or is b ...
2
votes
5answers
917 views

Distance between a point and a plane

I was just working on some review textbook problems in James Stewart's Multivariable Calculus when I encountered a problem that looked like the following: Find the distance between the point ...
9
votes
2answers
4k views

Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely

I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis. In section 2.7, he has the following exercise: Exercise 2.7.5 (a) Show that if $\sum{a_n}$ converges ...
3
votes
1answer
90 views

Why is echelon form important?

My professor gave us this definition for a system of equations in echelon form: A system of m linear equations in n variables is called an echelon system if m ≤ n. Every variable is the ...
2
votes
3answers
67 views

Will the rules of calculus stay the same when a real-valued function is defined over infinite number of variables?

So the question would be: Can we ever talk about a real-valued function that is defined over infinite number of variables? Will the rules of calculus remain the same for such functions described in ...
0
votes
1answer
82 views

Proving closed sets [duplicate]

Please may you help with the following question: let E be a non-empty subset of R. Let E' be its derived set(the set of all the limit points of E). How to prove that E' is a closed set. closed set ...
0
votes
1answer
281 views

Understanding quaternions & gradient descent in a paper on inertial / magnetic sensor arrays

I hope this question is appropriate here! I and a friend at work are trying to understand Sebastian Madgwick's paper, "An efficient orientation for inertial and inertial/magnetic sensor arrays" ...
12
votes
1answer
209 views

Characteristic of a finite ring with $34$ units

Let $R$ be a finite ring such that the group of units of $R$, $U(R)$, has $34$ elements. I would like to find the characteristic of $R$. Let $k:= \mathrm{Char}(R)$. If $\varphi$ denotes the ...
43
votes
2answers
909 views

This is stupid but I have a bad cold with cough

Can we have distinct positive real $x,y,z \neq 1$ with $$ x^{\left( y^z \right)} = y^{\left( z^x \right)} = z^{\left( x^y \right)} $$ in cyclic permutaion? It does not work well if any ...

15 30 50 per page