2
votes
4answers
361 views

Positive Definite Matrix Problem

Suppose that the Symmetric Matrix $$B=\left( \begin{array}{cc} \alpha & a^T \\ a & A \end{array} \right)$$ of order $n+1$ is positive definitive. (a) Show that the scalar $\alpha$ must ...
1
vote
3answers
535 views

Find a bijection between the Cartesian Product $[m] \times [n]$ and $[m n]$

Let $[n]$ denote $\{1,2,\dots n\}$. Show there is a bijection between $[m]\times[n]$ and $[mn]$. I was thinking about doing something related to induction to prove that there is some value $b$ ...
1
vote
1answer
121 views

Factoring polynomials of degree $a p^b$ over extension fields.

Let $f(x)$ be an irreducible polynomial with integer coefficients, which is irreducible over $\mathbb{Z}$ and has degree $a p^b > p$ with $p,a,b>0$ and $p$ a prime. It appears that $f(x)$ ...
2
votes
1answer
61 views

Summability of a function

If $u\in W^{1,p}(\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ and $\xi$ is a smooth compactly supported function in $\Omega$, is it true that $\xi u^{\beta-p+1} \in W^{1,p}_0$ if $\beta ...
3
votes
1answer
556 views

Canonical form of conic section

I have $x^2+2xy-2y^2+x-4y=0$ and I have to find its canonical form, but I'm a little confused.. I'd like to understand very well what I have to do.. Can you help me, please? Thanks!
2
votes
0answers
234 views

Field of Fractions for Commutative Ring with Identity

I'm trying to complete a problem that is walking me through creating a field of fractions $F$ from a commutative ring with identity (and no non-zero divisors) $R$. The construction first asks me to ...
1
vote
2answers
73 views

Simple Vector Question in $\mathbb{R}^3$

Two points $A$ and $B$ in $\mathbb{R}^3$ with origin $O$ are given in terms of a Cartesian coordinate system by $A = (1, 2, 3)$ and $B = (4, 5, −1)$. How do you find the point $C$, such that $OACB$ ...
4
votes
2answers
117 views

Proofs using linear congruences

We have just covered solving linear congruences, and I am confused about how to use them in proofs. I understand that the linear congruence $$cx \equiv b \pmod m$$ has a unique solution $\bmod m$ if ...
3
votes
1answer
151 views

Lattices inside matrix groups $SL_2(K)$

I am currently a second year undergraduate majoring in math and our university is offering an opportunity for undergraduates to do a project over the summer break. I have spoken to my professor who is ...
1
vote
0answers
34 views

To every Markov process corresponds a Q-matrix? [duplicate]

Possible Duplicate: Logarithm of a Markov Matrix It is known that to every Q-matrix corresponds a unique Markov process. Does the converse hold? Specifically, Can a (discrete state, ...
1
vote
1answer
391 views

Modules of finite type over Noetherian rings

Let M be a unitary module of finite type over a commutative Noetherian ring R with a unit. Can M then always be represented as a quotient of a pair of free R-modules of finite-type?
2
votes
2answers
104 views

bound the distance of two roots of multivariate polynomial systems

Consider a system of multivariate polynomial equations $\vec{x}= f(\vec{x})$ with integer coefficients, $f$ is at most of degree 2. Suppose $\vec{x}_1$ and $\vec{x}_2$ are two real roots of $f$, is ...
1
vote
1answer
28 views

How to extract a variable before differentation operator?

I am trying to make some derivations of open channel flow equations. And the problem is, I quite don't get some of the operations that are given in books on the following subject. For example: ...
0
votes
0answers
77 views

How to explain rigorously that quick sort is faster than say buble sort?

I can appreciate that in most instances, quick sort would avoid from repeated operations, in compare to an exhaustive method. But, how one can demonstrate this fact in a rigorous fashion, and ...
0
votes
2answers
450 views

graph theory and forests

We were given an this question in my class: Prove that a forest with n vertices and m components has n-m edges using induction on m. Induction is not my strongest point and I was wondering if anyone ...
-3
votes
2answers
263 views

Prove that 2x is always an even number.

How does one prove that for every value in $\Bbb N$ 2x = an even number?
1
vote
1answer
211 views

Linear Algebra basic notation question

My book writes: A vector in $F^n$ may be regarded as a matrix $M_{n\times 1}(F)$. (true / false) What is $F$ or $F^n$, and how does the notation $M_{m\times n}(F)$ work? The books also likes to use ...
0
votes
1answer
215 views

Image of open set through linear map

Suppose $f$ is a linear map between vector spaces, and whenever $U$ is an open set containing $0$, then $f(0)$ is an interior point of $f(U)$. Can we deduce that any open set containing $0$ has an ...
0
votes
2answers
272 views

Inequality. $\sum{\sqrt{x^2+xy+y^2}}\geq \sum{\sqrt{2x^2+xy}}.$

Let $x,y,z >0$. Prove that: $$\sum_{\text{cyc}}{\sqrt{x^2+xy+y^2}}\geq \sum_{\text{cyc}}{\sqrt{2x^2+xy}} .$$ Thanks for your help :)
3
votes
1answer
411 views

Uniform boundedness imply uniform convergence?

If ${f_n}$ is a uniformly bounded sequence of holomorphic functions in $\Omega$ such that ${f_n(z)}$ converges for every $z \in \Omega$, is the convergence is uniform on every compact subset of ...
15
votes
1answer
1k views

When is the image of a null set also null?

It is easy to prove that if $A \subset \mathbb{R}$ is null (has measure zero) and $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lipschitz then $f(A)$ is null. You can generalize this to $\mathbb{R}^n$ ...
5
votes
1answer
79 views

Bypassing a series of stochastic stoplights

In order for me to drive home, I need to sequentially bypass $(S_1, S_2, ..., S_N)$ stoplights that behave stochastically. Each stoplight, $S_i$ has some individual probability $r_i$ of being red, ...
0
votes
1answer
1k views

Do we divide first or multiply first if we don't have any other information [duplicate]

Possible Duplicate: What is 48÷2(9+3)? 6/2*(1+2) is 1 or 9? order of operations division Apparently a very simple question but My question basically is, whether the answer of the ...
0
votes
1answer
78 views

classical probability question

At the race course, Adam meets his friend Bruce. Bruce offers to receive bets from Adam for a race involved two horses named Gold and Diamond. Bruce suggests even odds for betting on both horses. ...
2
votes
1answer
329 views

$\sigma$ algebra of collection of random variables

Im doing a course on measure theory and I'm stuck on one of the exercises. Take $\{Y_{\gamma}:\gamma \in C\}$ as an arbitrary collection of random variables and $\{X_{n}: n \in N\}$ to be a countable ...
7
votes
1answer
266 views

If a sequence satisfies $\lim\limits_{n\to\infty}|a_{n+1} - a_n|=0$ then the set of its limit points is connected

Prove that if a sequence satisfies $\lim\limits_{n\to\infty}|a_{n+1} - a_n|=0$ then the set of its limit points is connected. My professor once mentioned a proposition likewise but I cannot find ...
0
votes
0answers
67 views

Extremely puzzling probability question… [duplicate]

Possible Duplicate: Conjunction fallacy Alice is thirty-one years old, single, outspoken and very bright. She majored in finance. As a student she was deeply concerned with issues of ...
1
vote
0answers
46 views

Invert big-O involving logarithms while retaining a good error term

I have an equation $$ y=\frac{kx}{\log(y/k)-1}+O(1) $$ which I would like to solve for $y$ in terms of $x$ ($k$ is constant). Clearly $y\sim kx/\log x$ but I would like to preserve the error term. ...
6
votes
2answers
98 views

Why is positivity of first entry sufficient for a matrix in $\mathrm{SO}(1,3)$ to be in $\mathrm{SO}^{+}(1,3)$?

This is a result physics books tell all the time, that the branch of proper Lorentz transformations with positive first entry forms the identity component of Lorentz group. In mathematical language, ...
1
vote
1answer
71 views

Transforming a Continuous Function

My math is quite limited so please bear with me. I will get to the point: Is there a way to transform a continuous function into a bounded one? In essence I have a normalized Gaussian distribution ...
2
votes
1answer
1k views

Laurent series for $\sin z \sin(1/z)$?

How can I find a laurent series for $\sin z \sin(1/z)$ for $z \neq 0$? Is it possible to multiply two laurent series? I saw that in wikipedia, it's not generally possible.
6
votes
5answers
7k views

How do you calculate the modulo of a high-raised number?

I need some help with this problem: $$439^{233} \mod 713$$ I can't calculate $439^{223}$ since it's a very big number, there must be a way to do this. Thanks.
5
votes
1answer
294 views

Could Residue theorem be seen as a special case of Stokes' theorem?

Residue theorem in complex analysis is seems like Stokes' theorem in real calculus, so a question arose that could Residue theorem be seen as a special case of Stokes' theorem?
2
votes
2answers
73 views

About $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$

Suppose $M$ is map from vector space $X$ to vector space $Y$, $M(0) =0$, and $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$. Does this mean that $M$ is a linear map? If not, could someone please give ...
1
vote
1answer
59 views

Nice example where $D^{\alpha}\Lambda_{f}\neq\Lambda_{D^{\alpha}f}$?

Let $\Omega\subset\mathbb{R}^n$ be open and $f$ be a locally integrable function. The distribution associated with $f$, $\Lambda_{f}\in D'(\Omega)$, is defined via \begin{equation} ...
3
votes
2answers
154 views

If $f\colon (0+\infty) \to \mathbb R$ and $f(x)=f(2x)$ what can we say about $f$?

Let $f \colon (0,+\infty) \to \mathbb R$ be a continuous function such that $$ f(x)=f(2x),\qquad \forall x \in \mathbb R. $$ What can we say about $f$? An easy induction shows that $$ ...
0
votes
1answer
52 views

How to resolve this?

I've the following problem to model and program it: suppose that we have a central server that provides 3 different services($S_1,S_2,S_3$), there are $N$ machines connected to this server: each ...
2
votes
4answers
271 views

Does there exists an abelian or nonabelian group G where $|Z(G)|=p^2$ for p a prime?

I have known cases of abelian and nonabelian groups that have a center of order p but never a power of p. Is there at least a known case where $|Z(G)|=p^2$?
1
vote
0answers
67 views

the 2-rank of field

Let the field $K=\mathbb{Q}(\sqrt{p_1}, \sqrt{p_2 q}, i)$ where $p_1, p_2 \equiv 1 \mod{4}$ and $q \equiv 3 \mod{4}$, kronecker(2,$p_1$)=1 and kronecker(2,$p_2$)=kronecker($p_1$,$p_2$) = ...
1
vote
0answers
20 views

Finding an element in a very specific set

I ran into the following problem during some self-motivated studies, and for the last 24 hours I have been unable to solve this problem. The problem arose by itself, meaning it doesn't have a source, ...
1
vote
0answers
106 views

Trading price of 2 consumers with the same utility function

Say that two consumers, A and B, have the same utility function, just $u(x) = (x_1)^2 + (x_2)^2$ for simplicity. If consumer A has endowment $x_A = (4, 3)$, and consumer B has endowment $x_B = (3, ...
1
vote
1answer
111 views

Linear Algebra Basics

I'm having my first linear algebra classes in college right now, and a few difficulties with the symbolism used. Missing some basics so to say. So I have a few small questions I will just ask here: ...
6
votes
3answers
135 views

Length of chord DE

In the given circle |AB| = 10 Units and AB || CD AB is the diameter of the circle. What is the length of the chord ED?
2
votes
3answers
94 views

Simple analytic geometry question I need help with

Give the equation of a circle with the center $ (a,0) $ which is tangent to the line $ y = x $ I now have $ (x-a)^2 + y^2 = r^2 $ but I don't know how to continue.. please help!
0
votes
2answers
114 views

What is this statement actually asking?

Let T: $\mathbb{R}^3\rightarrow \mathbb{R}$ be linear. Show that there exist scalars a, b, and c such that $T(x,y,z)= ax + by + cz$ for all $(x,y,z) \in \mathbb{R}^3$ Can I just say "you can pick ...
3
votes
1answer
2k views

General Proof for the triangle inequality

I am trying to prove: $P(n): |x_1| + \cdots + |x_n| \leq |x_1 + \cdots +x_n|$ for all natural numbers $n$. The $x_i$ are real numbers. Base: Let $n =1$: we have $|x_1| \leq |x_1|$ which is clearly ...
0
votes
2answers
204 views

Providing a counter example for a Logic Statement

How do I give a counter-example of the following logic statement (I think the statement is false): There exists $x$ $\geq$ 0 s.t. (For All real $y$, $x$ = $y$$^2$) Since the statement has a "There ...
1
vote
2answers
1k views

Elevator probability problem

An elevator in a building starts with five passengers and stops at seven floors. If each passenger is equally likely to get off on any floor and all the passengers leave independently of each other, ...
1
vote
1answer
68 views

Is $\overline F \setminus A$ perfect where $A$ is the set of isolated points of $F$?

Let $X$ be a metric space and $F\subset X$. I have proved that $\overline F \setminus A$ is closed. I'm having trouble with showing that $\forall x\in \overline F \setminus A$, $x$ is a limit point. ...
2
votes
1answer
115 views

Equivalence classes on Z

So I'm given that R is an equivalence relation on Z, and that adding classes in the natural way is well defined. That is, class(a)+class(b) = class(a + b). I want to show that R can only be equality ...

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