0
votes
0answers
2 views

Need to create a specific four variable Boolean function?

I need to create a Boolean function consisting of four inputs (a,b,c,d) that is 1 if no more than 2 of its inputs are 1. I have experienced great difficulty on this particular problem. If anyone ...
0
votes
0answers
4 views

Show that the family $E_\mu$ undergoes tangential bifurcation

Let $E_\mu(x)=\mu e^x$. Show that the family $E_\mu$ undergoes tangential bifurcation at $\mu=1/e$. In particular follow out the following steps: (a) Plot out the diagonal and the graph of $E_\mu ...
0
votes
0answers
4 views

a realtion about $R$- homomorphism

$R$ is abelian ring ,$I$ is ideal in $R$ , $M$ is $R$-module and includes finite generation. Let $\phi :M \rightarrow M$ : $R$- homomorphisms, and $\phi(M) \subset IM$. how to prove : $\exists a_1 ...
0
votes
0answers
5 views

proof of Laurents extension theorem

I found this theorem of Laurents expansion theorem online, but there is one thing I do not understand. How can he just say that $\gamma$ is a union of two cicles? Does he make some kind of ...
0
votes
0answers
4 views

Confused about addition in $p$-adic integers

I just learned $p$-adic integers and I am confused about something. I was wondering if someone could possibly explain me how it is done. Suppose I have $\bar{a} = 1 + 0 \cdot p + 0 \cdot p^2 + 0 ...
0
votes
0answers
5 views

$\mathbb{Q}(\sqrt{m}, \sqrt{n})$ : ring of integers, integral basis and discriminant

In the following document, http://people.math.carleton.ca/~williams/papers/pdf/033.pdf, I found three results about biquadratic fields and their ring of integers. It's the proof of the first theorem ...
0
votes
0answers
6 views

What is the variance of this random variable: number of items

Let us assume that we have a capacity $n$ which tends to infinity. We have an infinite number of random variables $X_1, X_2, \dotsc$, where each $X_i$ is independent and identically distributed with ...
0
votes
0answers
7 views

(complex variables) Expand $\frac{2z+3}{1+z}$ in a power series of $z-1$ and comment on its convergence

Question: Expand $\frac{2z+3}{1+z}$ in a power series of $z-1$. What can we say about its convergence? Attempt: First, notice $ \frac{2z+3}{1+z} = \frac{2z+3}{1} \frac{1}{1+z}$. Let $w = 1 -z$. Using ...
-1
votes
0answers
9 views

Irreducibility in Rings

$f(x,y)=x^{2}-y^{2}+1$. Is it irreducible in $C$? In $Z_{7}$? Please include the reason and the result. I have read Eisenstein Criterion and many other theorems. None of them directly addresses the ...
0
votes
0answers
6 views

covariance matrix in bivariate distribution

I hope stats questions are also allowed in this subsite, since they are related (or more) to math. I struggle to understand how exactly you get the covariance matrix in a bivariate normal ...
0
votes
4answers
15 views

Set builder of this set 0, 1, 3, 6, 10, 15

I have tried to create the set builder of this infinite set: 0, 1, 3, 6, 10, 15, 21, 28,... I have notice that n = (n - 1) + (N + 1) where ...
0
votes
0answers
20 views

Why can't solutions to Autonomous ODE intersect?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n $ be continuously differentiable. Why can't solutions to $x'=f(x) $ intersect? I use a proof by contradiction: Assume solutions can, and do intersect ...
1
vote
0answers
7 views

Epsilon-N method - Proof verification

Prove (using the epsilon-N method) that the sequence of numbers $\dfrac{5n^3-2}{n^3}$ converges. Calculate the limit first. First we calculate the limit: $\lim_{n \to \infty} \dfrac{5n^3-2}{n^3} ...
0
votes
2answers
17 views

What's the probability of rolling the same number twice with a pair of dice?

There are a number of questions similar to this, but I'm asking about rolling two dice twice, not one die two times. Or I guess you could think of it as 4 dice total, with each pair distinguishable. ...
0
votes
0answers
7 views

“locally finiteness” in coproducts

just to be sure: in any category of modules (as opposed to e.g. complete modules) any element $x$ in the (infinite) coproduct $X=\oplus_{\lambda\in\Lambda} X_\lambda$ can be written as a finite sum of ...
0
votes
0answers
5 views

Non linear Programming Problem

I am struggling with the following question: Solve the following programing problem: max $f(x_1,x_2)$= $ \sqrt{(x_1 + 1) (x_2+1)} $ subject to $x_2-(x_1-1)^2 \leq 0 $; $x_1+x_2 \leq 7 $; $x_1, x_2 ...
1
vote
1answer
10 views

Formal way of counting vertices

For a simple graph $G = (V,E)$ we know that there are $10$ edges, $|E|=10$. And we know that two vertices have a degree of $4$, and the rest have a degree of $3$. I know how to solve this, I'm just ...
0
votes
0answers
11 views

How many ways can 4 hands of 5 cards be selected from a deck of 52 cards?

1) How many ways can 4 hands of 5 cards be selected from a deck of 52 cards? 2) How many ways can 4 hands of 5 cards be dealt to 4 people?
0
votes
0answers
4 views

Closed Operator on a Sobolev spaces

I am wondering if the following differential operator $A:D(A)( \subset {\bf{H}}) \to {\bf{H}}$ defined on the sobolev space $\mathbf{H}=H_{0}^{k}(0,1)\times {{L}^{2}}(0,1)\text{ }$ is a closed ...
-1
votes
0answers
23 views

show that n is prime if and only if $S(n)0$

show that n is prime if and only if $S(n)0$ $S(n)={[\pi.\dfrac{(n-1)!+1}{n}]}$
2
votes
1answer
21 views

question about dual space and linear form

I stuck at the following linear algebra problem. Could you give me some hints? Let $V$ be a vector space. Given $g,\,f_1,\,f_2,\,...,f_r\in V^*$, prove that $g\in ...
1
vote
1answer
38 views

Divisibility !?? wihout mathematical induction if possible PLZ …

Hello I need to prove if $n\, |\, (x-a)$ and $n \, | \, f(x)$ then $n \, | \, f(a)$. It is true if $\operatorname{deg}(f)=1$ or $2$, but what for greater degree of $f(x)$? I don't know how to ...
3
votes
2answers
35 views

Prove that there are infinitely many integer solutions to a diophantine equation

Prove that there are infinitely many integer solutions to the diophantine equation: $(x-y)^7 = x^3y^3$
1
vote
1answer
57 views

intregration without substitution of $x^x \ln x$

How do i integrate this without any substitution, purely algebraically : $$x^x \ln ex$$ I've tried a lot but not have been able to: $$x^x \ln (x + 1) = \ln x^{x^x} + x^x$$ or $e^{x \ln x}\ln ...
3
votes
1answer
22 views

Biggest set such that sum of any pair is perfect square

What is the biggest set of positive integers such that the sum of any pair of them is a perfect square? (Or can we construct an infinite such set?) One such set of size $3$ is $\{6,19,30\}$, which ...
6
votes
0answers
58 views

Olympiad problem: Erdos-Selfridge

The following problem is a special case of Erdos-Selfridge theorem: http://projecteuclid.org/euclid.ijm/1256050816 Problem: Prove that for any positive integer $n$, the product $(n+1)(n+2)...(n+10)$ ...
0
votes
0answers
20 views

Induction proofs where the induction hypothesis is applied to “intermediate” values

An induction proof typically consists of one or more base cases in which some statement $B(n)$ is proven for small values of $n$, and an induction step, in which $B(n)$ is proven to be true under the ...
1
vote
1answer
15 views

Find volume of parallelepiped where all faces are rhombuses with edge $a$ and angle $60^\circ$.

Find volume of parallelepiped where all faces are rhombuses with edge $a$ and angle $60^\circ$. I first noticed that $\dfrac{h}a=\sin60^\circ$, so I easly found that height of the rhombus is ...
1
vote
2answers
56 views

Does a Group being Finite Imply that It Is Cyclic?

I have been studying Abstract Algebra, and all the finite groups that we have studied so far have also been cyclic. So, is it true that all finite groups are cyclic? If yes, what is the theorem? If ...
0
votes
1answer
22 views

Solving recurrence relation in form of $f(n)=f(n-1)+K-n$

I asked a question (now deleted it) on finding a relation between $$4,10,15,\cdots$$ I studied a a little about recurrence relation and solving them. for the above sequence I observed that $$f(n) = ...
1
vote
0answers
13 views

Need help checking my recurrence for a simple algorithm

All I'm writing to get a second opinion on the algorithm shown in this link. I'm pretty sure its supposed to be $T(n)=2T(n/2)+n$ but I can't see where I'm supposed to get the +n from. So far I'm ...
-1
votes
1answer
19 views

Question about isomorphism [on hold]

If we have to prove that U(8) is isomorphic to a set of matrices, are we allowed to define saying U(1)-> (..) and U(2)->(...) or does there have to be a set function with variables that works for each ...
1
vote
1answer
15 views

Measure on intersections of unions

Let $(X,\mathcal{A},μ)$ a measurable space and let $A_1,A_2,...∈\mathcal{A}$, assume that $\sum\limits_{j=1}^{\infty}=\mu (A_j)<\infty$ We have ...
1
vote
2answers
52 views

How many ways to seat 4 couple and 2 single around a round table

How many ways to seat 4 couple and 2 single around a round table, provided that each couple will sit together
2
votes
1answer
35 views

Is $\Bbb{R}[X,Y]/(X^2+Y^2)$ a UFD or Noetherian?

Hello everyone I would like to know if $R$:= $\Bbb{R}[X,Y]/(X^2+Y^2)$ is UFD or Noetherian. I'm not really confortable in seeing how $\Bbb{R}[X,Y]/(X^2+Y^2)$ looks like. From what i've ...
1
vote
1answer
16 views

Angles in a circle

I have troubles to prove the following: Let $\Gamma$ be a circle with center $O$, $a$ be a tangent to $\Gamma$, $A=a\cap \Gamma$, $D$ a point on $a$ and $B\in \Gamma$ such that $D$ and $B$ lies on ...
0
votes
1answer
19 views

Solving functional equation $f\left(\sum_{i=1}^n a_i^n\right)=\frac{1}{k} \sum_{i=1}^n f(a_i^n).$

Given natural number $n, k$ consider nondecreasing function $f:\mathbb{N}\cup {0} \to \mathbb{N}\cup {0}$ such that $$ f\left(\sum_{i=1}^n a_i^n\right)=\frac{1}{k} \sum_{i=1}^n f(a_i^n), $$ for ...
1
vote
0answers
8 views

Designing linear systems to respond to particular kinds of oscillations

Say that I have a linear system which is being perturbed by an oscillating signal of a single frequency, of the form $$ \dot{\vec{x}}(t) = A\vec{x} + B \sin(\alpha t), $$ where $B$ can only have a ...
0
votes
0answers
18 views

Correct proof relating to diagonalizability

Regarding the question Prove that if $B$ is diagonalzable then $A=0$ when $$B=\left(\begin{array}{cc} A & A\\ 0 & A \end{array}\right)$$ I have come up with the following proof: First ...
-1
votes
1answer
43 views

Weird limit problem

I cam across this unusual problem in a textbook and have no clue how to solve it, so any help would be appreciated. Find the value of the limit $$\lim_{m,n\to\infty}\cos^{2m}(n! \cdot \pi \cdot ...
4
votes
2answers
35 views

Expressing $1 + \cos(x) + \cos(2x) +… + \cos(nx)$ as a sum of two terms

Question in title, my progress: let $z = \cos(x) + i\sin(x)$ then $1 + \cos(x) + \cos(2x) +\dots + \cos(nx) = Re(1 + z + z^2 +\dots + z^n) = Re\left (\dfrac{1-z^{n+1}}{1-z} \right)$ by geometric ...
0
votes
0answers
6 views

Proof of distributive property of linear operator?

How can I show that for 2 linear operators $L$ and $M$ that transform some object $O$ into another object of the same type: $$(L(O)+M(O))*(L(O)+M(O)) = L(O)*L(O) + 2L(O)*M(O) + M(O)*M(O)$$ where $*$ ...
1
vote
0answers
11 views

Reflection in terms of simple reflections

Suppose $\beta=\sum_{i=1}^ka_i\alpha_i$, where $\alpha_i$ are simple roots. Is there any easier way to write the reflection corresponding to $\beta$ say $s_{\beta}$ in terms of $s_{\alpha_i}$'s. I ...
3
votes
3answers
27 views

Linear transformation onto and one to one?

(1)If a linear transformation $T:\mathbb{R}^n\rightarrow\mathbb{R^m}$ maps $\mathbb{R}^n$ onto $\mathbb{R^m}$ what is the relation between m and n? (2)If T is one to one what is the relationship ...
0
votes
0answers
10 views

Verification of convolution between gaussian and uniform distributions

Let $n \sim \mathcal{N}(\mu, \sigma^2)$ and let $u \sim \mathcal{U}(a,b)$, with $b>a>0$, and suppose that $n$ and $u$ are independent random variables. Let $g = n + u$. The probability density ...
0
votes
1answer
18 views

Powers of random variables always well-defined?

Given the random variable $X$, is $X^{0}$ a random variable? Can we take the expectation $E(X^{0})$? Is $X^{0}$ or its expectation defined or undefined under any conditions (say, on the sample space ...
0
votes
0answers
11 views

Find the two points that satisfy this equation where the tangent plane is normal to a given vector.

Find the two points on the ellipsoid $$\frac{x^2}{4}+\frac{y^2}{9}+z^2=1$$ where the tangent plane normal to $\vec{v}$ $= \langle1,1,-2\rangle$. I've tried everything I can think of, but I think I'm ...
0
votes
1answer
5 views

How many base-out configurations would be possible in sleazeball?

Problem: In a baseball there are 24 different "base out" configurations (runner on first - two outs, bases loaded- none out, and so on). Suppose that a new game, sleazeball, is played where there are ...
0
votes
0answers
18 views

Is this relation reflexive, symmetric, antisymmetric, transitive, or whether it is equivalence relation

Is this relation reflexive? symmetric? transitive? Is it an equivalence relation? Explain. I know that for it to be an equivalance relation, it has to be reflexive, symmetric and transitive.
0
votes
0answers
7 views

Solve set of poorly conditioned linear equations in block matrix form

I would like to solve the following set of linear equations where A, B, C and D are each 4x4 matrices. K is then an 8x8 matrix The values in A and D have magnitudes of $\approx 10^{17}$, B has ...

15 30 50 per page