## Show $f$ is infinitely differentiable February 7

Assume that $E$ is a compact Lebesgue measurable subset of $\mathbb{R}$ and let, $$f(t) = \int_E \cos(tx) \, dx, \hspace{1mm} t \in \mathbb{R}$$Show that $f$ is infinitely differentiable.I have a solution using the mean value theorem, induction, and

## Find the center of circle given two tangent lines (the lines are parallel) and a point. February 7    3

How to find the center of a circle if the circle is passing through $(-1,6)$ and tangent to the lines $x-2y+8=0$ and $2x+y+6=0$?Hint: The lines are perpendicular, and the center of the circl ...

## Differentiating a matrix product February 7

In one of the books I found that given that $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, $\frac{dV(x(t))}{dt}=\frac{dx^T}{dt}Qx+x^T Q\frac{dx}{dt}=x^T A^T Qx+x^T QAx$.I don't see how the right-most side was obt

## Zeros of this function February 7

Let $$f(z)=\gamma + z^{\beta_2-\beta_1}$$ where $\gamma\in \mathbb{R}$, $\beta_1\in \mathbb{Z}$, $\beta_2 \in \mathbb{Z}$ and $\beta_2 > \beta_1$. The variable $z$ takes complex values.Is there a way of expressing the zeros of $f(z)$ in terms of the

## Given 5 points on a sphere, divide the surface into 5 congruent connected regions containing one point February 7

There are $5$ points on the surface of a sphere. Is it always possible to divide the surface into $5$ connected congruent regions such that each region contains one of the $5$ points?

## Proof of independence of \bar{X} and S^2: Trouble understanding n-dimensional Jacobian Result February 7

I'm trying to work through the proof given here that the sample mean and sample variance of a random sample $X_1, X_2, ..., X_n \sim N(\mu,\sigma^2)$ are independent.The part I can't seem to follow is the assertion that the Jacobian of the transforma

I am trying to prove the following:Monomial Algebras (First Edition) Rafael H. Villarreal - Exercise 1.1.45 or Monomial Algebras (Second Edition) Rafael H. Villarreal - Exercise 6.1.26Let $\text{R }\!\!'\!\!\text{ }=\text{k}\left[ {{\text{x}}_{2}},\t ## Finding an orthogonal vector February 7 3 How will I be able to do this question?Find a vector C$\in\mathbb{C}^3$which is orthogonal to both A and B. Where$A = [2,0,i]^T$and$B = [i,-1,2]^T$. The answer is$C =c[i/2, 5/2, 1]^T$, but how did they get that? And does$\mathbb{C}^3$mean a ## find all vectors orthogonal to both: February 7 2 Find all vectors v = (x, y, z) orthogonal to both$u_1$= (2, -1, 3)$u_2$= (0, 0, 0) I'm not sure how to get to the answer of s(1, 2, 0) + t(0, 3, 1). I know how to find a vector orthogonal to just 1, getting confused with the both part.What vectors ## Monomial Algebras problem : lcm February 7 I am trying to prove the following:Monomial Algebras (Second Edition) Rafael H. Villarreal - Exercise 6.1.22Let I and J be two ideals generated by ﬁnite sets of monomials F and G, respectively, prove that the intersection I ∩ J is generated by the se ## Describe all vectors v =x y that are orthogonal to u = a b February 7 Describe all vectors v = [x y] that are orthogonal to u = [a b].I know that vectors that are orthogonal will have a dot product of 0. So here's what I was thinking:ax + by = 0 yb = -ax y = -ax/b I then looked up the answer to check if I was right and ## How to solve$\left\frac{x+4}{ax+2}\right \frac1x$February 7 2 How to solve: $$\left|\frac{x+4}{ax+2}\right| > \frac{1}{x}$$What I have done:I)$x < 0$:Obviously this part of the inequation is$x\in(-\infty, 0), x\neq \frac{-2}{a}$II)$x > 0$:$$\left|\frac{x+4}{ax+2}\right| > \frac1x$$ $$\frac{|x+4|}{|ax+ ## Relation on rational numbers that defines a total order February 7 1 Define the relation on \mathbb{Q} by$$[m,n]<[j,k]$$if and only if jn-mk belongs to \mathbb{N}, j and m belong to \mathbb{Z}, n and k belong to \mathbb{N}.(a) Show that < is well defined, that is if (m,n)\sim (m',n') and (j,k ## How do I show that the equivalence relation defining the rational numbers is transitive February 7 1 I apologize if this is a super easy question, but there is something fishy about my proof.I was to show:$$(p,q) \sim (m,n) \wedge (m,n) \sim (a,b) \implies (p,q) \sim (a,b) $$under the equivalences relation of rational numbers (i.e. (a,b) \sim (c,d) ## Having trouble proving transitivity February 7 1 We have a universal set of lowercase alphabet letters, U = \{a, b, ... , z\} . For sets A,B \subseteq U we can define a relation, A \sim B as long as the number of elements that are in either A or B, but not both, is even. Prove that \sim You Might Also Like • Questions:Show that is is theoretically possible to find a rational number that approximates \sqrt{3} with ... • This question is based on an exercise in Artin's Algebra:Which ideals in the polynomial ring R:={\Bbb C}[x, ... • I am trying the following exercise (I cannot use Cauchy's theorem, it's not in my course yet) : Let G be a ... • Assume \{k_n\}_{n\geq 0} a sequence of natural numbers such that k_0=0,k_n\leq k_{n+1}\leq k_n+1, and ... • How to get the solution set of the inequality$$\left ( \frac{\pi}{2} \right )^{(x-1)^2}\leq \left ( \frac{2 ... • I have a question about Novikov's condition.Let$L$be a local martingale such that either$\exp \left(\frac ...
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