$x$ is within 3 units of $c$ February 14

I want to use absolute value notation to represent the phrase "$x$ is within 3 units of $c$". Which of the following is the correct answer? (i) or (ii)? (i) $|x-c|<3$, (ii) $|x-c|\leq{3}$

expected and renewal process February 14

Let $N_t$ a renewal process with $T_i$ the jumps. I know $U(t)=E(N_t)$ on [0,t]. Let $B_t=T_{N_t}-t$ and $G_t(u)=P(B_t \leq u)$How to show on ]t,t+a]: $E(N_{t+a}-N_t)=\int_0^a U(a-u)G_t(du)=(G_t \star U)(a)=(U \star G_t)(a)$Thank you

For all $x$ close to $c$ February 14

What is the precise mathematical meaning of the phrase "for all $x$ close to $c$"? Does it mean that for all $x$ that satisfy the inequality $|x-c|<\epsilon$, where $\epsilon$ is a small positive number? This phrase is used very often.

Solving inequation where one of the terms is a log February 14    1

I trying to find the value for which $n^2 -n +1$ is less than $ 6n\log_{2}{n} +2n $ where n is a power of $2$.Trying it iteratively using a CAS you find that $n = 64$.How can $n^2 -n +1 < 6n\log_{2}{n} +2n $ where n is a power of $2$ be solved analyt

Proving $f(x)\leq x$ with some conditions February 14    2

Let $f:[0,1]\to [0,1]$ be a function such that$f(1)=1$ $f(x)+f(y)\leq f(x+y)$, for any numbers $x$ , $y$ , $x+y \in [0,1]$ Then we have to show that $f(x)\leq x$ for any $x\in [0,1]$.I can see that $f(0)=0$, and am able to get a few more properties b

How to find arbitrage for forward rate of exchange February 14

Assume that $S(0)$ is the current rate of exchange for foreign currency. Assume that and $K_h$ and $K_f$ are rates of return on home and foreign currency if it is invested over a period $T$. (A) Assume that the forward rate of exchange $F$ satisfies

For the ring $\mathbb{Z}\sqrt{d}$ February 14

For the ring $\mathbb{Z}[\sqrt{d}]$, we define the norm function $f:\mathbb{Z}[\sqrt {d}] \rightarrow \mathbb{N}\cup \{0\}$ by: $ f(a+b\sqrt d)=|a^2-db^2|$. How can I prove that $f(x)=0$ if and only if $x=0$?What I've got: Suppose $x=a+b\sqrt d$ and

Are these functions Lebesgue Integrable How to show this February 14

Are these functions Lebesgue Integrable How to show this
I have recently learnt the comparison test, MCT, Fatou's Lemma and DCT for Lebesgue integrals, but have been struggling with the details of the proofs.1) f is 0 a.e. so is integrable to 02) ...

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