5. How many years does the first 10 yards represent? What events are within the first 10 yards of the football field from the TODAY end zone? Explain what this means in terms of expansion of life on Earth.

Izabelle Lowery
2022-10-15
Answered

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silenthunter440

Answered 2022-10-16
Author has **19** answers

Based on the length of a football field and the timeline of earth's history, the first 10 yards represents 460 million years.

Beginning from 460 million years ago, there was massive explosion and expansion of life on earth.

What is a timeline of earth's history?A timeline of earth's history shows the events that occurred from the beginning when earth was formed 4.6 billion years ago until today.

In this activity, a football field is used to represent the timeline of earth's history.

Each end of the football field represents the two end zones, Earth's beginning and TODAY end zone.

A football field is 100 yards long.

The earth is 4,600,000,000 years old.

Each yard equals 46,000,000 years

Thus, the first 10 yards equals = 46,000,000 × 10 The The first 10 yards = 460,000,000 years.

From the TODAY end zone, the the first 10 yards = 460,000,000 years.

460,000,000 years ago marked the time from the appearance of the first plants until humans and all life on earth as it is today.

This, meant that within this period, there was massive explosion and expansion of life on earth.

Beginning from 460 million years ago, there was massive explosion and expansion of life on earth.

What is a timeline of earth's history?A timeline of earth's history shows the events that occurred from the beginning when earth was formed 4.6 billion years ago until today.

In this activity, a football field is used to represent the timeline of earth's history.

Each end of the football field represents the two end zones, Earth's beginning and TODAY end zone.

A football field is 100 yards long.

The earth is 4,600,000,000 years old.

Each yard equals 46,000,000 years

Thus, the first 10 yards equals = 46,000,000 × 10 The The first 10 yards = 460,000,000 years.

From the TODAY end zone, the the first 10 yards = 460,000,000 years.

460,000,000 years ago marked the time from the appearance of the first plants until humans and all life on earth as it is today.

This, meant that within this period, there was massive explosion and expansion of life on earth.

asked 2022-07-14

I read in an old book the following example of a measure: For the set $M=\mathbb{Q}\cap [0,1]$ denote with $S$ the set system of subsets of $M$ of the form $\mathbb{Q}\cap I$, where $I$ is any interval in [0,1]. Let us define the function $\mu :S\to \mathbb{R}$ as follows: for any set $A\in S$ of the form $A=\mathbb{Q}\cap I$ we set $\mu (A)=\ell (I)=b-a.$.

Then it said without proof that μ is finitely additive, but not σ-additive.

As I did not get why I tried to prove it by myself and I tried to show that $S$ is a semi-ring, I guess that is important before I start with the other proof.

We have $\mathrm{\varnothing}\in \mathbb{Q}$ and furthermore $(\mathbb{Q}\cap {I}_{1})\cap (\mathbb{Q}\cap {I}_{2})=\mathbb{Q}\cap {I}_{1}\cap {I}_{2}$ and the union of two closed intervals is either an interval or the disjoint union of two intervals. Then $(\mathbb{Q}\cap {I}_{1})\setminus (\mathbb{Q}\cap {I}_{2})$ is also an interval or the disjoint union of two intervals.

Now the proof. I do not quite understand how it cannot be $\sigma $-additive. Does it have something in common with Cantor sets? I don´t know how to start the proof here. Any help or explanation (maybe an idea for the beginning of a proof) is appreciated. If there is a proof...

Then it said without proof that μ is finitely additive, but not σ-additive.

As I did not get why I tried to prove it by myself and I tried to show that $S$ is a semi-ring, I guess that is important before I start with the other proof.

We have $\mathrm{\varnothing}\in \mathbb{Q}$ and furthermore $(\mathbb{Q}\cap {I}_{1})\cap (\mathbb{Q}\cap {I}_{2})=\mathbb{Q}\cap {I}_{1}\cap {I}_{2}$ and the union of two closed intervals is either an interval or the disjoint union of two intervals. Then $(\mathbb{Q}\cap {I}_{1})\setminus (\mathbb{Q}\cap {I}_{2})$ is also an interval or the disjoint union of two intervals.

Now the proof. I do not quite understand how it cannot be $\sigma $-additive. Does it have something in common with Cantor sets? I don´t know how to start the proof here. Any help or explanation (maybe an idea for the beginning of a proof) is appreciated. If there is a proof...

asked 2022-06-11

With the help of a suitable transformation and Fubini I want to determine the integral

${\int}_{V}{x}^{3}yd{\lambda}_{2}(x,y),$

where $V$ is the open subset of ${\mathbb{R}}_{+}^{2}$ bounded by the following curves:

$\begin{array}{rl}& {x}^{2}+{y}^{2}=4\\ & {x}^{2}-{y}^{2}=2\\ & {x}^{2}-{y}^{2}=1\end{array}$

I know how to do that. The only problem is finding $V.$ Is it

$V=\{(x,y)\in {\mathbb{R}}^{2}:1<{x}^{2}-{y}^{2}<2,0<{x}^{2}+{y}^{2}<4\}$

Because then I set

$\begin{array}{rl}& {x}^{2}+{y}^{2}=v\\ & {x}^{2}-{y}^{2}=u\end{array}$

and get

$(x,y)=(\sqrt{1/2(v-u}),\sqrt{1/2(u+v)})$

So either my transformation is wrong or the limits of the intervalls I chose.

Thanks for any kind of help.

${\int}_{V}{x}^{3}yd{\lambda}_{2}(x,y),$

where $V$ is the open subset of ${\mathbb{R}}_{+}^{2}$ bounded by the following curves:

$\begin{array}{rl}& {x}^{2}+{y}^{2}=4\\ & {x}^{2}-{y}^{2}=2\\ & {x}^{2}-{y}^{2}=1\end{array}$

I know how to do that. The only problem is finding $V.$ Is it

$V=\{(x,y)\in {\mathbb{R}}^{2}:1<{x}^{2}-{y}^{2}<2,0<{x}^{2}+{y}^{2}<4\}$

Because then I set

$\begin{array}{rl}& {x}^{2}+{y}^{2}=v\\ & {x}^{2}-{y}^{2}=u\end{array}$

and get

$(x,y)=(\sqrt{1/2(v-u}),\sqrt{1/2(u+v)})$

So either my transformation is wrong or the limits of the intervalls I chose.

Thanks for any kind of help.

asked 2022-05-31

In the beginning of the section the author of book i read states

If $\{{f}_{n}\}$ is a sequence of complex-valued functions on a set $X$, the statement "${f}_{n}\to f$ as $n\to \mathrm{\infty}$" can be taken in many different sense, for example pointwise or uniform convergence. If $X$ is a measure space, one can also speak of a.e. convergence or convergence in ${L}^{1}$.

Before moving further with the text I wanted to clarify what he means by ${f}_{n}\to f$ a.e. I am familiar with the notion of almost-everywhere, but what kind of convergence is he referring to exactly? Is this pointwise convergence almost everywhere, uniform convergence almost everywhere? Something else entirely? It is possible he spoke about this earlier in the text but I could not find it.

This passage is before he introduces convergence in measure, so it cannot be that. In fact, he has a corollary that if ${f}_{n}\to f$ in ${L}^{1}$, there is a subsequence $\{{f}_{{n}_{j}}\}$ such that ${f}_{{n}_{j}}\to f$ a.e., and follows this with a remark saying

If ${f}_{n}\to f$ a.e. it does not follow that ${f}_{n}\to f$ in measure.

If $\{{f}_{n}\}$ is a sequence of complex-valued functions on a set $X$, the statement "${f}_{n}\to f$ as $n\to \mathrm{\infty}$" can be taken in many different sense, for example pointwise or uniform convergence. If $X$ is a measure space, one can also speak of a.e. convergence or convergence in ${L}^{1}$.

Before moving further with the text I wanted to clarify what he means by ${f}_{n}\to f$ a.e. I am familiar with the notion of almost-everywhere, but what kind of convergence is he referring to exactly? Is this pointwise convergence almost everywhere, uniform convergence almost everywhere? Something else entirely? It is possible he spoke about this earlier in the text but I could not find it.

This passage is before he introduces convergence in measure, so it cannot be that. In fact, he has a corollary that if ${f}_{n}\to f$ in ${L}^{1}$, there is a subsequence $\{{f}_{{n}_{j}}\}$ such that ${f}_{{n}_{j}}\to f$ a.e., and follows this with a remark saying

If ${f}_{n}\to f$ a.e. it does not follow that ${f}_{n}\to f$ in measure.

asked 2022-07-09

Let $X$ be a set, $\mathcal{A}$ a ring of subsets of $X$, $\mu :\mathcal{A}\to {\overline{\mathbb{R}}}_{\ge 0}$ a premeasure and ${\mu}^{\ast}$ the outer measure generated by $\mu $. (By Caratheodory)

If $E\in {\mathcal{M}}_{{\mu}^{\ast}}$ and satisfacies ${\mu}^{\ast}(E)<\mathrm{\infty}$, then for each $\epsilon $ existx $A\in \mathcal{A}$ such that ${\mu}^{\ast}(A\mathrm{\u25b3}E)<\epsilon $

I try to test this, my first idea was to give a cover of $E$, I just don't know if I can find said cover in $\mathcal{A}$, as the measure is not finite, so the extension is not unique, someone can give me a hint how to proceed?

If $E\in {\mathcal{M}}_{{\mu}^{\ast}}$ and satisfacies ${\mu}^{\ast}(E)<\mathrm{\infty}$, then for each $\epsilon $ existx $A\in \mathcal{A}$ such that ${\mu}^{\ast}(A\mathrm{\u25b3}E)<\epsilon $

I try to test this, my first idea was to give a cover of $E$, I just don't know if I can find said cover in $\mathcal{A}$, as the measure is not finite, so the extension is not unique, someone can give me a hint how to proceed?

asked 2022-05-21

I have an electronic weighing-machine, which I believe to be internally very accurate. It will weigh up to 100 kg, but not activate below 10 kg. The digital display reports to one decimal place. The problem is that I don't know whether the reading is rounded (with worst error $\pm 50$50 g) or truncated (with worst error −100 g and expected bias −50 g). I have a great quantity of books and papers that can be stacked on the machine to make any weight within its limits, but nothing of accurately known weight.

I guess that any solution must be statistical. A good solution would minimize the number of weighings, given a tolerance probability of a false indication. (Assume a 50/50 prior distribution for rounding/truncation; for illustration, a targeted probability could be 0.1%.)

I guess that any solution must be statistical. A good solution would minimize the number of weighings, given a tolerance probability of a false indication. (Assume a 50/50 prior distribution for rounding/truncation; for illustration, a targeted probability could be 0.1%.)

asked 2021-02-04

A certain scale has an uncertainty of 3 g and a bias of 2 g.

a) A single measurement is made on this scale. What are the bias and uncertainty in this measurement?

b) Four independent measurements are made on this scale. What are the bias and uncertainty in the average of these measurements? c) Four hundred independent measurements are made on this scale. What are the bias and uncertainty in the average of these measurements?

d) As more measurements are made, does the uncertainty get smaller, get larger, or stay the same?

e) As more measurements are made, does the bias get smaller, get larger, or stay the same?

a) A single measurement is made on this scale. What are the bias and uncertainty in this measurement?

b) Four independent measurements are made on this scale. What are the bias and uncertainty in the average of these measurements? c) Four hundred independent measurements are made on this scale. What are the bias and uncertainty in the average of these measurements?

d) As more measurements are made, does the uncertainty get smaller, get larger, or stay the same?

e) As more measurements are made, does the bias get smaller, get larger, or stay the same?

asked 2022-06-02

I have a random variable $\xi :\mathrm{\Omega}\to \mathbb{R}$ which distribution function has a density, so by definition I have that the probability measure of each $(-\mathrm{\infty},x]$ can be calculated by:

${F}_{\xi}(x)={P}_{\xi}(-\mathrm{\infty},x]={\int}_{-\mathrm{\infty}}^{x}{f}_{\xi}(y)\mathrm{d}y\phantom{\rule{1em}{0ex}}(1)$

where the integral above is in the Lebesgue sense, with respect to the Lebesgue measure in $\mathbb{R}$.

My book says that a wider formula holds, that is:

${P}_{\xi}(B)={\int}_{B}{f}_{\xi}\mathrm{d}x,\phantom{\rule{1em}{0ex}}\mathrm{\forall}B\in \mathcal{B}(\mathbb{R})$

How can I use (1) in order to obtain this last formula? In other words, how can I extend (1) to every Borel set?

${F}_{\xi}(x)={P}_{\xi}(-\mathrm{\infty},x]={\int}_{-\mathrm{\infty}}^{x}{f}_{\xi}(y)\mathrm{d}y\phantom{\rule{1em}{0ex}}(1)$

where the integral above is in the Lebesgue sense, with respect to the Lebesgue measure in $\mathbb{R}$.

My book says that a wider formula holds, that is:

${P}_{\xi}(B)={\int}_{B}{f}_{\xi}\mathrm{d}x,\phantom{\rule{1em}{0ex}}\mathrm{\forall}B\in \mathcal{B}(\mathbb{R})$

How can I use (1) in order to obtain this last formula? In other words, how can I extend (1) to every Borel set?