necessaryh
2020-11-12
Answered

Find the least common multiple of ${x}^{3}-{x}^{2}+x-1\text{}and\text{}{x}^{2}-1$ . Write the answer in factored form.

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yagombyeR

Answered 2020-11-13
Author has **92** answers

Factor the given polynomials:

${x}^{3}-{x}^{2}+x-1={x}^{2}(x-1)+(x-1)$

$=x(-1)({x}^{2}+1)$

${x}^{2}-1=(x(-1)(x+1)$

The lowest common multiple is the product of all the factors of each polynomial to the highest degree. Each of the factors$x-1$ , and ${x}^{2}+1$ all have a degree of 1 so the lowest common multiple is
$(x-1)(x+1)({x}^{2}+1)$ .

The lowest common multiple is the product of all the factors of each polynomial to the highest degree. Each of the factors

asked 2021-01-04

Write in words how to read each of the following out loud.

a. $\{x\in {R}^{\prime}\mid 0<x<1\}$

b. $\{x\in R\mid x\le 0{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}x\Rightarrow 1\}$

c. $\{n\in Z\mid n\text{}is\text{}a\text{}factor\text{}of\text{}6\}$

d. $\{n\in Z\cdot \mid n\text{}is\text{}a\text{}factor\text{}of\text{}6\}$

asked 2021-08-11

A ball is tossed upward from the ground. Its height in feet above ground after t seconds is given by the function $h\left(t\right)=-16{t}^{2}+24t$ . Find the maximum height of the ball and the number of seconds it took for the ball to reach the maximum height.

asked 2022-07-06

There is a special non-commutative group related to the isometry $\mathrm{\u266f}:\mathbb{H}/\mathrm{\u266f}\to {\mathcal{P}}_{2}\left({\mathbb{R}}^{d}\right)$, namely the set $\mathcal{G}(\mathrm{\Omega})$ of Borel maps $S:\mathrm{\Omega}\to \mathrm{\Omega}$ (they lie in $\mathbb{H}$ ) that are almost everywhere invertible and have the same law as the identity map id.

Here

1/ $\mathrm{\Omega}$ is the ball of unit volume in ${\mathbb{R}}^{d}$, centered at the origin.

2. $\mathbb{H}:={L}^{2}(\mathrm{\Omega},\mathrm{d}x,{\mathbb{R}}^{d})$

My naïve guess is that "almost everywhere invertible" means the Lebesgue measure of $\{\omega \in \mathrm{\Omega}\mid \mathrm{card}({S}^{-1}(\omega ))\le 1\}$ is 1.

Could you elaborate on this notion?

Here

1/ $\mathrm{\Omega}$ is the ball of unit volume in ${\mathbb{R}}^{d}$, centered at the origin.

2. $\mathbb{H}:={L}^{2}(\mathrm{\Omega},\mathrm{d}x,{\mathbb{R}}^{d})$

My naïve guess is that "almost everywhere invertible" means the Lebesgue measure of $\{\omega \in \mathrm{\Omega}\mid \mathrm{card}({S}^{-1}(\omega ))\le 1\}$ is 1.

Could you elaborate on this notion?

asked 2022-05-15

Consider the following stochastic differential equation:

$d{Y}_{t}={Z}_{t}d{W}_{t}$

and terminal condition ${Y}_{T}=b,$ for which holds: $E[|b{|}^{2}]<\mathrm{\infty}$ Furthermore b is adapted to the filtration generated by the Browian motion only at terminal time $T$.

${Z}_{t}$ is a predictable square integrable process. So the right hand side ${Z}_{t}d{W}_{t}$ is martingale.

Why is the solution ${Y}_{t}$ adapted to the underlying filtration If I rewrite the equation, I get:

${Y}_{t}={Y}_{T}-{\int}_{t}^{T}{Z}_{s}d{W}_{s}=b-{\int}_{t}^{T}{Z}_{s}d{W}_{s}$

Since $b$ is only mb w.r.t to terminal time $T$, ${Y}_{t}$ cannot be adapted.

Where did I do a mistake?

$d{Y}_{t}={Z}_{t}d{W}_{t}$

and terminal condition ${Y}_{T}=b,$ for which holds: $E[|b{|}^{2}]<\mathrm{\infty}$ Furthermore b is adapted to the filtration generated by the Browian motion only at terminal time $T$.

${Z}_{t}$ is a predictable square integrable process. So the right hand side ${Z}_{t}d{W}_{t}$ is martingale.

Why is the solution ${Y}_{t}$ adapted to the underlying filtration If I rewrite the equation, I get:

${Y}_{t}={Y}_{T}-{\int}_{t}^{T}{Z}_{s}d{W}_{s}=b-{\int}_{t}^{T}{Z}_{s}d{W}_{s}$

Since $b$ is only mb w.r.t to terminal time $T$, ${Y}_{t}$ cannot be adapted.

Where did I do a mistake?

asked 2022-05-26

How do you solve ${x}^{3}>-1$ ?

asked 2022-03-27

Construct a sample (with at least two differentvalues in the set) of 66 measurements whose median is smaller than the smallest measurement in the sample. If this is not possible, indicate "Cannot create sample".

asked 2022-06-15

Let's say we conduct a random experiment. The possible outcomes may be too complicated to describe, so we instead take some measurement (in terms of real numbers) from it which are of interest to us. Mathematically we have defined a random variable X from the sample space/measurable space $(\mathrm{\Omega},\mathcal{A})$ to $(\mathbb{R},\mathcal{B})$ where $\mathcal{B}$ is the Borel sigma field. We then choose an appropriate probability for each Borel set, via a distribution function. Then, how does that distribution function determine the probability space $(\mathrm{\Omega},\mathcal{A},P)$ in theory or proves its existence? What result is implicitly used here?

I understand the probability space on $(\mathbb{R},\mathcal{B})$ is well defined, and that is what practically concerns us, but in theory we also have a probability space $(\mathrm{\Omega},\mathcal{A},P)$ which pushes forward its measure to the Borel sets. Hence given the distribution on $X$ we are pulling back to $(\mathrm{\Omega},\mathcal{A})$ via $P(X\in B)={P}_{X}(B)$. But what theorem guarantees that such a space exists?

Edit for clarification:For example, let us take weather $\mathrm{\Omega}$ of a city as the original sample space and the recorded temperature $X$ as the random variable. Suppose the distribution of $X$ is decided upon, based on empirical data. That's fine and now we can answer questions like 'what is $P(a<X<b)$'. We may not care about the probability space on $\mathrm{\Omega}$ since all the probabilities we wish to know relate to $X$, but how do we know for sure that there exists a probability space on $\mathrm{\Omega}$ in the first place, and further, how do we know that a probability space on $\mathrm{\Omega}$ exists which would push its probability to yield the distribution of $X$? Unless we know that there exists such a probability space in theory, we will not be able to talk about expectation of $X$, so the knowledge that such a space exists is crucial.

*Further edit: Note that we may not have an explicit complete mathematical model of the weather ever, but that does not bother us. We do have the ability however, of taking measurements of different types and thereby getting distributions related to temperature, pressure, wind speed, historical records etc. So practically we can answer questions related to these measurements. This is so what happens in practice I think. What I want is some mathematical theorem which assures me that there exists a model of the weather space, even though it is hard/difficult/impossible to find, which pushes its probability on to these distributions.

I understand the probability space on $(\mathbb{R},\mathcal{B})$ is well defined, and that is what practically concerns us, but in theory we also have a probability space $(\mathrm{\Omega},\mathcal{A},P)$ which pushes forward its measure to the Borel sets. Hence given the distribution on $X$ we are pulling back to $(\mathrm{\Omega},\mathcal{A})$ via $P(X\in B)={P}_{X}(B)$. But what theorem guarantees that such a space exists?

Edit for clarification:For example, let us take weather $\mathrm{\Omega}$ of a city as the original sample space and the recorded temperature $X$ as the random variable. Suppose the distribution of $X$ is decided upon, based on empirical data. That's fine and now we can answer questions like 'what is $P(a<X<b)$'. We may not care about the probability space on $\mathrm{\Omega}$ since all the probabilities we wish to know relate to $X$, but how do we know for sure that there exists a probability space on $\mathrm{\Omega}$ in the first place, and further, how do we know that a probability space on $\mathrm{\Omega}$ exists which would push its probability to yield the distribution of $X$? Unless we know that there exists such a probability space in theory, we will not be able to talk about expectation of $X$, so the knowledge that such a space exists is crucial.

*Further edit: Note that we may not have an explicit complete mathematical model of the weather ever, but that does not bother us. We do have the ability however, of taking measurements of different types and thereby getting distributions related to temperature, pressure, wind speed, historical records etc. So practically we can answer questions related to these measurements. This is so what happens in practice I think. What I want is some mathematical theorem which assures me that there exists a model of the weather space, even though it is hard/difficult/impossible to find, which pushes its probability on to these distributions.