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Recent questions in Algebra

Zhyar Hoshyar Wali 2022-05-25

vaganzahi 2022-05-24 Answered

Equation simplification, can't get it right
$\frac{1}{\frac{x - 1}{x + 2}} - \frac{2}{x^{2} - 1}$
should be simplified into
$\frac{x^{2} + 3 x}{x^{2} - 1} .$
However, when I try to do it (tried several times), I fail to get it done right:
$\frac{1}{\frac{x - 1}{x + 2}} - \frac{2}{x^{2} - 1} = 1 * \frac{x + 2}{x - 1} - \frac{2}{x^{2} - 1}$
$= \frac{x (x + 2) - 2}{x^{2} - 1} = \frac{x^{2} + 2 x - 2}{x^{2} - 1}$
What am I doing wrong?
Thanks a lot for the help

Brice Colon 2022-05-24 Answered

Find the water pumped to flowers using fractions.
A certain tank has water filled to $\frac{1}{5}$ of it's capacity. A persons opens a tap and allows the tank to fill upto to $\frac{3}{4}$ of it's capacity. Then he uses the water in the tank and finally $\frac{1}{3}$ water remains in the tank.
What I have found :
Water that was added to the tank after opening the tap : $\frac{11}{20}$
Fraction of water he used was : $\frac{5}{12}$
Total capacity of water in the tank : 1800 litres
Problem:
Find the amount water he used up in litres
How many more litres remained in the tank in the end when compared to the beginning

cricafh 2022-05-24 Answered

Could anyone help prove the following claim?
For $Z$ a random variable, choose $X, Y$ appropriately in Holder's inequality to show that
$f (t) = \log (E | Z^{t} |)$
is a convex function on the interval of $t$ where $E | X^{t} | < \infty$ .
I'm confused on the part of choosing appropriate $X, Y$ in Holder's inequality. I tried using only X and Z and rearranging for the $\log (E | Z^{t} |)$ term but it feels headed in the wrong direction.

Charity Daniels 2022-05-24 Answered

Let $(X_{n})_{n \geq 0}$ be a sequence of random variables and $τ, t$ stopping times with respect to the sequence $(X_{n})_{n \geq 0}$
$\begin{aligned} {τ + t = n} = {τ + t = n} \cap {t \leq n} & = ⋃_{k = 0}^{n} {τ + t = n} \cap {t = k} \\ = ⋃_{k = 0}^{n} {τ = n - k} \cap {t = k} . \end{aligned}$
As ${τ = n - k} \in F_{n - k} \subseteq F_{n}$ and ${t = k} \in F_{k} \subseteq F_{n}$ for any $k \leq n$ , this implies that ${τ + t = n} \in F_{n}$ , and so $τ + t$ is a stopping time.
Now, my question is, let assume that I am considering $τ - t$ I know that in general, $τ - t$ is not a stopping time. However, if I were to consider my birthday this year (a stopping time), which is a deterministic stopping time. At any time, I know exactly when my birthday occurs. Also, I know two days before my birthday i.e, $τ - 2$ . What kind of a formulated counterexample will show that $τ - 2$ is indeed a stopping time in this setting.

herbariak1 2022-05-24 Answered

How to sum up these fractions?
Found this question in a competitive math test for elementary students. The long way is to add all the decimal values but is there a pattern/trick to solve this question (or these types)? I don't know how to solve this except by the long method of adding all the decimal equivalents.
The Answer is 1
Compute:
$\frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{11} + \frac{1}{12} + \frac{1}{14} + \frac{1}{15} + \frac{1}{18} + \frac{1}{22} + \frac{1}{24} + \frac{1}{28} + \frac{1}{33} = ?$

Camille Flynn 2022-05-24 Answered

How do you tell whether the given value $r = 8$ is a solution to $r - 3 \leq 9$ ?

prensistath 2022-05-24 Answered

How do you grap $- y \leq 3 x - 5$ ?

Monserrat Sawyer 2022-05-24 Answered

Negative Indices
I am not exactly good at evaluating negative indices--can someone please show me how to work out this expression:
$\frac{m^{- 3} n^{- 2}}{m^{- 5} n^{6}}$
Both m's and the top n have negative indices.
Thanks!

Jazmine Bruce 2022-05-24 Answered

How to simplify a diabolical expression involving radicals
A friend and I have been working on this problem for hours - how can the following expression be simplified analytically?
It equals $\frac{1}{2},$ , and we have tried the following to no avail:
1. Substitution of $x = \sqrt{5}$
2. Substitution of $x = 2 \sqrt{5}$
3. Substitution of $x = 5 + \sqrt{5}$
4. Substitution of $x = \sqrt{5 + \sqrt{5}}$
5. Manipulations by substituting the golden ratio
Here goes:
$\frac{\frac{\sqrt{5 + 2 \sqrt{5}}}{2} + \frac{\sqrt{5 (5 + 2 \sqrt{5})}}{4} - \frac{\sqrt{10 + 2 \sqrt{5}}}{8}}{\frac{\sqrt{5 (5 + 2 \sqrt{5})}}{4} + 5 \cdot \frac{\sqrt{5 + 2 \sqrt{5}}}{4}}$
Thanks in advance for any help.

Nylah Burnett 2022-05-24 Answered

Suppose $Y_{1}, Y_{2}, \dots$ is any sequence of iid real valued random variables with $E (Y_{1}) = \infty$ . Show that, almost surely, $\underset{n}{lim sup} (| Y_{n} | / n) = \infty$ and $\underset{n}{lim sup} (| Y_{1} + . . . + Y_{n} | / n) = \infty$ .
I have solved the first part by considering non-negative integer iid r.vs $X_{n} = floor (| Y_{n} |)$ and using $E (X) = \sum_{0}^{\infty} P (X \geq n)$ then doing some clever tricks so I can apply the (2nd) Borel-Cantelli lemma, but I'm not really sure how I can use the same approach to solve the second part seeing as it is tempting to set $S_{n} = floor (| Y_{1} + . . . + Y_{n} |)$ but then the $S_{i}$ are not iid. I'm pretty sure its gonna be Borel-Cantelli again (since limsup) so I need to come up with the right events. Please can someone nudge me in the right direction.
Hints only please
EDIT: Suppose $\underset{n}{lim sup} | a_{1} + . . . + a_{n} | / n < \infty$ . Then set $S_{n} = \sum_{1}^{n} a_{k}$
$\frac{| a_{n} |}{n} = \frac{| S_{n} - S_{n - 1} |}{n} \leq \frac{| S_{n} |}{n} + \frac{| S_{n - 1} |}{n - 1}$
bounded

istupilo8k 2022-05-24 Answered

The standard form of a linear firs-order DE is
$\frac{d y}{d x} + P (x) y = Q (x)$
I think the equation is separable if and only if $P (x)$ and $Q (x)$ are constants, but I'm not sure. (Haven't found any counterexamples but also can't seem to prove it.) Can anyone confirm or deny that this is correct?

Thomas Hubbard 2022-05-24 Answered

Vance wants to have pictures framed. Each frame and mat costs $32 and he has at most $150 to spend. How do you write and solve an inequality to determine the number of pictures he can have framed?

Jaidyn Bush 2022-05-24 Answered

Extract from a proof in a measure theory script:
For each set $A \subset R^{n}$ and each $r > 0$ , we set.
$A_{r} := {x \in A : B (x, r) \subseteq A}$
Let $Σ$ be an algebra in $R^{n}$ containing all open subsets of $R^{n}$ , then for any set $A \in Σ$ and any $r > 0$ , we have $A_{r} \in Σ$ .

Eliaszowyr1 2022-05-24 Answered

Q is a rational function with $x \cdot Q (x + 2018) = (x - 2018) Q (x), \forall x \notin 2018 and 0$ . If $Q (1) = 1$ . what is Q(2017)?
I tried substituting values in for Q(x) but there is no way to have both Q(1) and Q(2017) in the same equation, perhaps im missing a property of a rational function since i haven't used it yet.

Waylon Ruiz 2022-05-24 Answered

I met an excercise in the book by Rabi Bhattacharya and Edward C. Waymire. Suppose that $μ, ν$ are probbaility measures on $R^{d}$ , with $ν$ absolutely continuous with pdf $f$ , i.e., $d ν = f (x) d x$ . How to show that the convolution, $μ * ν$ , is also absolutely continuous? Thanks!

Antoine Hill 2022-05-24 Answered

If I have two algebraic numbers $α$ and $β$ and a rational function $w$ with rational coefficients (a function that's the ratio of two rational polynomials) that relates the two $α = w (β)$ if I were to substitute $β$ with one of it's algebraic cojugates $β^{^{'}}$ in the rational function will I get a conjugate of $α$ or rather is $w (β^{^{'}})$ an algebraic conjugate of alpha? I feel like there is a very obvious counter example but I've been struggling to find one.

Brooke Kramer 2022-05-24 Answered

In book i read now, the author defines a field just like this:
A field is a non-empty class of subsets of $Ω$ closed under finite union,finite intersection and complements.
In the following, he says like this:
A minimal set of postulates for $A$ to be a field is (i) $Ω \in A .$ (ii) $T \in A$ implies $\bar{T} \in A .$ (iii) $S, T \in A$ implies $S \cup T \in A$
Though I know why it exactly defines a field, the author says nothing about why they are minimal. He even uses the word minimal without an exact definition. Can anyone help me with this problem?

Pitrellais 2022-05-24 Answered

How to find an irrational number in $Q \cap [0, 1]$ ?

Isaiah Owens 2022-05-24 Answered

I have some observed values with measurement errors . I know the error distribution is gaussian with zeros mean and variance $σ^{2}$ . What are the ways to compensate for this error?
Any help is appreciated.

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