Every Cayley-Dickson algebra can be viewed as a $(+,-,*,0,1)$ algebra. The reals and the complexes share the same equational identities. The quaternions have a subset of the equational identities, because they lose commutativity of multiplication. The octonions lose associativity, and the sedenions lose even more. Is the sedenions truly the last stop. Do all further Cayley-Dickson algebras have the same equational identities in that signature as the sedenions. This question is different from the tagged question, because I didn't see an answer for whether there are any equational identities that the sedenions have that any higher Cayley-Dickson algebras don't. I know that the sedenions are a power associative algebra.
In the construction of types of numbers, we have the following sequence:  $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$  or:  $$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions} \subset 2^3 \mathrm{-ions} \subset 2^4 \mathrm{-ions} $$  or:  "Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions"  With the following "properties":   From $\mathbb{R}$ to $\mathbb{C}$ you gain "algebraic-closure"-ness (but you throw away ordering). From $\mathbb{C}$ to $\mathbb{H}$ we throw away commutativity. From $\mathbb{H}$ to $\mathbb{O}$ we throw away associativity. From $\mathbb{O}$ to $\mathbb{S}$ we throw away multiplicative normedness.   The question is, what lies on the right side of $\mathbb{S}$, and what do you lose when you go from $\mathbb{S}$ to one of these objects ?
Missing abstract with amsart class I'm not very good at LaTeX but I've been using it to write my dissertation. Have just come to the end and trying to format everything correctly. Whenever I try to put my Abstract paragraph into the form \begin{abstract} (writing here) \end{abstract} and click play the whole abstract section just disappears. What am I doing wrong?  \documentclass[12pt]{amsart}  \begin{document}  \begin{abstract} (my abstract here) \end{abstract} \end{document} In the simple linear regression model, $\hat{\beta}$ is the sum of independent normally distributed random variables.  Is it false because in linear regression there is $\beta$ and not $\hat{\beta}$? Would time freeze if you could travel at the speed of light? Let's say we have $2$ particles facing each other and each traveling (almost) at speed of light.    Let's say I'm sitting on #$1$ particle so in my point of view #$2$ particle's speed is (almost) $c+c=2c$, double light speed? Please say why I am incorrect :)    EDIT: About sitting me is just example, so in point of view of #1 particle, the second one moves at (almost) $c+c=2c$ speed? Seeking birth record for Jacob Schulthess from Wangen an der Aare in Switzerland? tmux: enable mouse only when shift pressed booting windows 10 spends too long Closing Streams in Java Where do I ask where to ask? Bootcamp: The startup disk does not have enough space to be partitioned When clicking continue on the first Boot Camp screen,  I am presented with a dialog that displays:     The startup disk does not have enough space to be partitioned.   You must have at least 39 GB of free space available.     This is despite "About this Mac" reporting 271GB of free space:    And "Disk Utility" reporting 268GB of free space:    Searching online I found  which suggested it was the result of Time Machine which i recently enabled and suggests to run sudo tmutil thinlocalsnapshots / 999999999999 which I did, it ran sucessfully, and I restarted, yet to no avail.  What is the solution to this instance of Apple's poor engineering?  Running macOS High Sierra 10.13.4 (17E202) How much creative independence are the Show runners accorded in Game of Thrones? Since Season 3, we have been observing that the show was taking a different direction than the books. Many important characters never made it into the show, others who are still alive were killed off.   As we know the show is supposed to be based on the books, not to be direct adaption of the books from print to television media, there's bound to be a certain degree of creative independence accorded to them by the author of the series, George R.R. Martin.   The question is, how much independence do they have? Do they still have to take GRRM on board with whatever changes they might intend to make? By "they" I mean D&amp;D and the whole writers crew for the show. Given the number of significant changes they have made, I reckon the answer would be a lot. Has GRRM or HBO ever commented on that? I have a query pattern that must be very common, but I don't know how to write an efficient query for it.  I want to look up the rows of a table that correspond to "the most recent date not after" the rows of another table.  I have a table, inventory say, which represents the inventory I hold on a certain day.  date       | good | quantity ------------------------------ 2013-08-09 | egg  | 5 2013-08-09 | pear | 7 2013-08-02 | egg  | 1 2013-08-02 | pear | 2   and a table, "price" say, which holds the price of a good on a given day  date       | good | price -------------------------- 2013-08-07 | egg  | 120 2013-08-06 | pear | 200 2013-08-01 | egg  | 110 2013-07-30 | pear | 220   How can I efficiently get the "most recent" price for each row of the inventory table, i.e.  date       | pricing date | good | quantity | price ---------------------------------------------------- 2013-08-09 | 2013-08-07   | egg  | 5        | 120 2013-08-09 | 2013-08-06   | pear | 7        | 200 2013-08-02 | 2013-08-01   | egg  | 1        | 110 2013-08-02 | 2013-07-30   | pear | 2        | 220   I know one way of doing this:  select inventory.date, max(price.date) as pricing_date, good from inventory, price where inventory.date &gt;= price.date and inventory.good = price.good group by inventory.date, good   and then join this query again to inventory. For large tables even doing the first query (without joining again to inventory) is very slow.  However, the same problem is quickly solved if I simply use my programming language to issue one max(price.date) ... where price.date &lt;= date_of_interest ... order by price.date desc limit 1 query for each date_of_interest from the inventory table, so I know there is no computational impediment.  I would, however, prefer to solve the whole problem with a single SQL query, because it would allow me to do further SQL processing on the result of the query.  Is there a standard way to do this efficiently? It feels like it must come up often and that there should be a way to write a fast query for it.  I'm using Postgres, but an SQL-generic answer would be appreciated. Access to original contents of mount point Is there canon info on shelf life of various potions in Harry Potter? Do we know how long potions (in general, or at least specific ones) are potent (no pun intended) in Harry Potter canon?  E.g. do they last a couple of hours? Minutes? Years?   I seem to vaguely remember some comments of Snape's about Wolfesbane potion therfore I think at least some information exists. Functional equation $f(r \cos \varphi)+f(r \sin \varphi)=f(r)$ Find all such monotonous function $f:[0,+\infty)\to \mathbb R$, such that for any real $r\ge0$ and $\varphi \in \left[\frac{\pi}6, \frac{\pi}4\right]$, $$f(r \cos \varphi)+f(r \sin \varphi)=f(r)\text.$$  My work so far:  If $\varphi \in \left[\frac{\pi}6, \frac{\pi}4\right]$ then $$\frac{\sqrt2}2\le\cos \varphi\le\frac{\sqrt3}2$$ $$\frac12\le\sin \varphi\le\frac{\sqrt2}2$$  If $r=0$ then $f(0)=0$  If $\varphi= \frac{\pi}{4}$ then $$f(r)=4f\left(\frac r2\right)$$  $f(x)=cx^2$ for $x\ge0$ and $c\in\mathbb R$ Should methods in a Java interface be declared with or without a public access modifier? Should methods in a Java interface be declared with or without the public access modifier?  Technically it doesn't matter, of course. A class method that implements an interface is always public. But what is a better convention?  Java itself is not consistent in this. See for instance Collection vs. Comparable, or Future vs. ScriptEngine. How do you deserialize json properties that are reserved words in apex? Give a sequence $(f_n)_{n\in \mathbb{N}}$ of differentiable functions which uniformly converge to $0$, but for which the seqeunce $(f_n')_{n\in \mathbb{N}}$ of the derivatives isn't even pointwise convergent.    I found this one in my textbook marked as "Fun things to solve" and although it may be fun, it's kind of hard to do.   To be honest I already failed at the first hurdle. I couldn't even find a sequence of functions which uniformly converges to $0$.   Is there a certain way of dealing with this kind of problem? Because it seems kind of hard for me to come up with sequences without a mathematical of way of doing so (or maybe there is a mathematical way I just don't know yet). What is the idiom, proverb for "Little problems often become big problems if no one takes the initiative to correct them" What is the idiom, proverb for "Little problems often become big problems if no one takes the initiative to correct them"  Which means in an example that If the employees don’t bother to report a malfunctioning machine or a slip-and-fall hazard, serious injuries could occur – injuries that could have been preventable. Ignoring any hazard can be detrimental to one's business and the safety of their employees. What is the practical difference between a differential and a derivative? I ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative:  So, a derivative is the rate of change of a function with respect to changes in its variable, this much I get.  Thing is, definitions of 'differential' tend to be in the form of defining the derivative and calling the differential 'an infinitesimally small change in x', which is fine as far it goes, but then why bother even defining it formally outside of needing it for derivatives?  And THEN, the bloody differential starts showing up as a function in integrals, where it appears to be ignored part of the time, then functioning as a variable the rest.  Why do I say 'practical'? Because when I asked for an explanation from other mathematician parties, I got one involving the graph of the function and how, given a right-angle triangle, a derivative is one of the other angles, where the differential is the line opposite the angle.   I'm sure that explanation is correct as far it goes, but it doesn't tell me what the differential DOES, or why it's useful, which are the two facts I need in order to really understand it.  Any assistance?