tag:blogger.com,1999:blog-85767302018-04-20T11:57:30.472-05:00SedenionDavid Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.comBlogger1307125tag:blogger.com,1999:blog-8576730.post-57261491060288440952018-04-20T11:57:00.000-05:002018-04-20T11:57:30.441-05:00This semester teaching topology,
I told the students that my
"no holes" informal interpretation of compactness
can be formalized as simply, chains of
nonempty closed sets have nonempty intersection.
(I also taught the open cover definition.)
<p>I haven't seen this chain characterization
in the textbooks I've used,
but it's trivial to prove using a
known result from infinitary order theory.
(A poset is chain-complete iff it is
directed-complete.)
Surely, this has been done before.
I just don't know where.
<p>Rather than develop a bunch of order
theory I didn't have time for in my
topology class, I typed up an "elementary"
<a href="http://dkmj.org/5365s18/other/chain-compact.pdf">proof</a>,
using "just" a well-ordering,
that chain compactness is equivalent
to the compactness in the usual sense.
<p>If you know a thing or two about posets,
you will recognize that the proof trivially
generalizes into a proof that
posets are chain-complete if and only if
directed-complete. I don't know if my
proof approach is new. But my recollection is
that the standard proof uses induction
on cardinality of the chains and directed sets,
which I think is conceptually more elaborate
than my approach of using a well-ordering to
extract a minimal bad chain from a bad
directed set.
David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-5656760433554501672018-04-20T11:28:00.002-05:002018-04-20T11:28:26.377-05:00It seems timely to post a link to <a href="https://en.wikipedia.org/wiki/Noether%27s_theorem">Noether's Theorem</a>.David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-33615034551376986352018-03-24T15:31:00.001-05:002018-03-24T15:31:22.105-05:00This semester teaching undergraduate linear algebra,
I <a href="http://dkmj.org/3310s18/board/02.zip">defined</a>
a <i>pivot</i> of a matrix as:
<blockquote>a one
with nothing but zeros to the left,
nothing but zeros above, and
nothing but zeros below.
</blockquote>
Some textbooks don't require a pivot
to be a one, just to be nonzero. But in any case,
the books I've seen state the requirement
about surrounding zeros in more complicated way.
<p>The same day,
I gave definition of
<i>reduced row echelon form</i> simpler
than what I'd seen in textbooks.
<blockquote>
A matrix is in RREF if every nonzero row
contains a pivot, the pivot rows are above
the zero rows, and the pivots descend to
the right.
</blockquote>
Maybe I just haven't been reading
the right textbooks. I was given
a review copy of Gareth Williams' textbook,
but too late to use it this semester.
It doesn't define pivots, but its definition
of RREF matches mine, with instances of "pivot"
expanded into a definition of pivot that
matches mine.
David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-65713195416108738442018-02-06T22:24:00.002-06:002018-02-06T22:24:09.315-06:00Some <a href="http://philsci-archive.pitt.edu/2673/">implications</a> of Haag's Theorem.David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-38369708184341243782018-01-29T18:05:00.001-06:002018-01-29T18:05:02.466-06:00<a href="https://news.ycombinator.com/item?id=16256701">Linear logic, differentiable programming, and deep learning</a>.David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-59099304685952370352018-01-25T22:51:00.000-06:002018-01-25T22:51:13.859-06:00An amateur <a href="https://skyriddles.wordpress.com/2018/01/21/nasas-long-dead-image-satellite-is-alive/">found</a> a satellite NASA lost a dozen years ago.David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-39892190422957300192018-01-12T11:36:00.003-06:002018-01-12T13:08:15.436-06:00Tensor squares entangle.Reading (<a href="https://arxiv.org/abs/1208.0928">arXiv:1208.0928</a>)
about quantum error correction,
I was very confused for a while when the paper described
a <a href="https://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_linear_maps">tensor product</a>
of measurement operators as "simultaneous" measurement.
Actually, a tensor product A⊗<!---->B
corresponds to measuring the arithmetic product
of an A measurement value and a B measurement value.
<p>The context involves a pair of particles
which we will assume to be electrons for simplicity.
Let X<sub>n</sub> measure the spin of the nth electron
with respect to the x-axis.
Applying X<sub>n</sub> will put the nth electron's spin
in the +x or -x direction;
the corresponding macroscopic observation
will be +1 or -1 (ignoring physical units).
Let Y<sub>n</sub> be the analog of X<sub>n</sub> for the y-axis.
If we measure with X<sub>n</sub> then Y<sub>n</sub> then X<sub>n</sub> again,
then the first and second X<sub>n</sub> measurements
merely have a 1/2 chance of being the same.
In general, if spin of an electron is along one axis
and we measure its spin with respect to another axis,
we effectively destroy information.
(Technically, the information is not destroyed.
But recovering it is like unscrambling an egg.)
<p>The above information loss is algebraically manifested
as X<sub>n</sub> and Y<sub>n</sub> not commuting. However,
<a href="https://en.wikipedia.org/wiki/Pauli_matrices">these operators</a>
do <i>anti-commute</i>: X<sub>n</sub>Y<sub>n</sub>=-Y<sub>n</sub>X<sub>n</sub>.
Therefore, assuming our two electrons' spin states
are independent of each other
(which is approximately true if the electrons are not too close together),
a little algebra shows that the tensor products
X=X<sub>1</sub>⊗<!---->X<sub>2</sub> and
Y=Y<sub>1</sub>⊗<!---->Y<sub>2</sub>
do commute.
Physically, this means that if we measure with X then Y then X again,
the two X measurements will agree with probability 1, not 1/2.
If we physically interpret a tensor product
as simply performing two measurements at the same time,
then this makes no sense.
<p>But X actually measures the product of
a potential X<sub>1</sub> measurement value and
a potential X<sub>2</sub> measurement value.
This will be +1 if the X<sub>1</sub> and X<sub>2</sub> both output +1 or both output -1,
and will be -1 otherwise. In other words, X is measuring merely whether
the two particles have the same or opposite spin with
respect to the x-axis. Unlike an X<sub>n</sub>,
the act of measuring X does not align either particle's
spin to the x-axis (unless it was already there). Instead,
applying X merely changes the joint state of the particles
such that the results of potential future X<sub>1</sub> and X<sub>2</sub>
measurements are now either perfectly correlated or
perfectly anti-correlated, depending on whether
X measured +1 or -1.
<p>The bottom line is that, by measuring with X and then Y,
that is, by measuring with respect to each of two perpendicular axes
merely whether our two particles have the same
or opposite spins, we put the two particles into one of four
<a href="https://en.wikipedia.org/wiki/Bell_state">maximally entangled</a>
joint states with the very nice property that
repeated measurements of X <i>and</i> Y
will preserve the joint state of the electrons.
If the X output changes or the Y output ever changes
when performing these repeated measurements,
that indicates outside "noise."
This is a simple instance of quantum error detection for two qubits.
With more electrons, the paper explains
how to achieve quantum error <i>correction</i>.David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-74484357375849863892018-01-08T22:13:00.000-06:002018-01-08T22:13:03.652-06:00Following up on <!--selfref--><a href="http://sedenion.blogspot.com/2018/01/you-know-new-class-of-vulnerabilities.html">this post</a>, more <a href="https://webkit.org/blog/8048/what-spectre-and-meltdown-mean-for-webkit/">ugly hacks</a> to fight Spectre and Meltdown.David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-2187684908473813972018-01-06T13:24:00.003-06:002018-01-06T13:24:56.697-06:00<a href="https://www.sociologicalscience.com/articles-v4-28-686/">Pruning.</a>David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-84171489352933466262018-01-04T14:27:00.000-06:002018-01-04T14:27:14.223-06:00You know a <a href="https://meltdownattack.com/">new class of vulnerabilities</a> is really bad
when the LLVM compiler is <a href="http://lists.llvm.org/pipermail/llvm-commits/Week-of-Mon-20180101/513630.html">desperately resorting</a> to <a href="https://support.google.com/faqs/answer/7625886">this ugly hack</a>.David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-82325882596371361502017-11-24T22:23:00.003-06:002017-11-24T22:23:29.178-06:00<a href="https://distill.pub/2017/feature-visualization/">Neural net feature visualization by optimization.</a>David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-48904963856087400432017-11-21T14:00:00.000-06:002017-11-21T14:00:40.405-06:00<a href="https://www.nytimes.com/2017/11/15/opinion/roy-moore-evangelical-politics.html">Preach it, brother!</a>David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-34513320682402834712017-11-14T22:47:00.001-06:002017-11-14T22:47:18.540-06:00Mere existence of God is a powerful
but actually relatively weak in its
logical consequences, much like the
existence of an <a href="http://cantorsattic.info/Inaccessible">inaccessible</a>
is a relatively weak large cardinal axiom
in set theory. On the other hand,
<a href="http://cantorsattic.info/Rank_into_rank">I0</a>
is a very strong axiom
only one step removed from logical impossibility,
Perhaps the analogously strong axiom of the faith
is the Incarnation.
David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-86965055959631394272017-09-09T11:54:00.002-05:002017-09-09T11:54:21.287-05:00<a href="https://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking">Spontaneous symmetry breaking</a> is much easier to wrap my mind around using a
<a href="http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node140.html">Newtonian</a> example.David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-21545662723014092022017-09-05T21:00:00.000-05:002017-09-05T21:00:08.372-05:00Regarding the <a href="https://cbmw.org/nashville-statement">Nashville statement</a>, anthropology:chastity::keep:castle. It's good but insufficient to fortify the keep. Two of the castle walls are respectively located at divorce and contraception.
David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-31010979503216708272017-07-12T00:27:00.000-05:002017-07-12T00:27:11.417-05:00<a href="https://fredrikdeboer.com/2017/07/10/study-of-the-week-to-remediate-or-not-to-remediate/">On a randomized control trial at CUNY on replacing remedial college algebra with introductory college statistics:</a>
<blockquote>
[T]here is a significant and fairly large (16% without covariates in the model, 14% with) difference in odds of passing the course for those randomized to the intro stats course compared to the elementary algebra course...
<p>I will hold with Hacker in suggesting that this does represent a lowering of standards, and that this is a feature, not a bug. That is, I think we should allow some students to avoid harder math requirements precisely because the current standards are too high. Students in deeply quantitative fields will have higher in-major math requirements anyway.
</blockquote>
Yep. And statistics is more directly useful for non-STEM majors who, for example, want to understand the news.
<p>I also found it interesting that "support workshops" for remedial college algebra students didn't measurably improve pass rates in this study. I'm not sure what to make of that. I suppose it's consistent with my anecdotal, non-randomized-control-trial experience that more conscientious students are both more likely to pass and more likely to take the time to get help from tutors and/or their professors on a regular basis.David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-32752853774680233062017-07-08T17:27:00.001-05:002017-07-08T17:27:14.308-05:00<a href="https://nationalaffairs.com/publications/detail/federalism-in-blue-and-red">Replace</a> the federal income tax deduction of state income taxes with block grants for states with below-average per capita income. Such "equalization grants" work well in Canada and Australia.David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-16530870248873835392017-07-01T16:47:00.000-05:002017-07-01T16:47:08.558-05:00I've long been a fan of Scott Sumner's <a href="http://econlog.econlib.org/archives/2017/06/my_views_on_mon.html">views</a> on monetary policy. I recently <a href="http://www.arnoldkling.com/blog/scott-sumners-macroeconomics-my-thoughts/#comment-474147">commented</a> on Arnold Kling's blog in defense of Sumner. In summary:
<blockquote>
If the Fed wanted 2018 US NGDP to be more than $20T (for example), it could say, "whenever betting markets generally predict 2018 US NGDP under $20T, we will start to buy and hold assets of our choosing, at a rate of $1B the first day and doubling the rate every day thereafter, until betting markets generally predict 2018 US NGDP over $20T."
<p>When there is a large negative expected NGDP, the mismatch between downward-sticky wages and much more plastic hiring/firing/layoffs aggregates into a large short-term misallocation of real resources.
<p>Therefore, the Fed should buy whatever it takes to reverse large downward surprises in expected NGDP.
</blockquote>David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-4750286192853109082017-06-26T15:08:00.001-05:002017-06-26T15:08:46.738-05:00News of <a href="http://blog.achernya.com/2017/06/by-installing-nat-mit-stifles-innovation.html?view=classic">MIT's</a> <a href="https://en.wikipedia.org/wiki/Network_address_translation">NAT</a> saddens me.
<p>I remember when the names iteration.mit.edu and recursion.mit.edu
were mine; they pointed to my public IPv4 address 18.238.3.106.
I don't claim to have done anything innovative with them,
but what I did was very educational.
I wrote a minimal <a href="http://dkmj.org/sigmafield/webserver/webserver.html">http server</a>
in C++ and hosted a static website
<a href ="http://dkmj.org/mathimages/">with</a> fractal <a href="http://dkmj.org/mathimages/z%281+.5%28z-exp%281.4i%29%29%29-zoomed-in.jpg">images</a> I created and a Mandelbrot set Java applet I wrote.
<p>I probably never would have been motivated to learn
how to write an http server without such public visibility.
Security risks?
<a href="https://slashdot.org/comments.pl?sid=28902&cid=3103368">Yes.</a>
(I promptly fixed that one.)
Despite this, MIT thrived without a NAT for many years.
Even if security risks are greater today,
firewalls can be made arbitrarily strict without a NAT.
David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-33845270170599634662017-06-19T17:42:00.001-05:002017-06-19T17:45:25.107-05:00A bit of TeX hacking<p>I've decided that if I state, say, Theorem 2.2.15 but don't prove it until many pages and lemmas later in Section 5,
then, for the reader's sake, I should repeat the theorem verbatim, including the original theorem number, immediately before the proof, as opposed to going straight from the proof of Lemma 5.2.33 to "Proof of Theorem 2.2.15." (Granted, if the delay between statement and proof is just for one "Main Theorem" whose statement is easy to remember, then this is not necessary. My decision is in the context of revising a longer paper with several theorems stated early and proved much later.)
<p>
<pre>
%#1 theoremstyle inpute (plain/definition/remark/...)
%#2 type of theorem (Theorem/Lemma/Corollary/...)
%#3 label of theorem to be repeated
%#4 unique new input for \newtheorem
%#5 statement of theorem
\def\repeattheoremhelper#1#2#3#4#5{
\theoremstyle{#1}
\newtheorem*{#4}{#2 \ref{#3}}
\begin{#4}
#5
\end{#4}
}
\def\repeattheorem#1#2{
\repeattheoremhelper{plain}{#1}{#2}{repeat#2}{\csname state#2\endcsname}
}
%usage example
%\theoremstyle{plain}
%\newtheorem{thm}{Theorem}
%...
%\def\statemytheorem{blah blah}
%\begin{thm}\label{mytheorem}\statemytheorem\end{thm}
%...
%\repeattheorem{Theorem}{mytheorem}
%\begin{proof}
%...
%\end{proof}
</pre>David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-4133593436792746812016-11-19T10:14:00.002-06:002016-11-19T10:14:34.887-06:00I think I need a red ink pad and stamp with
"<a href="http://algebrarules.com/">AlgebraRules.com</a>"
on it.
David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-59107390276196303962016-11-03T08:33:00.000-05:002016-11-03T08:33:15.449-05:00<a href="http://www.themoneyillusion.com/?p=32058">"I always say that politics takes 30 points of[f] a personâ€™s IQ, including me."</a>David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-52328293859102270452016-10-18T14:58:00.001-05:002016-10-18T14:58:34.411-05:00<a href="https://imgcert.com/image/yJzh">The old ways are the best ways.</a>
David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-29641514440932904942016-10-08T14:04:00.000-05:002016-10-08T14:04:05.183-05:00<a href="http://www.nytimes.com/interactive/2014/12/22/upshot/what-2000-calories-looks-like.html?contentCollection=smarter-living&_r=0">What 2000 calories looks like.</a>David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0tag:blogger.com,1999:blog-8576730.post-31197016290098367972016-09-22T19:16:00.003-05:002016-09-22T19:16:30.995-05:00<a href="http://theresurgent.com/coming-to-terms-with-trump/">I just cannot vote for either one.</a>David Milovichhttps://plus.google.com/109637306067900610878noreply@blogger.com0