timelets

timelets @ 2022-03-30T00:25:00

2022-03-30T07:31:39Z

The wolf is a tortilla [functor], while the house is not. Rather, the house is a nullifier. That is,
hwh = hw.

see Burritos for the Hungry Mathematician, by Ed Morehouse April 1, 2015.

The prince is a tortilla.

Prince vs Wolf is like freedom-for vs freedom-from.

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timelets @ 2021-03-07T18:37:00

2021-03-08T02:53:55Z

Usually the key aspect of an action of some

X

is that

X

itself carries an algebraic structure, such as being a group (or just a monoid) or being a ring or an associative algebra, which is also possessed by

Y^Y

and preserved by the curried action

\widehat{act}

. Note that if

Y

is any set then

Y^Y

is a monoid,

https://ncatlab.org/nlab/show/action

also see Lawvere, 1986

"Historically the notion of monoid (or of group in particular) was abstracted from the actions, a pivotally important abstraction since as soon as a particular action is constructed or noticed, the demands of learning, development, and use mutate it into: 1) other actions on the same object, 2) actions on other related objects, and 3) actions of related monoids. "

===
MxA-> A

UxA -> A

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timelets @ 2020-04-01T20:57:00

2020-04-02T03:59:06Z

This is a really neat example of a quotient.

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timelets @ 2020-03-17T13:07:00

2020-03-17T20:16:20Z

Our foundry used to be much like other foundries. When we cast the first “Model T” cylinders in 1910, everything in the place was done by hand; shovels and wheelbarrows abounded. The work was then either skilled or unskilled; we had moulders and we had labourers.

Now we have about five per cent. of thoroughly skilled moulders and core setters, but the remaining 95 per cent. are unskilled, or to put it more accurately, must be skilled in exactly one operation which the most stupid man can learn within two days. The moulding is all done by machinery.

--- Henry Ford, My Life and Work.

As a result of this knowledge environment, most of How To books are written "for dummies."

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An alternative to Hitler's thinking on monads

2019-06-13T03:58:00Z

I don't know much about programming but mathematically the concept is quite beautiful. Overall, monoidal categories are a great way to model consistent processes. Kant would love it because it incorporates the notion of unity, which reason seeks to apply to the world.

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TIL: NFL careers

2018-08-16T18:01:43Z

The average NFL career is short — about 3.3 years.
Quarterbacks fare longer, with an average career span of 4.4 years.

https://fivethirtyeight.com/features/the-backup-qb-who-gets-paid-a-lot-to-never-play/

That would explain the importance of systems and set plays

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Servantes and free graph (monoid)

2018-07-20T16:20:55Z

This would be a good example of how great literature describes a narrative as a technology that can be further described by a mathematical object.

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timelets @ 2018-04-04T14:50:00

2018-04-04T21:58:45Z

Bicycle is a device for mapping a monoid to a linear order.

upd: stationary bicycle, aka the hamster wheel, is a device for mapping a monoid to a monoid.

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Really good stuff

2018-03-22T20:29:17Z

...the most basic questions about a resource theory: When are two resources equivalent under the free operations, in the sense that one can convert one to the other and vice-versa? What are the necessary and sufficient conditions on two resources such that the first can be converted to the second (possibly irreversibly) under the free operations? How can we find measures of the quality of a resource, that is, functions from the resources to the reals which are nonincreasing under the transformations allowed by the resource theory? ... Can a given resource conversion be made possible by the presence of a catalyst, i.e. a resource which must be present but is not consumed in the process?

We refer to this minimal framework as a theory of resource convertibility. The requisite mathematical formalism turns out to be that of a commutative (equivalently, Abelian or symmetric) preordered monoid.

First, consider the preorder. This is the order over resources that captures whether one resource can be converted to another or not. Recall that the first of the questions above called us to characterize the equivalence classes of resources under the free operations, and the second called us to find the partial order over these equivalence classes that is induced by the free operations. But this is nothing more than a call to find the preorder over the resources induced by the free relation of convertibility. Once the preorder is identified, one can define measures of the resource, or “resource monotones”, as any map from the resources to the reals which respects the preorder.

BFS. https://doi.org/10.1016/j.ic.2016.02.008

Just a few days ago I wanted to write that I can simulate any system of interest with a preorder and a monoid :)

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CT-based Resource Theory

2018-03-22T14:48:11Z

Bob Coecke, Tobias Fritz, and Robert W. Spekkens. “A mathematical theory of resources”. In: Inform. and Comput. 250 (2016), pp. 59–86. issn: 0890-5401. url: https://doi.org/10.1016/j.ic.2016.02.008.

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timelets @ 2018-03-20T11:55:00

2018-03-20T19:02:43Z

For every set M we may construct the monoid of endomorphisms (M → M,◦,id), where ◦ is function composition and id is the identity function.

Theorem 1.1 (Cayley representation for (Set) monoids)
Every monoid (M,⊕,e) is a sub-monoid of the monoid of endomorphisms on M.

Exequiel Rivas, Mauro Jaskelioff. 2014.
https://arxiv.org/abs/1406.4823v1

My hunch about the relationship b/w design and artefact turns out to be true. It also represents a mathematical version of Henry Ford's saying, "You won't get rich through inventions. You get rich through improvements."

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Immersion

2018-02-20T18:09:22Z

I've settled into a routine now: watch CT videos before sleep, discover problems in my models while sleeping or in the wee hours of the morning, solve the problems in the morning or during the day, repeat the next day. I feel like a cyclical monoid :)

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timelets @ 2018-01-23T14:29:00

2018-01-23T22:31:08Z

I think this is better than Spivak's approach

https://arxiv.org/pdf/0905.3010.pdf
Categories for the practising physicist
Bob Coecke and Eric Oliver Paquette

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TIL: evaluation of a composite

2017-12-25T09:13:23Z

In a system with perfect trust a composite can be implemented through division of labor wrt judgement, e.g. Ray Dalio uses competence weighted judgement.

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timelets @ 2017-12-21T10:16:00

2017-12-21T18:22:31Z

Допустим, я хочу построить штуку, которая реализует правило "больше - лучше."

Тогда мне нужен моноид с функцией добавления, посет и стрелка (value judgement) между ними.

f: + -> >

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More on the Portuguese

2017-12-12T04:25:04Z

The general idea was that if the altura, or height above the horizon, of the sun or a star (normally the Pole Star) could be determined and compared with the known altura of the port of destination, then the ship could sail north or south until it reached that latitude, and then sail, as appropriate, east or west in the certainty of finding its point of destination. The measurement of the altura was possible with the use of either the quadrant or the astrolabe.

Both devices were standard university instruments of astronomy and astrology that carried a great deal of information that was both unnecessary to the calculation of latitude and simply incomprehensible to the layman. On the back of the astrolabe there was, however, an alidade, which was a rule on a swivel with two pinhole sights. The observer held the instrument upright by a swivel suspension ring, peered along the alidade, and measured the altura of the star by reading off the position of the alidade on a scale marked on the rim ( figure 3 ). The quadrant was an instrument with similar functions.

But instruments, inscriptions, and stars were not enough. Part of the association of elements to convert stars into latitudes lay in the practices of the mariners, and it was this element that was the most prone to distortion. It was difficult, although not ultimately impossible, to create a new social group necessary for closure: the astronomical navigator.

- John Law, ibid

The navigator embodied a (cyclical?) monoid, i.e. he could start applying technology arrows (device-based mappings) from any object in the graph: Stars -> Latitude -> Port -> Course -> Stars.

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Tradeoff of the day

2017-12-11T20:05:10Z

As a war machine in the relatively sheltered waters of the Mediterranean the galley was a great success. As a cargo carrying vessel, however, it had its drawbacks. Its carrying capacity was extremely limited. The features that made it a good war ship—it was slim and low and could carry a large crew that might repel boarders—were an impediment to the carriage of cargo (Lane 1973, p. 122; Denoix 1966, p. 142). Furthermore, the endurance of the galley was restricted by the size of its crew. It could not pass far from the sight of land and the possibility of water and provisions.

....

Accordingly, it is a paradigmatic case of the fundamental problem faced by system builders: how to juxtapose and relate heterogeneous elements together such that they stay in place and are not dissociated by other actors in the environment in the course of the inevitable struggles—whether these are social or physical or some mix of the two.

-- ibid

Three types of technological innovation were important.

-- The first of these took the form of a revolution in the design of the sailing ship in the fourteenth and early fifteenth centuries. ... a mixed-rigged seagoing vessel ( figure 2 ) that had much greater endurance and seaworthiness than its predecessors, one that was able to convert winds from many directions into forward motion. There were no rowers, so manpower was reduced, and it was thus possible to carry sufficient stores to undertake a considerable passage without foraging.

-- The second was the fact that the magnetic compass became generally available in Christian Europe in the late twelfth century. ... it allowed a reasonably consistent heading to be sailed in the absence of clear skies. Combined with dead reckoning and a portolano chart, 9 the magnetic compass took some of the guesswork out of long-distance navigation, and in particular it meant that the sailor did not need to hug the coast to have some idea of his location.

-- It was the invention of this circle, called the volta by the Portuguese, that marks the decisive third step. Ships were no longer forced to stay close to the coast. Cape Bojador, the classic point of no return, was no longer the obstacle it had previously been. The masters could sail beyond it and expect to be able to return.

The volta can thus be seen as a geographical expression of a struggle between heterogeneous bits and pieces assembled by the Portuguese system builders and their adversaries, that is, the winds, the currents, and the capes.

--
System builders seek to create a network of heterogeneous but mutually sustaining elements. They seek to dissociate hostile forces and to associate them with their enterprise by transforming them.
====

It looks like a battle of free monoids composed of different elements. Juxtaposition, i.e. the idea of the opposing system, makes a big difference here.

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сны по теории услуг как стрелок

2017-10-08T16:28:35Z

Снилось, что я взял в библиотеке какую-то старую книгу. Еду по крученой дороге вдоль океана, но читаю, не могу оторваться. Еду слишком быстро, меня заносит на повороте в соседнюю полосу и вот-вот врежусь в грузовик. Притормаживаю. С дороги открывается вид на город с огромным готическим собором. Паркуюсь в двух кварталах от собора и иду гулять. Во время прогулки читаю книгу, но не понимаю, о чем она.

Потом вдруг вспоминаю, что парковка, наверное, платная. Возвращаюсь, вижу знак платной парковки. И тут до меня доходит, что это пример простейшей бизнес модели услуг, которую можно описывать стрелкой А -> В. Стелка обозначает перемещение предмета во времени, причем, еще одна стрелка показывает, сколько стоит услуга в зависимости от размера перемещения. Работу этой стрелки делает счетчик.

Соответственно, появляется параллель между банковским делом и автомобилями. Очевидно, что авто-авто приведут к эффекту "капитализма" в перемещениях. Первые признаки видны по Uber.

О чем была книжка так и не вспомнил.

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timelets @ 2017-09-29T22:44:00

2017-09-30T05:50:24Z

Пока вы тут о политике спорите, люди делом занимаются. Даже Fox News иногда понимает толк в свободных моноидах и посетах :)

http://video.foxnews.com/v/5593344599001/?#sp=show-clips

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Technology to business and back

2017-08-08T22:38:06Z

Here's two categories: Technology and Business

Technology is represented by a product of desires DxD. Desires conflict; therefore, one desire D is a PreOrder, while another is a PreoOrder Op. That is, improving our position on one desire makes another one worse off. Each desire has a terminal object 1, which is the lowest performance threshold, respectively. The thresholds form technical specification 2.

Business is represented by a Utility Monoid, e.g. a combination of Revenues (or expenses). Business has a terminal object 1, which represents profitability requirements, i.e. "the bottom line." The business would like to increase Utility, but it is constrained by the conflict of desires in Technology.

We can probably build adjoints, but I don't know how to specify them.

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timelets @ 2017-07-10T12:14:00

2017-07-10T19:30:14Z

Another interesting quote from McLane&Moerdijk, 1999.

In short, we can create two contrasting categories: a poset P, which has one morphism and many objects, and a group G, which has one object and many morphisms.

Now, by thinking in topos terms, we can see that both categories have a terminal object ( "the arrow", "the object") and can be used in combination with a subobject classifier. The existence of a subobject classifier creates an opportunity for a simple decision making process. Then the question arises, "Decision making about what?"

One way to answer it would be to go back to Adam Smith and say, "Specialization!" That is, posets P enable us to specialize in arrows ("the arrow"), while groups G (including monoids) let us specialize in objects ("the object".) A combination (e.g. a disjoint union) of these two categories helps us build a broad range of large, highly specialized categories with relatively simple decision making rules. I bet most industrial systems can be modeled this way.

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Rationality

2017-06-28T17:56:02Z

Restricting the model from category A to category B simplifies choices because it requires only one type of rationality, i.e. function m: HxH -> H.

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Why do we see free monoids all around us

2017-06-19T17:22:42Z

A shout is a monoid because we perceive it as a single complex sound. Animals can do that.
SxS -> S

A spelled out word is a free monoid on a set of symbols because it has a distinct temporal/spatial structure.

Language underlines a unique human ability to create free monoids on arbitrary sets.

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Quote of the Day: free monoids

2017-06-05T11:56:26Z

Marshall McLuhan, The Gutenberg Galaxy, p 219

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Yin - Yang

2017-02-23T06:24:28Z

Я опять все понимаю, даже лучше прежнего.

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