isomorphs

sciencedaily.com
Physics determined ammonite shell shape

Ammonites are a group of extinct cephalopod mollusks with ribbed spiral shells. They are exceptionally diverse and well known to fossil lovers. Researchers have developed the first biomechanical model explaining how these shells form and why they are so diverse. Their approach provides new paths for interpreting the evolution of ammonites and nautili, their smooth-shelled distant “cousins” that still populate the Indian and Pacific oceans.

We have to reject the intuitively appealing idea that the brain is storing an accurate and strictly isomorphic representation of the world. To some degree, it is storing perceptual distortions, illusions, and extracting relationships among elements. It is computing reality for us, one that is rich in complexity and beauty. A basic piece of evidence for such a view is the simple fact that light waves in the world vary along one dimension -wavelength -and yet our perceptual system treats color as two dimensional… Newton was the first to point out that light is colorless, and that consequently color has to occur inside our brains. He wrote, “The waves themselves are not colored.” Since his time, we have learned that light waves are characterized by different frequencies of oscillation, and when they impinge on the retina of an observer, they set off a chain of neurochemical events, the end product of which is an internal mental image that we call color. The essential point here is:  What we perceive as color is not made up of color. Although an apple may appear red, its atoms are not themselves red. Similarly with pitch: From a one-dimensional continuum of molecules vibrating at different speeds, our brains construct a rich, multidimensional pitch space with three, four, or even five dimensions (according to some models). If our brain is adding this many dimensions to what is out there in the world, this can explain the deep reactions we have to sounds that are properly constructed and skillfully combined…If I put electrodes in your visual cortex (the part of the brain at the back of the head, concerned with seeing), and I then showed you a red tomato, there is no group of neurons that will cause my electrodes to turn red. But if I put electrodes in your auditory cortex and play a pure tone in your ears at 440 Hz, there are neurons in your auditory cortex that will fire at precisely that frequency, causing the electrode to emit electrical activity at 440 Hz- for pitch, what goes into the ear comes out of the brain!
—  Daniel J. Levitin
Model-theoretic Galois theory, 3:

 We’ve seen that the action of the automorphism group \(\operatorname{Aut}(\mathbb{M}/A)\) on the space of strong types over \(A\) yields a profinite group isomorphic to the action of \(\operatorname{Aut}(\mathbb{M}/A)\) on the algebraic closure of \(A\). Our goal is the following theorem, due to Poizat.

Theorem. If \(T\) eliminates imaginaries, then the two maps \(\operatorname{Fix}(-)\) and \(\operatorname{G}(\operatorname{acl}(A)/-)\) between the closed subgroups of \(G(\operatorname{acl}(A)/A)\) and the definably-closed subsets of \(\operatorname{acl}(A)\) containing \(A\), as in the following diagram:

are inverse to each other.

This is sort of amazing—all coding of definable sets tells us is that the orbit of definable sets under the automorphism group is controlled by a single element. Furthermore, because of the \((-)^{\operatorname{eq}}\)-construction, every first-order theory can be interpreted inside one that has EI, thus one which admits a Galois theory.

To get to showing this, let us recall some facts about profinite groups.

Definition. A profinite group is a topological group isomorphic to the projective limit (taken in the category of topological groups) of a cofiltered diagram of finite groups, each with the discrete topology. Profinite groups can be computed by taking the projective limit of the diagram of underlying topological groups and checking that this naturally inherits a subgroup structure from the product of all the finite groups in the diagram. Profinite groups have the projective topology induced by the limit, so that if \[G = \underset{\longleftarrow}{\operatorname{lim}} \left(\left(G_i\right)_{i \in I}, \left(f_{ij}: G_i \to G_j \right)_{i,j \in I} \right)\] with canonical projections \(G \overset{\pi_i}{\to} G_i\), then \[G \simeq \{(g_i)_{i \in I} \in \displaystyle \prod_{i \in I} G_i \operatorname{ | } \left(f_{ij}(g_i)\right) = \left(g_j\right), \forall i, j \in I\}.\] The topology on \(G\) is the one generated by the cylinder sets, and since the topologies on the \(G_i\) are discrete, the cylinder sets are just preimages of the \(\pi_i\) of arbitrary subsets. Therefore an open set in \(G\) is an arbitrary union of a finite intersection of cylinder sets. Dually, a closed set in \(G\) is an arbitrary intersection of a finite union of negations of cylinder sets. The topology is Hausdorff, compact, and totally disconnected: a Stone space.

Proposition. The clopen sets of a profinite group \(G\) are the cylinder sets.

Proof. To see this, note first that the intersection of cylinder sets is again a cylinder set, since if \(U \subseteq G_i\) and \(V \subseteq G_j\), we can pass to some appropriate \(W \subseteq G_k\) admitting a map to both \(G_i\) and \(G_j\). Hence, the cylinder sets are closed under finite intersection and form a basis, so that any clopen set is the union of cylinder sets. Since the space is compact, clopen sets are compact, hence clopens are finite unions of cylinder sets. By the same reasoning for intersections, finite unions of cylinder sets are again cylinder sets, so clopens are cylinder sets.

Corollary. All open subgroups of a profinite group are closed of finite index. All closed subgroups are intersections of open subgroups.

Proof. Clear.

Definition. If \(E\) is a finite equivalence relation with classes \(C_1, \dots, C_n\), with \(\Gamma\) a subgroup of \(G(E/A)\), define the \(\mathbb{M}\)-definable relation \(r_{\Gamma}\) as follows: \[r_{\gamma}(x_1, \dots, x_n) \iff \bigvee_{\sigma \in \Gamma} \bigwedge_{i \leq n} x_i \in \sigma(C_i).\] This is invariant under \(\Gamma \subseteq G(E/A)\), and so invariant under any \(\sigma \in G \left(SF_s(\mathbb{M}/A)/A\right)\) that projects to \(\Gamma\) in \(G(E/A)\). Write the subgroup of such \(\sigma\) in the profinite group as the stabilizer of the relation \(r_{\Gamma}\), \(\operatorname{Stab}(r_{\Gamma})\).

Right then. Now we can prove the theorem.

Proof of theorem. Consider the action of \(G\left( \operatorname{SF}_s(\mathbb{M}/A)/A \right) \simeq G \left(\operatorname{acl}(A)/A \right)\) on \(\operatorname{acl}(A)\). Since algebraic elements corresponds to certain strong types, the stabilizer of any point \(b \in \operatorname{acl}(A)\) is open. Therefore, \(G(\operatorname{acl}(A)/B)\) is an intersection of open sets, therefore closed. (This works for any \(B\).)

Also if \(H \subseteq G(\operatorname{acl}(A)/A)\), then \(\operatorname{Fix}(H)\) is definably closed (here we use the fact that we’re in a monster model: if \(m \in \mathbb{M} \backslash \operatorname{Fix}(H)\), then there is some \(\sigma \in H\) fixing \(\operatorname{Fix}(H)\) but moving \(m\), so \(m\) can’t possibly be in \(\operatorname{dcl}(\operatorname{Fix}(H))\), since elements definable over a set \(B\) have to be fixed by every automorphism fixing \(B\).

So our diagram is well-defined. Now, for the same reason as above, if \(b\) is not definable over \(B\), then there is a \(B\)-automorphism moving \(b\), so that \(B\) must be exactly \(\operatorname{Fix}(G\left(\operatorname{acl}(A)/B\right))\). Hence, \(\operatorname{Fix}(-)\) left-inverts \(G \left(\operatorname{acl}(A)/(-)\right)\).

Now, let \(H \subseteq G(\operatorname{acl}(A)/A)\) be a closed subgroup. Then \(H\) is an intersection of open subgroups, each of the form \(\operatorname{stab}(r_{\Gamma_i})\) for some sequence of finite subgroups\(\Gamma_i\). Since each \(r_{\Gamma_i}\) as a set has only finitely many conjugates under \(A\)-automorphisms (because \(A\)-automorphisms can only permute the classes of \(E\), of which there are finitely many), its code \(c_i\) is \(A\)-algebraic. Then by definition of a code: \(\operatorname{stab}(r_{\Gamma_i}) = G(\operatorname{acl}(A)/(A \cup \{c_i\})\).

Hence, \(H = G\left(\operatorname{acl}(A)/A \cup \bigcup_{i} \{c_i\} \right)\). Throwing in all the other codes (since there may be more than one associated to given definable set, depending on the formulas you use to define it) completes \(A \cup \bigcup_{i} \{c_i\}\) to a definably closed set \(B\). (This is because anything interdefinable with a code is also a code, and codes for \(A\)-definable \(X\) are \(A\)-interdefinable.) By the definition of codes, \(B\) must be \(\operatorname{Fix}(H)\). So \(\operatorname{Fix}(-)\) is right-inverse to \(G\left(\operatorname{acl}(A)/(-)\right)\), completing the proof.

Corollary. Let \(\mathbb{M}\) model the theory of algebraically closed fields. Since that theory (regardless of characteristic) codes definable sets, we recover the classical fundamental theorem of (infinite) Galois theory.

Remark. To help translate the abstract theorem displayed at the beginning of the post to field-theoretical terms, note that definable closure is sort of a closure-under-functions, and that the definable closure of a parameter set \(A\) in a model of an algebraically closed field is precisely the subfield (not necessarily algebraically closed) generated by \(A\), so that the Corollary produces a group-theoretic classification of intermediate field extensions after all (as expected).


To prove this corollary proper, we need to show that \(\mathsf{ACF}\) codes definable sets. That it codes finite sets is easy: treat your finite set as the zeroset of some polynomial (in a monster); the coefficients of this polynomial will be elementary symmetric functions, so fixed pointwise by any automorphisms fixing the roots setwise. Amplifying this to arbitrary definable sets is reduces to showing that \(\mathsf{ACF}\) has weakly codes definable sets, which is just to say that an the orbit of an arbitrary definable set over \(A\) under \(A\)-automorphisms is controlled by a finite set that must be fixed setwise, which will be covered (maybe—it’s a little tedious, but much cleaner overall than developing all that theory about field extensions) in a future post.

Category theory in theory:

“Yoneda’s Lemma asserts that an object X of a category is determined (up to unique isomorphism) by the functor that records morphisms from X to each of the objects of that category. Or, in more evocative terms, a mathematical object X is best thought of in the context of a category surrounding it, and is determined by the network of relations it enjoys with all the objects of that category. Moreover, to understand X it might be more germane to deal directly with the functor representing it. This is reminiscent of Wittgenstein’s ‘language game’; i.e., that the meaning of a word is–in essence–determined by, in fact is nothing more than, its relations to all the utterances in a language.“

Category theory in practice:

Okay so one of the designers I’m going to look at is Kate Moross, a ‘Jack of all trades’ from London. She studied an art foundation at Wimbledon School of Art, and went on to complete her BA Degree at Camberwell School of Art in 2008. Throughout her years at uni, she’s always been doing freelance work and then carried this work out after she graduated. 

In 2007 Kate launched Isomorph Records, a vinyl only record label. This was set up to explore the relationship between design and music - after I read this I knew Kate would be a great designer to base my case study on. ‘With each release Moross collaborated closely with the artists in order to create the definitive visual representation of their sound, that encapsulates the bands ideas and Moross’ vision for them. In 2008 Kate Moross was named at number 18 in the NME’s Future 50 innovators driving music forward.’ Unfortunately in 2012 Isomorphic Records stopped releasing music and took their website down, in a video Moross said this was due to her realising that vinyl record labels were a dark hole and ended up in negative bank numbers.

In 2012 Moross founded Studio Moross, which focuses on music based projects - again, what I’ve always been interested in. Moross currently acts as art director for Jessie Ware and has directed several music videos for her including Running and Wildest Moments.

sciencedaily.com
Fractal geometry: Finding the simple patterns in a complex world

A mathematician has developed a new way to uncover simple patterns that might underlie apparently complex systems, such as clouds, cracks in materials or the movement of the stockmarket. The method, named fractal Fourier analysis, is based on new branch of mathematics called fractal geometry. The method could help scientists better understand the complicated signals that the body gives out, such as nerve impulses or brain waves.

Isomorphism, Information, and Qualia

(from a thread about physicalism)

To assume that a computer behaves like a human mind is begging the question. It is like saying that if pixels on a TV screen behave like a rose then saying that they are still a-signfifying pixels makes no sense. By assuming that it is the pixels which ‘behave like a rose’ you are contaminating your reasoning before you are even aware that you have begun to reason. The pixels behave like pixels, not a rose. The computer behaves like a machine, not a mind.

It is only in our mind that we make an isomorphic connection between the two. The isomorphism is a comparison between Q and p where Q is the qualia (thoughts, feelings, sensations, will) and p is forms and functions (physics, information, logic, processing). The problem is that Q and p are not transitive. Q partially reduces to p but p does not inflate to Q. There is only an isomorphism Q(Q→p) but there is no p(Q) at all, and nothing to attach an isomorphism to other than p(p→p’).

[github-trend]meatier

2016/02/12 editor’s choice ( 490★ )

meatier : 기가 많이 든, 충실한.

이란 이름의 프로젝트가 올라왔습니다. 해외에서는 엄청나게 많은 호평을 받았고 github에서 아직도 수위권에 들어있는 meteor라는 프로젝트의 대안이라고 자신을 밝히고 있네요.

링크 : https://github.com/mattkrick/meatier

meteor 링크 : https://github.com/meteor/meteor

meteor에 대한 내용은 링크(클릭) 를 방문하시면 자세히 볼 수 있습니다. 일찍이 Isomorphic Javascript 구조를 갖고 있어서 많은 프로젝트에 지대한 영향을 끼쳤습니다.

다만 국내에서는 생각보다 관심이 많아 보이지 않습니다.

하지만, 오늘 소개 드릴 이 meatier는 그간 meteor가 가지고 있었던 pain point를 먼저 지적을 하면서 시작합니다.

  • Built on Node 0.10, and that ain’t changing anytime soon
  • Build system doesn’t allow for code splitting (the opposite, in fact)
  • Global scope (namespacing doesn’t count)
  • Goes Oprah-Christmas-special with websockets (not every person/page needs one)
  • Can’t handle css-modules (CSS is all handled behind the scenes)
  • Tied to MongoDB for official support

그리고 여러가지 feature들을 언급하는데, 제 눈에는 front end 단이 react로 되어 있는 점이 눈에 띄고, db도 mongoDB가 아니라 RethinkDB라는 점, 무엇보다 node5를 지원한다는 점이 눈에 띄네요.

1. 설치

  • brew install rethinkdb
  • npm i -g webpack@2.0.2-beta (optional, but recommended)
  • rethinkdb (in second terminal window)
  • git clone this repo
  • cd meatier
  • npm install
  • npm run quickstart

2. 실행

실행에는 두가지 옵션이 있습니다. 클라이언트 사이드 개발을 위해서는

$npm start

서버사이드 개발은

$npm run prod

명령을 치면 된다고 하는구요.

둘다 실행시키고 localhost:3000을 접속하면

와 같은 화면을 볼 수 있습니다.

redux 모니터가 켜져 있는 것을 볼 수 있네요.

arxiv.org
[1602.03355] Freezing and melting line invariants of the Lennard-Jones system

[ Authors ]
Lorenzo Costigliola, Thomas B. Schrøder, Jeppe C. Dyre
[ Abstract ]
The invariance of several structural and dynamical properties of the Lennard-Jones (LJ) system along the freezing and melting lines is interpreted in terms of the isomorph theory. First the freezing/melting lines for LJ system are shown to be accurately approximated by isomorphs. Then we show that the invariants observed along the freezing and melting isomorphs are also observed on other isomorphs in the liquid and crystalline phase. Structure is probed by the radial distribution function and the structure factor and dynamics is probed by the mean-square displacement, the intermediate scattering function, and the shear viscosity. Studying these properties by reference to the isomorph theory explains why known single-phase melting criteria holds, e.g., the Hansen-Verlet and the Lindemann criterion, and why the Andrade equation for the viscosity at freezing applies, e.g., for most liquid metals. Our conclusion is that these empirical rules and invariants can all be understood from the isomorph theory and that the invariants are not peculiar to the freezing and melting lines, but hold along all isomorphs.

Rapid Web Application Development With Meteor
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Meteor provides you with a fast development workflow that will create isomorphic web apps that ‘just work’. The Meteor architecture is truly beautiful in that it will update all clients connected to your app simultaneously, straight out of the box.

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Shift operators defined in the Riordan group and their applications

Publication date: 1 May 2016Source:Linear Algebra and its Applications, Volume 496
Author(s): Tian-Xiao He
In this paper, we discuss a linear operator T defined in Riordan group R by using the upper shift matrix U and lower shift matrix UT, namely for each R∈R, T:R↦URUT. Some isomorphic properties of the operator T and the structures of its range sets for different domains are studied. By using the operator T and the properties of Bell subgroup of R, the Riordan type Chu–Vandermonde identities and the Riordan equivalent identities of Format Last Theorem and Beal Conjecture are shown. The applications of the shift operators to the complementary Riordan arrays and to the Riordan involutions and Riordan pseudo-involutions are also presented.

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense.

Functional Analysis
Dr Joel Feinstein
Genre: Mathematics
Price: Get
Publish Date: July 9, 2010

Functional analysis begins with a marriage of linear algebra and metric topology. These work together in a highly effective way to elucidate problems arising from differential equations. Solutions are sought in an infinite dimensional space of functions. This module paves the way by establishing the principal theorems (all due in part to the great Polish mathematician Stefan Banach) and exploring their diverse consequences. Topics to be covered will include: – norm topology and topological isomorphism; – boundedness of operators; – compactness and finite dimensionality; – extension of functionals; – weak*-compactness; – sequence spaces and duality; – basic properties of Banach algebras. Suitable for: Undergraduate students Level Four Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951 Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.

Copyright © University of Nottingham

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense.

Functional Analysis
Dr Joel Feinstein
Genre: Mathematics
Price: Get
Publish Date: July 9, 2010

Functional analysis begins with a marriage of linear algebra and metric topology. These work together in a highly effective way to elucidate problems arising from differential equations. Solutions are sought in an infinite dimensional space of functions. This module paves the way by establishing the principal theorems (all due in part to the great Polish mathematician Stefan Banach) and exploring their diverse consequences. Topics to be covered will include: – norm topology and topological isomorphism; – boundedness of operators; – compactness and finite dimensionality; – extension of functionals; – weak*-compactness; – sequence spaces and duality; – basic properties of Banach algebras. Suitable for: Undergraduate students Level Four Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951 Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.

Copyright © University of Nottingham

Man Ray

Though informally related to the Dada and Surrealist movements, the American artist Man Ray contributed impressively to avant-garde, fashion and portrait photography, in particular with his solarised and isomorphic portraits of Lee Miller. Ray’s photomontages play with femininity and form, as in his multiple exposures of Alice Prin, better known as Kiki de Montparnasse, and Dora Maar.

Web developer at Rabalder Media (Stockholm, Sweden) (allows remote)

We’re looking for curious developers with a sharp eye for JavaScript. The most important characteristic we’re looking at is if you’re eager to learn and want to try new technologies. Rabalder Media hardly ever stands still in terms of technology and we want you to help us choose the right path to move forward on.
The minimum requirement is that you’ve worked extensively with JavaScript and know your way around the language, along with basic Git knowledge. Do check out our current stack and see if you find any match.


Merits if any of the items below matches you:


  • Experience with Node.js, MongoDB/CouchDB, Redis/Aerospike & Neo4j/PostgreSQL.
  • Experience with Aurelia/Angular/React/etc.
  • Experience with Stylus/LESS/SASS and Jade.
  • Tried out ES6 & ES7.
  • Understands the concept of Isomorphic code & Reactive Programming.
  • Experience with Gulp/Grunt and understands how they work.
  • Good understanding of how the browser works (headers, protocols, etc).
  • Experience with the Meteor framework is very valuable to us.
  • Experience with native app development using Apache Cordova.
  • Experience with optimizing for performance, stability & security.
  • Experience with Adobe Photoshop/Illustrator & UX.

Even if you don’t have experience with any of the above but only the bare minimum, don’t be afraid to apply - if you’re a fast learner we’re sure you’ll catch on real quick.
Education is always a plus, but we value self-taught developers equally as that shows passion and dedication.


With Rabalder Media, you’ll get the chance to develop your skills and learn lots of new ones. You’ll be working closely with everyone on our team, sit in Google Hangouts/Teamspeak throught the day chatting with the team about anything & everything. Our team is very nice, open and laid back - just be yourself.


Schedule is standard office hours (Swedish timezone) and we appreciate ability to work from our office but it is not required. We have extensive experience with co-workers working remotely.
We are currently only looking for Swedish candidates.



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Shift operators defined in the Riordan group and their applications

Publication date: 1 May 2016Source:Linear Algebra and its Applications, Volume 496
Author(s): Tian-Xiao He
In this paper, we discuss a linear operator T defined in Riordan group R by using the upper shift matrix U and lower shift matrix UT, namely for each R∈R, T:R↦URUT. Some isomorphic properties of the operator T and the structures of its range sets for different domains are studied. By using the operator T and the properties of Bell subgroup of R, the Riordan type Chu–Vandermonde identities and the Riordan equivalent identities of Format Last Theorem and Beal Conjecture are shown. The applications of the shift operators to the complementary Riordan arrays and to the Riordan involutions and Riordan pseudo-involutions are also presented.

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