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Richard's posts

The infatuation of a nation for a foolish minister is like that of a lover for an ugly woman: when he opens his eyes, he wonders what the devil bewitched him.

From a letter Sep/30/1755 from Horace Walpole to Richard Bentley.

From a letter Sep/30/1755 from Horace Walpole to Richard Bentley.

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It's alive! Let me present to you the

http://dmishin.github.io/hyperbolic-ca-simulator/index.html

Help page is here: http://dmishin.github.io/hyperbolic-ca-simulator/help.html

Though I hope that GUI should be self-describing. Use left mouse button to drag view, middle or left+shift to edit cells.

I humbly believe that it is the first simulator of such kind that supports:

- Arbitrary regular tilings

- Infinite world, limited only by memory

Additionally, it does not require installation and runs directly in the browser.

Of course, it is an open software, sources are on Github: https://github.com/dmishin/hyperbolic-ca-simulator

In the nearest few months, I am planning to spam from time to time about the simulator, its algorithms and my (not-so-exciting-yet) findings.

~~~

It was a long journey. Working in short spurs with long pauses, it took more than 2 years to bring the simulator to

its current more or less finished form. It is, without doubt, the most knowledge-loaded program I ever wrote.

It uses math from almost every area I ever touched: non-euclidean geometry (well, that's natural), linear algebra, numeric optimization,

finitely generated groups and triangle groups, and other things I forgot to mention.

While writing it, I learned a lot about groups, their presentations, word problem and related things, all new to me.

A big thanks to Markus Pfeiffer, whose DiplomarbeitPfeiffer.pdf actually helped me a lot, being more digestible source of information.

**simulator of cellular automata on hyperbolic plane**, supporting arbitrary regular tilings.http://dmishin.github.io/hyperbolic-ca-simulator/index.html

Help page is here: http://dmishin.github.io/hyperbolic-ca-simulator/help.html

Though I hope that GUI should be self-describing. Use left mouse button to drag view, middle or left+shift to edit cells.

I humbly believe that it is the first simulator of such kind that supports:

- Arbitrary regular tilings

- Infinite world, limited only by memory

Additionally, it does not require installation and runs directly in the browser.

Of course, it is an open software, sources are on Github: https://github.com/dmishin/hyperbolic-ca-simulator

In the nearest few months, I am planning to spam from time to time about the simulator, its algorithms and my (not-so-exciting-yet) findings.

~~~

It was a long journey. Working in short spurs with long pauses, it took more than 2 years to bring the simulator to

its current more or less finished form. It is, without doubt, the most knowledge-loaded program I ever wrote.

It uses math from almost every area I ever touched: non-euclidean geometry (well, that's natural), linear algebra, numeric optimization,

finitely generated groups and triangle groups, and other things I forgot to mention.

While writing it, I learned a lot about groups, their presentations, word problem and related things, all new to me.

A big thanks to Markus Pfeiffer, whose DiplomarbeitPfeiffer.pdf actually helped me a lot, being more digestible source of information.

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**Not Even Right**

A cautionary tale of just how good a model can be and still be wrong. The two galaxies on the right are a very famous pair known as Markarian's Eyes. It seems logical and sensible to assume that the weird structures have been formed by a collision between the two galaxies. Indeed, simulations (such as the one on the left) managed to reproduce very similar features using nothing more than two virtual galaxies and the absolute minimum of physics (i.e. gravity). Using more sophisticated models gave even better results (see link).

But this extremely simple and consistent picture turned out to be wrong - or at least woefully incomplete. Observations later revealed that there's a huge and spectacular stream of ionized gas linking these two galaxies to a third much larger galaxy, which was previously thought not to have been involved at all.

It's all too easy to assume that just because a model gets very precise details right, it must be the correct solution. In fact that's a necessary condition of a good model - but even reproducing very precise details is sometimes not enough to guarantee that you've come up with the true explanation. Sometimes even being right isn't good enough.

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A favorite book if mine for a long time.

Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games

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6:15 this morning at the "turn round and go back" point of my morning walk in South Redlands.

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Nice.

You have be careful how you define substitution when are variables bound inside the expression -- in classic math: integration and summation. In logic when there are quantified and definite descriptions. In CSci: in the lambda calculus. Found in several logic text books.

Also -- I've thought it would be nice to have a notation for equality under a mapping: x =_f y (Forgive my ΤεΚ)

Meaning f(x)=f(y).

Exercise: prove relation is an equivalence relation.

You have be careful how you define substitution when are variables bound inside the expression -- in classic math: integration and summation. In logic when there are quantified and definite descriptions. In CSci: in the lambda calculus. Found in several logic text books.

Also -- I've thought it would be nice to have a notation for equality under a mapping: x =_f y (Forgive my ΤεΚ)

Meaning f(x)=f(y).

Exercise: prove relation is an equivalence relation.

As a mathematician, it is sometimes hard to keep in mind that certain basic terms that have a very precise and accepted meaning in mathematics, can be interpreted differently by non-mathematicians. I was reminded of this recently when discussing one of the most fundamental mathematical notions, that of

0. Equality is a binary mathematical relation. That is to say, if x and y are two mathematical objects, then x=y is a mathematical statement (either true or false, depending on the precise values of x and y).

1. Equality is reflexive: for any mathematical object x, we have x=x.

2. The law of substitution (Leibniz's law): if x and y are mathematical objects with x=y, P is a statement, and Q is a statement formed from P by replacing one or more occurrences of x with y, then P is logically equivalent to Q.

From these laws one can deduce some further basic properties of equality, in particular

2'. If x and y are mathematical objects with x=y, A is a mathematical expression, and B is a mathematical expression formed from A by replacing one or more occurrences of x with y, then A=B.

3. Equality is symmetric: if x and y are mathematical objects with x=y, then y=x.

4. Equality is transitive: if x, y, and z are mathematical objects with x=y and y=z, then x=z.

These properties are completely self-evident and internalised to any working mathematician, to the point where one is probably not even aware of the dozens of times one uses these properties when writing a mathematical argument. However, these axioms serve a critical function of separating the mathematical conception of equality - in which x=y means that x and y have exactly the same value, and are interchangeable for each other in any mathematical expression or statement - with more informal interpretations of "x equals y" or "x is y" which are commonly accepted in English usage, but violate one or more of the above axioms of equality. For instance:

(a) Informally, "equals" or "is" is often used in the sense closer to the mathematical concepts of "is a subset of" or "is an element of", as in "cats are animals" or "36 = square number". This notion violates properties 2, 3, and 4 above: for instance, "cats are animals" and "dogs are animals" do not imply "cats are dogs". (But in analysis, we do "abuse" the equality sign in this fashion through notations such as the big-O notation, e.g. "X = Y + O(1)", or the ± notation, e.g. "x = ( - b ± sqrt(b^2-4ac))/2a". As with many other common abuses of notation, this is a situation in which the convenience and efficiency of the abuse can outweigh the technical violation of the formal rules of mathematical logic.)

(b) As a variant of (a), the symbol "=" is sometimes used informally to mean something like "has an attribute with value", e.g. "Alice = 23" to denote "Alice has age 23", or "AB=3" to denote "the line segment from A to B has length 3". As with (a), this usage tends to violate properties 2-4 above, but is often seen for instance in the homework assignments of undergraduate maths students, as a notational shorthand for a slightly longer, but more precise, English or mathematical sentence (e.g. writing |AB|=3 instead of AB=3); it may be formally incorrect usage, but the violation is often fairly harmless and can be properly understood from context.

(c) As further variant of (a), the symbol "=" is sometimes used informally to mean something like "is transformed into" or "has a consequence of", e.g. "Passing Go = $200", "water + heat = steam", or (as is sometimes seen in calculus homework assignments) "x^3 = 3x^2" or even "x^3 = 3x^2 = 6x". Again, this violates properties 2 and 3, and perhaps also 1, though one could argue that property 4 still survives.

(d) In some computer languages, the equals sign = is used as an assignment operator rather than a binary relation, basically violating all of the properties 0-4 above, e.g. "x=x+1" would be interpreted to mean "replace the value of x with the value of x incremented by 1". As this type of variable assignment is so at odds with the laws of first-order logic, its use is discouraged in mathematics (except perhaps in portions of a mathematical argument that are explicitly designated as "pseudocode" for describing an algorithm).

(e) The laws of equality prohibit distinct objects from being set equal to the same mathematical object; for instance, one cannot have a (non-degenerate) triangle whose vertices are given by A, A, and B, because the "location of A" then becomes ambiguous, in violation of the law of substitution. Similarly, one cannot discuss the concept of a set {2,2,3} consisting of three elements, two of which are the number 2 and the third is the number 3. However, it is perfectly permissible to have distinct objects being

(f) Occasionally, one sees the notion of equality used in a stronger sense than the mathematical notion of having equal value, in that one also requires equality of

A practising mathematician would implicitly know that the usage of the equality symbol = in mathematics is usually not intended to be of any of the interpretations (a)-(f) given above, but this seemingly obvious fact is not necessarily evident to non-mathematicians. Given how fundamental equality is to mathematics, this issue can be a real obstacle if one wishes to explain a mathematical argument to a non-mathematician...

*equality.*According to the laws of first-order logic, the equality symbol = obeys the following axioms:0. Equality is a binary mathematical relation. That is to say, if x and y are two mathematical objects, then x=y is a mathematical statement (either true or false, depending on the precise values of x and y).

1. Equality is reflexive: for any mathematical object x, we have x=x.

2. The law of substitution (Leibniz's law): if x and y are mathematical objects with x=y, P is a statement, and Q is a statement formed from P by replacing one or more occurrences of x with y, then P is logically equivalent to Q.

From these laws one can deduce some further basic properties of equality, in particular

2'. If x and y are mathematical objects with x=y, A is a mathematical expression, and B is a mathematical expression formed from A by replacing one or more occurrences of x with y, then A=B.

3. Equality is symmetric: if x and y are mathematical objects with x=y, then y=x.

4. Equality is transitive: if x, y, and z are mathematical objects with x=y and y=z, then x=z.

These properties are completely self-evident and internalised to any working mathematician, to the point where one is probably not even aware of the dozens of times one uses these properties when writing a mathematical argument. However, these axioms serve a critical function of separating the mathematical conception of equality - in which x=y means that x and y have exactly the same value, and are interchangeable for each other in any mathematical expression or statement - with more informal interpretations of "x equals y" or "x is y" which are commonly accepted in English usage, but violate one or more of the above axioms of equality. For instance:

(a) Informally, "equals" or "is" is often used in the sense closer to the mathematical concepts of "is a subset of" or "is an element of", as in "cats are animals" or "36 = square number". This notion violates properties 2, 3, and 4 above: for instance, "cats are animals" and "dogs are animals" do not imply "cats are dogs". (But in analysis, we do "abuse" the equality sign in this fashion through notations such as the big-O notation, e.g. "X = Y + O(1)", or the ± notation, e.g. "x = ( - b ± sqrt(b^2-4ac))/2a". As with many other common abuses of notation, this is a situation in which the convenience and efficiency of the abuse can outweigh the technical violation of the formal rules of mathematical logic.)

(b) As a variant of (a), the symbol "=" is sometimes used informally to mean something like "has an attribute with value", e.g. "Alice = 23" to denote "Alice has age 23", or "AB=3" to denote "the line segment from A to B has length 3". As with (a), this usage tends to violate properties 2-4 above, but is often seen for instance in the homework assignments of undergraduate maths students, as a notational shorthand for a slightly longer, but more precise, English or mathematical sentence (e.g. writing |AB|=3 instead of AB=3); it may be formally incorrect usage, but the violation is often fairly harmless and can be properly understood from context.

(c) As further variant of (a), the symbol "=" is sometimes used informally to mean something like "is transformed into" or "has a consequence of", e.g. "Passing Go = $200", "water + heat = steam", or (as is sometimes seen in calculus homework assignments) "x^3 = 3x^2" or even "x^3 = 3x^2 = 6x". Again, this violates properties 2 and 3, and perhaps also 1, though one could argue that property 4 still survives.

(d) In some computer languages, the equals sign = is used as an assignment operator rather than a binary relation, basically violating all of the properties 0-4 above, e.g. "x=x+1" would be interpreted to mean "replace the value of x with the value of x incremented by 1". As this type of variable assignment is so at odds with the laws of first-order logic, its use is discouraged in mathematics (except perhaps in portions of a mathematical argument that are explicitly designated as "pseudocode" for describing an algorithm).

(e) The laws of equality prohibit distinct objects from being set equal to the same mathematical object; for instance, one cannot have a (non-degenerate) triangle whose vertices are given by A, A, and B, because the "location of A" then becomes ambiguous, in violation of the law of substitution. Similarly, one cannot discuss the concept of a set {2,2,3} consisting of three elements, two of which are the number 2 and the third is the number 3. However, it is perfectly permissible to have distinct objects being

*labeled*by the same label, so long as one keeps the label of the object distinct from the object itself. For instance, one can certainly talk about an ordered triple (A,A,B), which can be thought of as a labeling of the set {1,2,3} in which 1 and 2 are assigned the label A and 3 is assigned the label B. (Similarly, given a careful definition of the notion of multiset, one can talk about a multiset {2,2,3} in which the number 2 occurs with multiplicity two and the number 3 occurs with multiplicity one.) So in formal mathematics one sometimes has to make a careful distinction between an object, and the name or label given to that object.(f) Occasionally, one sees the notion of equality used in a stronger sense than the mathematical notion of having equal value, in that one also requires equality of

*form*as well as*value*. For instance, with this notion, the fractions 2/4 and 3/6 would not be equal, because their forms are different (they have different numerators and denominators). As another example, with this interpretation, 2+3 is now not equal to 5 (the former is a sum, and the latter is not). This stronger notation of equality still obeys all the laws of mathematical equality, and can be accommodated by the device of introducing notions of "formal objects" (e.g. formal fractions, formal power series, formal polynomials, formal strings, etc.), and carefully distinguishing the formal object from their evaluations, e.g. distinguishing the formal fractions 2/4 and 3/6 from the real number 0.5 that they both evaluate to, or the formal string "2+3" from the number 2+3=5 that it evaluates to. This is necessary in order to preserve the law of substitution for mathematical concepts such as "numerator of a fraction" (or, for an example from more advanced mathematics, "degree of a polynomial" when working over a finite field). (One could argue that the failure to distinguish between a formal numeral and the number that it evaluates to is one of the reasons why mathematical identities such as 0.999... = 1 or 1+2+3+... = -1/12 can cause so much confusion.)A practising mathematician would implicitly know that the usage of the equality symbol = in mathematics is usually not intended to be of any of the interpretations (a)-(f) given above, but this seemingly obvious fact is not necessarily evident to non-mathematicians. Given how fundamental equality is to mathematics, this issue can be a real obstacle if one wishes to explain a mathematical argument to a non-mathematician...

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I guess I am to old but astrobiology is interesting..,

#REU : Summer Research Experience for Undergraduates - Applications are now being accepted!

General Information about the SETI Institute Astrobiology REU Program

What is it?

Students will work with scientists at the SETI Institute and at the nearby NASA Ames Research Center on projects spanning the field of astrobiology from microbiology to planetary geology to observational astronomy.

Who should apply

Current Sophomore and Junior Undergraduate Students who are United States Citizens or Permanent Residents

When

Applications for summer 2016

Program dates: June 12 - August 19, 2016.

Financial support

$5000 ($500/week for the 10 weeks of the program). In addition, participants will be provided with dorm housing. Travel reimbursement is up to $600 for travel from home or campus to the San Francisco Bay Area.

More info: http://buff.ly/1NgyMeL

General Information about the SETI Institute Astrobiology REU Program

What is it?

Students will work with scientists at the SETI Institute and at the nearby NASA Ames Research Center on projects spanning the field of astrobiology from microbiology to planetary geology to observational astronomy.

Who should apply

Current Sophomore and Junior Undergraduate Students who are United States Citizens or Permanent Residents

When

Applications for summer 2016

Program dates: June 12 - August 19, 2016.

Financial support

$5000 ($500/week for the 10 weeks of the program). In addition, participants will be provided with dorm housing. Travel reimbursement is up to $600 for travel from home or campus to the San Francisco Bay Area.

More info: http://buff.ly/1NgyMeL

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Get' em Thanksgiving !

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Trip to Disney Concert Hall yesterday afternoon. Incredible good acoustics. Incredible design for an organ.

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