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数学笔记

Constructor theory counterfactuals global causality. Constructor is a coordinate space. It allows for remeasurement.

S3 portrayed by Three card Monte.

What group do we get if instead of shuffling three different chips, we let two of the chips be indistinguishable? We get S3 acting on the 3 possible states in 6 different ways. But if all three chips are the same, then we get just the identity.

Are there group representations that grow like the combinatorics of the Hermitian polynomials? (And thus the most trivial Zeng trees?)

Given R-modules (where R=CH) and S-modules (where S=CG) we get two adjoints that turn an H-module into a G-module. The right adjoint, the produced representation HomR(S,W) lets you add S homomorphism extensions externally. It takes W to the fixed points under G. If G is the trivial group, then derived functors are called cohomology of H. The left adjoint, the induced representation S XR W lets you multiply by S on the left, so that works internally, in the mathematical structure. It takes W to the quotient W/(1-g)W. If G is the trivial group, then derived functors are called homology of H. Left adjoints are right exact and right adjoints are left exact, so if the left adjoint equals the right adjoint, then they are exact, they preserve exact sequences. Exact sequences express divisions of everything internally, and adjoint strings express them externally. Are these two representations, in terms of increasing slack and decreasing slack?

So the equivalence of the left-adjoint and right-adjoint for turning an H-module into a G-module is the equivalence of external relations and internal structure. And this is the point of Schur-Weyl duality and it is reflected in the construction of the representation theory of the symmetric group and of the general linear group.

An inner product like <x,x'> from V and its dual V' divides up space into two subsystems. Adjoint functors coordinate the division.

Adjunctions are the basis for recurring chemical processes, biological, neurological processes.

Chaos is natural, such as the ugly variety of possible bases for a vector space. Order is not natural. Unitary matrix or operator implies orthogonality of basis which is orderly and not natural. It is also a sign of there being an observer. Adjunctions break down the orderly components needed to generate a coordinate system and impose order, to relate two subsystems, and so on.

Adjoint string - divisions, adjunction - topology - one perspective in a division, one variable added.

Consider how entropy relates to the transformation of chaos to order by way of adjunctions.

Math consists of a small set (twelve?) of ways of adding a node (variable, label, etc.) Consider the kinds of variables. And in each case removing that variable is trivial because the information is already explicit (forgetting it), whereas the adjoint functor of adding the variable is quite involved (a free construction). And each adjoint functor is in a different branch of math, thus they distinguish the branches of math. And in that branch of math you have to spend a month mastering the absolute basics to understand that trivial adjunction. But if you knew all of those trivial adjunctions then you could build up everything, all of math. You could also explain what is involved in building up, for example, coordinate systems, order out of chaos.

povms are a kind of generalization of observables - positive-operator valued measurements - where the condition of orthogonality for measurement observables is dropped

What does it mean that we establish a "best" basis so that a unitary matrix is possible? Does that happen with the collapse of the wave function? Does orthogonality make sense before the collapse?

A complex number is an example of the twosome. In the real term "all is the same" (it is self-conjugage) whereas in the imaginary term "opposites coexist" (there are distinct conjugates).

How do the Zeng growth trees code the decomposition of representations?

The permutation representation is just one dimension different than the (geometrical) standard representation. The former is the basis for the root system {$x_i$} and the latter is the basis for the root system {$x_i-x_{i+1}$}. How do the classical Lie families relate them? Is that related to Jucys elements? Given a vector subspace, its complement is not unique. Likewise, bases are not unique. However, there is a unique "best basis" which is orthonormal. (In what sense is that a universal property?) And with regard to that basis, the representation is in terms of unitary matrices.

Adjoint functors relate viewing forward and viewing backward as occurs with shifts of perspective in the foursome, fivesome, sixsome, sevensome. Are these shifts representations? Are adjoint functors representations? Do they represent divisions? What then are divisions?

The relation between self-adjoint (reals) and adjoint (complexes) gets repeated with the complexes and the quaternions, and with the quaternions and the octonions.

Is there a way that the octonions get identified with the reals? The eightsome = nullsome gets understood as a onesome? And the identification of nullsome with onesome is related to the field with one element. And we are left with exceptional Lie structures.

Self-adjoint operators play the role of the real numbers (being equal to their own "complex conjugate") and form a real vector space. They serve as the model of real-valued observables in quantum mechanics.

Hermitian adjoint The adjoint of an operator plays the role of the complex conjugate of a complex number.

If we think of the scalar product as summing over all elements of the group and thus ensuring group invariance, and if that scalar product is semilinear in the second argument, then inverting all of the group elements is the same as transposing and taking the complex conjugate of the matrix. Thus adjunction expresses this duality.

If a representation is defined with regard to an orthonormal basis, then it is a unitary matrix. We can think of a unitary matrix as expressing that a representation is defined with regard to an orthonormal basis defined with regard to the scalar product.

Self-adjoint means that it is "observable" because observability requires being able to approach it from both directions of causality, forwards and backwards, as per the critical point - the ambiguous point - of the fivesome.

GOD, CONTRADICTION AND PARACONSISTENCY- Sept 16 - 4pm CET - LARAWebiner Is the concept of God contradictory? But what a contradiction really is? Is it a basis or limitation of our thinking? Is the principle of non-contradiction a key to the understanding of reality? If it is, how can we use it properly? Can logical systems that relativize this principle, paraconsistent logics, as promoted by Newton da Costa, help us to clear the way and have a better understanding of God? These are some of the questions that will be addressed in this session, which will be based on some recent works by the speakers: - N.A.da Costa and J-Y-Beziau "Is God Paraconsistent?" http://www.jyb-logic.org/GOD - Jc Beall, "The Contradictory Christ" https://global.oup.com/academic/product/the-contradictory-christ-9780198852360 - Paul Weingartner, "Theodicy - From a Logical Point of View" https://www.peterlang.com/view/title/75192

Paraconsistent logic

JC Beall logician website