Theory U Example

[Lorriane Nasuti]

Responding analytically to texts is a complex activity containing multiple components, many of which are abstruse for novice learners. If you try to describe these elements, you are forced to use abstract phrases like sophisticated analysis, judicious use of quotations and, in the absence of examples, these terms merely serve to mystify the process further. This is the language of mark schemes, terminology that may make sense to experts but leaves novices confused. Creating worked examples-in English this may mean sentences, paragraphs or essays-exemplifies these opaque terms, converting the abstract into the concrete.

If we accept the notion that short-term memory capacity is pretty much fixed as well as the idea that we cannot really teach generic higher order thinking skills , then building domain specific background knowledge may be our most important job as teachers. Studying worked examples is more effective and efficient than merely attempting problems: deconstructing and studying model sentences, paragraphs and essays should, in the long run, be superior to merely writing them.

Great blog Tom though I must say that my own investigations into CLT urge more caution. It works in specific domains of learning but to apply it across the board is unproven to say the least.
IMO teaching is a balance of skulls.

Although theory in terms of science is used to express something based on extensive research and experimentation, typically in everyday life, theory is used more casually to express an educated guess.

In the scientific method, the hypothesis is constructed before any applicable research has been done, apart from a basic background review. You ask a question, read up on what has been studied before, and then form a hypothesis.

A theory, in contrast, is a principle that has been formed as an attempt to explain things that have already been substantiated by data. It is used in the names of a number of principles accepted in the scientific community, such as the Big Bang Theory. Because of the rigors of experimentation and control, it is understood to be more likely to be true than a hypothesis is.

This mistake is one of projection: since we use theory in general to mean something lightly speculated, then it's implied that scientists must be talking about the same level of uncertainty when they use theory to refer to their well-tested and reasoned principles.

There are many shades of meaning to the word theory. Most of these are used without difficulty, and we understand, based on the context in which they are found, what the intended meaning is. For instance, when we speak of music theory we understand it to be in reference to the underlying principles of the composition of music, and not in reference to some speculation about those principles.

In science, however, a theory is much more than just a guess. A theory presents a concept or idea that is testable. Scientists can test a theory through empirical research and gather evidence that supports or refutes it.

There are five major types of psychological theories: behavioral, cognitive, humanistic, psychodynamic, and biological. Let's take a closer look at each of these psychological theories and how they work.

Cognitive-behavioral therapy (CBT) is an important type of therapy that is rooted in these cognitive psychological theories. CBT focuses on helping people change their thoughts, which can help alleviate behavioral and emotional problems.

I am reading texts about (co)ends, and everywhere I see a lack of examples. I am not an expert in this area, and without examples it is difficult for me to use my intuition to grasp the idea. MacLane and Loregian mention some examples in passing, in particular, the so-called "geometric realization"$$\int^n (Sn)\cdot\Delta n, $$and the "integral against measure"$$\int_X f(x) d\mu, $$but they don't give details, and I don't understand what is meant here.

Not necessarily the examples that MacLane and Loregian mention, but just examples with accurate formulations: "if we take this bifunctor, then this construction will be the (co)end"... Of course, the more examples, the better. Thank you.

Edit. From people's comments here I see that I have to clarify that by examples I mean examples for non-specialists, namely, the constructions from other fields of mathematics that could be interpreted as (co)ends. The examples that I see up to now in the texts are methodical, they explain to specialists the details of the definition, and do not rouse interest of non-specialists.

At the same time, what could rouse this interest, are not examples of (co)ends: as an illustration, in Example 1.4.5 Fosco Loregian mentions the Stokes theorem, but, as far as I understand the construction he considers there is a cowedge, not a coend. So it remains unclear why (co)ends are important.

Gregory Arone gave a link to a thread where some informative examples are discussed. It will take me some time to analyze this, but I hope, people will add some more examples here if there indeed are some.

From what I understand, the OP wants a list of examples of (co)ends, where the notion is something familiar that doesn't require a background in category theory to understand. Something like this list of examples of adjoint functors, including free functors, functors of the form $-\otimes X$, induction $Ind_H^G$ from a subgroup $H

The geometric realization functor from topological spaces to simplicial sets, left adjoint to the singularization functor, is a coend $\int^n\in \Delta \Delta^n \times X_n$. Similarly, the categorical realization of a simplicial set, arising from $\tau_1: sSet\leftrightarrows Cat: N_i$, is a co-end, as is the coherent nerve. See Sec 3.1.2 and Examples 3.2.5, 3.2.6, 3.2.7 of Loregian's book. This is also a left Kan extension.

Reconstructing a $G$-space $X$ from its fixed points is a co-end $\int^H\in Orb(G) X^H \times G/H \cong X$. Similarly, one can apply co-ends to extensions in Galois theory; see Exercise 1.8 of Loregian's book.

Let $V$ be a finite dimensional $k$-vector space and let $V^\vee$ be the dual vector space of linear transformations $V\to k$. Then the co-end reconstructs $k$ from the vector spaces, as $\int^V V^\vee \otimes_k V \cong k$. This is Example 2.3.12 of Loregian's book.

As Loregian points out in a comment, "(Pointwise) Kan extensions are ends (on the right) or coends (on the left)" - see Chapter 2 of Loregian's book. So, the Dold-Kan correspondence from simplicial abelian groups to non-negatively graded chain complexes, is a left Kan extension, hence a co-end. See Example 3.2.10 of Loregian's book. Similarly, subdivision functors in simplicial contexts.

Building on the above, there are many examples of Kan extensions, including limits of functors (as right Kan extensions, hence ends), colimits of functors, direct sum, kernels and cokernels, extending presheaves (Example 2.7 in the link), etc. As the link says "any list is necessarily wildly incomplete."

The convolution product is a co-end; see Loregian Prop 6.2.1. Also, it's a left Kan extension. The composition product for symmetric sequences (used to define operads) is a co-end; see Chapter 6 of Loregian, and exercise 6.8.

Obviously, there are many, many more examples. But, I've run out of steam for now. I may add more later. Ends and co-ends are also useful for what they can prove, e.g., the Bousfield-Kan formulas for homotopy (co)limits.

I'm trying to learn class field theory and I'm wondering if anyone knows of any good sources with a bunch of examples on how to actually use it? This can be anything from books to course notes to course websites with solved homework. Interesting examples would be something like constructing specific extensions of $\mathbbQ$ and $\mathbbQ_p$, determining splittings of primes in more complicated extensions than the quadratics or anything else that is "concrete" where it might be useful.

My problems seems to be that while I can understand the actual statements it still seems like I can't see how to actually use it for any practical computations. By looking at the definitions it just seems like most objects aren't terribly computable. Most books just seem to have just some fairly trivial examples like e.g. finding the Hilbert Class Field of something like $\mathbbQ(\sqrt-5)$ and the examples exists to just give an example of some defined object and do not actually use the theorems for anything.

As I have written in your question on SE, if you want to know how to actually compute polynomials that give you ring class fields for a given modulus, then Cohen's Advanced Topics in Computational Number Theory is a very good resource. For a "real life example", you can have a look at section 3.1 of this paper, where I spell out how to find dihedral extensions of $\mathbbQ$ with a given intermediate quadratic (again, if you want polynomials generating these extensions, then see Cohen). See also the articles by Yui with various collaborators, many of which use and make explicit the constructions of class field theory.

In the London proceedings (Cassels-Froehlich), Tate and Serre have written some (classical) exercises regarding CFT (i.e. deducing higher reciprocity laws from Artin's reciprocity law, the Hasse-Minkowski theorem and a few others I can't recall right now).

I suggest taking a look also at the two books "A classical invitation to algebraic numbers and class fields" and "Introduction to the construction of class fields" by Harvey Cohn. While dealing only with global class field theory, they adopt a very concrete approach (much in the same spirit as the book of Cox in Some guy's answer) and (if memory serves me well) offer explicit examples of construction of class fields which are quite hard to find elsewhere.

You can prove that the class number of a cyclotomic number field of an odd prime order is divisible by that of its subfield by using class field theory.Since it has the unique quadratic subfield and its class number can be relatively easily computed when the discriminant is small, you can get useful information of the class number of the cyclotomic number field.

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