soft question – Uses of the term "canonical"

This question is formulated jointly with Neil Barton.

In several mathematical fields, the term "canonical" appears with
with regard to objects, maps, structures and presentations. It is not
clear if there is something unmistakable signified by this term through
math, or if people just mean different things in different
contexts by the term. Some examples:

  1. In category theory, if we have universal property, the
    the single card is canonical. It seems here that the fact is that the card
    is only determined by certain data in the category. In addition, this type of schema can be used to select objects with certain properties which are canonical in the sense that they are unique up to isomorphism.

  2. In set theory, L is a canonical model. Here it is unique and definable. In addition, its construction depends only on the ordinals – two models of ZF with the same ordinals build the same version of L.

  3. In set theory, other models are called "canonical" but this is not
    clearly how it can be so, since they are not unique in some
    manners. For example, there is no analog of the above fact for L with respect to CFA models with an unlimited number of measurable cardinals. No matter how we extend the ZFC + theory "There is an appropriate class of measurables", there will be no single model of this theory until the specification of the ordinals plus a defined size parameter. See here.

  4. Presentations of objects can be canonical: being the simplest
    that of fractions, the presentation of which is canonical in case the
    the numerator and denominator have no common factors (for example,
    presentation of 4/8 is 1/2). But this also applies to other areas; see here.

  5. Sometimes the canonicity seems to be relative. Given a finite dimensional vector space, there is a canonical way to define an isomorphism between V and its double V * from a choice of a base for V. This determines a base for V *, and therefore the initial basic choice for V gives a canonical isomorphism from V to V **. But two stages can be more canonical than one: the resulting isomorphism between V and V ** does not vary with the choice of the base, and can indeed be defined without reference to any base. See here.

Our sweet questions:

(a) Does the term "canonical" appear in your field? If yes, what is the
meaning of the term? Is it relative or absolute?

(b) What role does canonicity play in your field? For example, does this help solve problems, set research goals or just make the results more interesting?

soft question – What is the mathematical branch on which one should focus more to better understand the other mathematical branches?

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How to modify the sunsets and sunrises to obtain a soft peach pink color compared to the usual orange?

Can you please help me figure out how to get the soft peachy pink coloring on the sand and light in the pictures below?

Usually my sunsets / sunrises are much more orange and when I try to edit them, I end up too reddish pink.

Peachy Pink Sunrise here

robots.txt – Google Search Console reports Soft 404 on an Excel file

since the middle of this month, an Excel file that was indexed in Google search is reported as a Soft 404 on the Search Console


We have several Excel, Powerpoint, Word, … files indexed and working according to so we do not know what is the origin of & # 39; error is for this specific file.

What we did to fix the problem

  1. Download file in multiple browsers by inspecting network traffic and response headers – same file returned, same response headers (as expected)
  2. Inspected URL in Search Console – this shows the Soft 404
  3. Checked with Google tester robots.txt to make sure Google sees the file – the answer is yes
  4. Checked HTTP response codes using – returns status 200 and the same headers we saw in step 1.
  5. Submitted the file for validation to Google – Google Search Console returned with "Failed to validate".

Before resubmitting for re-validation, I would appreciate if you have any ideas on what we may have missed in our testing process?

thank you so much

soft question – What are the research problems in functional analysis that can possibly be solved by a person with a basic knowledge of the subject?

I wanted to know if there were any problems in functional analysis (AF) that could be successfully solved by someone like me who had no expertise in this area but who only knew a few subjects basic that you would find in most undergraduate studies. Classes?

I wanted to mention that I browsed the web before posting here. There doesn't seem to be much that an undergraduate student can do in this area (or almost any other area), but I have seen articles from time to time. other researchers who, at the end of their articles, mention how their work can be used to do something (usually these are concrete applications or suggestions for working on specific examples), but the author doesn & # 39; Did not find the time or did not have the resources to do the work and it was left to the interested reader. I wanted someone to help me find these kinds of problems that it would be easy to work on if I give it time.

My goal is to write a research paper and have it published in an appropriate journal. I am not in school at the moment and I would like to be admitted to a good doctoral program. It is very difficult for someone like me to get the attention of a professor to take me as a doctoral student without first proving that I am motivated and able to do the job. work in FA.

Thank you for your time and your help.

usability – Doubt whether Soft UI meets accessibility criteria

If you just want to check the visual side of accessibility (since your examples are not interactive prototypes, so they cannot be tested for accessibility completely), you can go to WCAG 2.1, by example to see if the colors and sizes are good enough.

Target sizes

For sizes, to reach the "AAA" level (which, no doubt, means that the buttons have good accessibility), they say:

The target is available via a link or equivalent control on the same page which is at least 44 by 44 CSS pixels;

The two of your examples are 148 × 148 pixels CSS, so the two buttons pass this test.

Color contrast

For color contrast, there are two tests. Level AA requires:

The visual presentation of text and text images has a contrast ratio of at least 4.5: 1, with the exception of the following:

Large-scale text and large-scale text images
contrast ratio of at least 3: 1;


So, for the first example (the "blend" button), the contrast ratio between text and background is 5.09: 1, which goes easily.

For the second ("pure soft UI"), it's 4.79: 1, which is good too.

Now there is a second test, for the AAA level:

The visual presentation of text and text images has contrast
report at least 7: 1, with the exception of the following:

Large-scale text and large-scale text images have a contrast ratio
at least 4.5: 1;


And your two buttons fail.

I would worry about the lack of cue linked in the second example – the borders help users understand how big the target is, so they certainly help make your design more accessible. You should probably consider testing at least the contrast for these too.

So, in the end, I would say that these two button visuals fail at the "AAA" level of WCAG 2.1, but could conform to the "AA" level.

Please note that the accessibility of a UX is not only about visuals, if we test buttons, the test requires interactivity and, preferably, user tests.

soft question – Polynomial regression with periodicity constraints. Can we find an optimal basis?

Imagine that we want to make linear least squares adjusted over a certain interval $ t in (0,1) $,
$$ min _ { bf v} { | { bf Phi v -d} | _2 ^ 2 } $$
for a polynomial $ t to p (t) $ :

$$ p (t) = sum_ {k = 0} ^ M { bf v} _k t ^ k $$

but with additional constraints:

$$ p ^ {(k)} (0) = p ^ {(k)} (1) hspace {1cm} forall k in \ {0, cdots, N \} $$

In other words, no "jump" in the derivative of the function or it is $ N $ first derivatives.

Assuming that the degree of polynomial is $ M $, what values $ N $ would be possible?

How restrictive would that be for our polynomial?

Given $ M $ and $ N $, can we find a "better basis" in a certain sense to represent this polynomial?

I guess in the extreme case that $ N $ goes to infinity, $ M $ will necessarily have to grow to infinity also and only the remaining degrees of freedom would be equivalent to the expansions of power series for the basic function of the expansion of the Fourier series over this interval.

rt.representation theory – indecomposable modules of soft algebras

Let $ A = mathcal {k} Q / I $ to be a soft algebra (let $ mathcal {k} $ be algebraically closed). In the following article:

Butler and Ringel show that the chain and band modules classify the indecomposable modules of A (pages 157 – 161). To add a little more detail, for each string $ c $ of $ Q $ they produce a $ textit {string module} $ $ M (c) $. And for each cyclic chain $ b $ they produce a family of $ textit {tape modules} $ $ M (b, k, n) $ or $ k in mathcal {k} ^ * $ and $ n geq 1 $.

I'm trying to compare this to the classification of indecomposable representations of the 2-Kronecker quiver. But as an example, I don't see where the unstoppable representation $$ mathcal {k} xrightarrow (1) { xrightarrow {0}} mathcal {k} $$

appears in the classification of Butler and Ringel. What am I missing?

soft question – Modeling in pure mathematics

We all know that models play a major role in scientific practice. (By "model" here I mean different types of entities that function as representations or descriptions of real world phenomena. This includes images, diagrams, equations, concrete physical objects, fictitious or imaginary systems, etc.) Many models are valuable because they are simpler than their target systems, but they also generate intuition, understanding, predictions or useful explanations about the nature or behavior of these systems.

I am sure that mathematicians use models in a similar way, for similar reasons. But there is virtually no academic literature (to my knowledge) on the types of models found in pure mathematics, how and why they are used, how modeling practices in mathematics compare to those in empirical science, etc. (By contrast, philosophers have written extensively on scientific models.) As a philosopher interested in mathematical practice, this is something I would like to understand better.

So my question is: In what cases do mathematicians use models to better understand, predict or explain mathematical phenomena?

Some details on what I am looking for:

  1. I only ask for models in pure mathematics. In other words, the models in question should represent a mathematical object, fact or state of affairs, not an empirical object.
  2. I'm not necessarily nor even primarily interested in cases involving model theory. My notion of model is broader and more informal: pretty much anything M which can be used to give us a better idea of ​​a system of interest S, unless M satisfies a set of sentences in formal language associated with S.
  3. Models can be (but need not be) mathematical objects themselves.
  4. I have no particular preference for elementary vs sophisticated examples. Glad to see good clear cases.
  5. It would be nice to see a published source where a mathematician explicitly describes their methods as involving some kind of modeling, but this is not necessary.

gui design – Soft entry panel for 480×320 LCD screen

I am working on a C ++ project based on Qt. The project includes a 480×320 TFT LCD screen for display. The screen is also capacitive and used for touch input. The screen is the size of a credit card.

Qt does not provide a software input panel (SIP, the virtual keyboard). Qt offers a complete example of a QWERTY keyboard. Obviously, a full QWERTY will not be enough due to screen size constraints. I need to implement the SIP for the project.

I have consulted Google Scholar and articles on user interface design, small touch screens and virtual keyboards. I did not find any advice on the layout of the input panels. The touch interfaces for small touch screens from Poupyrev and Maruyama look very promising, but they lack useful details.

My question is, what do UI users recommend for SIP? More precisely:

  • How many screens
  • Key layout on the screen
  • How to transition between screens

The touch input is used for the initial configuration. Users will not need to enter data regularly while using it. That is to say., not a texting app.

The character set entered is that of a usual "strong" password.

  • Characters A – Z
  • Characters a – z
  • Characters 0 – 9
  • Characters with diacritical marks
  • Punctuation
  • Backspace and enter

Based on the initial design and testing, I think the screen limits the keys to a 9×6 arrangement, using 51 square pixels per key. Anything smaller than 51 pixels causes missed touch events with the help of a finger.

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I have no affinity with a QWERTY keyboard. In fact, I think it is unintuitive and difficult to use for someone who does not have frequent interactions with him, like an elder. I think there is a better way to group key collections (screens) and select groups (screen transitions).