User:NeilOnWiki/sandbox
This page contains my current list of tasks and draft edits. It may use WP:List of discussion templates. I tell myself it's a FILO queue; though it's starting to look like a FINO one (First In Never Out).
Title
Should we change the title to eg. Erlang traffic theory, or split off Erlang's formulae as a separate article?
Telecomms vs. more general queues?
Where's the context for this article? Telecomms or more general queues?
Reconciling with M/M/c queue
Regarding the present text on Erlang's equations:
- Erlang (unit) seems more for engineers, ie. a reader looking for practical applied solutions.
- M/M/c queue seems more for mathematicians and queue theorists: more abstract and within the historically more recent, wider context of queueing theory.
These are two articles with ostensibly the same subject, from two very different perspectives. For that reason, I'm not proposing to merge them, but others might argue differently.
(I'm also unsure that Extended Erlang B has a natural place in M/M/c queue; while it does seem to fit naturally in an article that features Erlang-B.)
Other issues
- Rename Section 1, eg. Forms or types?
- Importance of statistical equilibrium.
- Erlang (unit)#Calculating offered traffic is awkward. Better in Offered load?
- See Traffic measurement (telecommunications).
- Reconcile with Traffic intensity!
NeilOnWiki (talk) 17:39, 16 September 2019 (UTC)
Refer to Sylvester to show:
- diagonalisation of symm quad form;
- 'orthonormal' basis;
- k is unique.
Cf. Sylvester's law of inertia
Check Quadratic form article:
- compare with Pseudo-Euclidean space, esp Quadratic space;
- add link to Pseudo-Euclidean space.
NeilOnWiki (talk) 17:39, 16 September 2019 (UTC)
See Encyclopedia of Math which suggests:
- it's a general method for variational problems in an infinite- or high- dimensional target space;
- which gets an approx solution by searching for a best candidate in a smaller-dimension space spanned by a set of test functions.
Is it really for "boundary value problems"? — This seems too broad.
The Wiki article overlaps with Rayleigh–Ritz method. Both articles could be a lot more accessible.
To my mind, when the functional is a Rayleigh quotient, it makes more sense to use the name Rayleigh–Ritz method, which more people are likely to hear about.
The Talk suggests a merge:
FOR: the example minimises a Rayleigh quotient;
AGST: the Intro works;
AGST: Ritz is an umbrella that covers Rayleigh–Ritz, so they're not equivalent;
UNK: perhaps QM practitioners call it Ritz rather than Rayleigh–Ritz;
UNK: check link with FEM.
This is a quite demanding example:
- in a fairly specialised field, ie. QM;
- and not fully worked through.
NeilOnWiki (talk) 17:39, 16 September 2019 (UTC)
See Blade element momentum theory
This seems to start well but loses its umph.
See Wind Turbine Blade Analysis using the BladeElement Momentum Method as a possible source doc.
NeilOnWiki (talk) 17:39, 16 September 2019 (UTC)
See Feed-in tariffs in the United Kingdom
Out of date and historical - warning box?
NeilOnWiki (talk) 17:39, 16 September 2019 (UTC)
Information system framework
Should {\displaystyle I:\mathbb {U} \rightarrow V_{a}} as written actually be {\displaystyle a:\mathbb {U} \rightarrow V_{a}} ? NeilOnWiki (talk) 21:43, 22 May 2020 (UTC)
Possible tasks
- Done! - reasons on Talk: Revert proof
- Done! - asked on Talk: Raise to at least Mid Importance?
- Done! - added minor headings: Consider comments in proof
- Consider small restructuring:
- Done! applications
- generalisations?
- infinite vs. finite?
- Done! Make Head more accessible (ie. less technical)
- Done! Shift formal statement from Head to Definition section in main body
- Rephrase Banach stuff in Head. Should it be there at all?
- Incorporate H's comments in Head
- bring integral forward
- mass example asap?
- lake example?
- unique ae illustrated by vertical drops in lake bed?
- Consider making Measurable space more accessible
- Done!-ish Bring finite g forward in proof:
- to ensure g ∈ F for F as is;
- o'wise allowing F to include extended functions is more delicate and less robust against future edits (BUT this is what links below seem to do); eg. showing g + ε 1B ∈ F for extended F cleaner (may need to show g finite ae)??;
- affects proof versions with and without Hahn decomposition.
- Note: I think it's possible to decouple g from F, hence make F a set of simple functions; which is closer to how I think a scientist would visualise what's going on.
- See eg:
- ...and:
- http://www.stat.yale.edu/~pollard/Courses/600.spring2017/Handouts/projection_6march.pdf — Neumanny
- http://www.dam.brown.edu/people/huiwang/classes/Am264/Archive/cond_expe.pdf — Neumanny
- http://www.dmi.unipg.it/candelor/lavori/lavoro44.pdf — Hahn (interesting; but dense)
- https://www.math.cuhk.edu.hk/course_builder/1415/math5011/MATH5011_Chapter_5.2014.pdf — Neumann
- https://www.cmi.ac.in/~prateek/measure_theory/2010-10-13.pdf — g maximal; sidesteps Hahn (clear; good comments; use link!)
- http://www.math.unl.edu/~s-bbockel1/922-notes/Radon_Nikodym_Lebesgue.html — Hahn; g maximal (poor formatting)
- https://www.math.ksu.edu/~nagy/real-an/4-04-rn.pdf — just Real or Complex functions?
- http://pioneer.netserv.chula.ac.th/~lwicharn/materials/Radon-Nikodym%20Theorem.pdf — Neumanny
NeilOnWiki (talk) 22:37, 16 November 2020 (UTC)
Drafts for Radon–Nikodym theorem
∅
Possible tasks
- Make more accessible
- Red Child's bike example — available in a Store for 3 sets
- Mathematical meaning/properties: eg. common multiples; ideals???
- Fix empty intersection
- algebraic identities eg. Cap (A union B) = (Cap A) cap (Cap B); or finite version
- Collins on intersection
- Add algebraic identities, eg. commutative, associative, distributive over union
- Subsume Simple theorems in the algebra of sets under Algebra of sets?
- See:
- Union (set theory) — poss exemplar
- Empty product
- Von Neumann–Bernays–Gödel set theory
NeilOnWiki (talk) 18:56, 13 December 2020 (UTC)
Drafts for Intersection
Yuk! RETHINK THIS! Eg. check Collins ref on Maths proj Talk page.
Talk...
Possible original research in "Nullary intersection"
I've tagged this with the Template:Original research inline as it isn't sourced and contradicts some of the established properties of sets, including those discussed for null intersection earlier in the article. In particular, if M is empty then it isn't true that "the intersection over a set of sets is always a subset of the union over that set of sets", so there's a strong ingredient of circularity here. Instead, we'd expect that intersecting over fewer sets would produce a more populous result and this would be maximised when M=∅.
We would, in contrast, expect that intersecting over fewer sets produces a more populous result: the elements have fewer conditions to fulfil, so we expect to end up with more of them. To explore this more formally, define a set function R(M) = ⋂M = ⋂ {A|A ∈ M}; so the original text is asserting that R(∅) exists and equals ∅. But, logically, the definition of arbitrary intersection implies that ⋂ (M ∪ {B}) = (⋂M) ∩ B = R(M) ∩ B ⊆ R(M).
When M is empty, ⋂ (M ∪ {B}) equates to B; so choose B non-empty, such as B = {Georg Cantor}, the singleton containing the single element Georg Cantor. Suppose R(∅) = ∅ as asserted, then combining with the right-most relation implies B ⊆ ∅, hence {Georg Cantor} = ∅. Not only is this set-theoretic heresy, but it is a contradiction.
NeilOnWiki (talk) 12:54, 17 January 2021 (UTC)
Nullary intersection
Considerations in the null case Note that in the previous section, we excluded the case where M was the empty set (∅). The reason is as follows: The intersection of the collection M is defined as the set (see set-builder notation)
If M is empty, there are no sets A in M, so the question becomes "which x's satisfy the stated condition?" The answer seems to be every possible x. When M is empty, the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set (the identity element for the operation of intersection) [1], which needs to be treated with some care.
Where the universe is not defined Unfortunately, attempting to construct a universal set in naive set theory leads to contradictions, such as Russell's paradox. In consequence, the most commonly adopted formalised set theory (ZFC) constrains which sets are allowable and the universal set does not exist. This line of reasoning means that, in the most general case, the intersection over an empty collection of sets is undefined.
Within a well-defined universe It may however be the case that only a specific universe of sets is being considered, such as the subsets of an existing set S.
Comparing with a nullary union
Unfortunately, according to standard (ZFC) set theory, the universal set does not exist. A fix for this problem can be found if we Another approach is to note that the intersection over a set of sets is always a subset of the union over that set of sets[original research?]. This can symbolically be written as
Therefore, we can modify the definition slightly to
In general, no issue arises if M is empty. The intersection is the empty set, because the union over the empty set is the empty set. In fact, this is the operation that we would have defined in the first place if we were defining the set in ZFC, as except for the operations defined by the axioms (the power set of a set, for instance), every set must be defined as the subset of some other set or by replacement.
NeilOnWiki (talk) 12:54, 17 January 2021 (UTC)
esp Definition for filter bases — limit points, "similar to". NeilOnWiki (talk) 10:24, 14 January 2021 (UTC)
eg. Pairing NeilOnWiki (talk) 10:24, 14 January 2021 (UTC)
Draft for WP:MSM
(See WP:Policies and guidelines.)
Definition of the limit of a function
There are two differing definitions of the limit of a function.
The preferred
NeilOnWiki (talk) 18:41, 11 March 2021 (UTC)
Draft for WT:MSM
From the WT:MATH discussion, now archived:
Definition of the limit of a function
NeilOnWiki (talk) 18:41, 11 March 2021 (UTC)
Notes
WT:MATH#Proposal: Demystify math written in symbols by including programming language style code side-by-side With respect to a style guide, that doesn't matter for your proposal yet. Style guides attempt to encourage consistency with what we have: the rules can only be made when the practice exists. — Charles Stewart (talk) 08:33, 22 April 2021 (UTC)
In (ε, δ)-definition of limit, limit of a function and several other articles, the limit of a function is defined as
Because of the condition I call this definition the "punctured definition".
The definition that I have learnt more than 40 years ago, is the "unpunctured definition"
From the WT:MATH discussion, now archived:
- Wikipedia readers must be warned that both definitions are commonly used.
- Both definitions are discussed in one of the articles, see Limit of a function#Deleted versus non-deleted limits where it is asserted (with citation) that punctured limits are "most popular".
- The question here is rather whether (e.g.) the Kronecker delta function δ 0 , x {\displaystyle \delta _{0,x}} {\displaystyle \delta _{0,x}} has a limit as x → 0 {\displaystyle x\to 0} {\displaystyle x\to 0} or not.
- in the unpunctured definition, the condition "the limit exists at a point of the domain" means the same as "the function is continuous at that point".
- Limit of a function#Deleted versus non-deleted limits link observes that the unpunctured definition interacts more nicely with function composition (my wording).
- generalised to open sets, neighbourhoods, nets, etc). The non-standard analysis topics
- the punctured definition vacuously implies (I think!) that if c is an isolated point, then any L in the codomain of f (and not just in the image of f) is a limit as x approaches c. The unique unpunctured limit is f(c).
- in modern English sources, the punctured definition is used almost universally, at all levels of mathematics.
- for functions . This definition means that every real number is a limit of at every point.
- we now have a continuous f where there's a limit but not a unique one (even though R is a Hausdorff space).
- we may need to pause before writing that in general a Real-valued function f is continuous at c iff the limit exists and equals f(c), because we might have to choose our phrasing more carefully to account for non-uniqueness if there's a possibility that c is an isolated point (as happens with c ∈ Z above).
- the Net article has a definition of limit with a punctured flavour for a function from a metric space to a topological space, which does ensure uniqueness when the codomain is Hausdorff. It agrees with the punctured ε-δ definition when c is a cluster point (limit point), but not when c is isolated. Instead, in effect it avoids the vacuous condition for an isolated point and implies the limit either doesn't exist or uniquely equals f(c). (As far as I can tell, although it's not developed there, the obvious unpunctured counterpart would be fully consistent with the unpunctured ε-δ definition for both kinds of point.)
- MOS:MATHS has a section on Mathematical conventions.
- help future editors by adding a summary of some of the less obvious implications of the punctured definition (notably for function composition and isolated points)
Question: Must c be a limit point of the domain D? And is the limit undefined if not (ie. if it's a an isolated point?
- Precise statement and related statements in (ε, δ)-definition of limit says that c is a limit point for both the Real and Metric Space cases. There's no explicit mention of isolated points.
- There's the same restriction and omission in Limit of a function § More general subsets.
NeilOnWiki (talk) 19:26, 10 March 2021 (UTC)
Question: Pole vs. princ value: see Euler–Mascheroni constant
NeilOnWiki (talk) 18:03, 4 May 2021 (UTC)
See Talk:Foundations of mathematics on:
Done Eudoxus in Foundations of mathematics#Ancient Greek mathematics — cite Russell or copy of Euclid so this isn't the editor's interpretation;
Done use of ZF, ZFC;
- any other stuff!
Consider:
- section on set theory;
- non-neutral framing as a crisis plus resolution.
Possible copyright concerns:
- https://en.wikipedia.org/w/index.php?diff=512057455 Revision as of 19:08, 12 September 2012 Spoirier~enwiki (BIG DEVELOPMENT first step)
- https://xtools.wmflabs.org/articleinfo/en.wikipedia.org/Foundations_of_mathematics
- https://sigma.toolforge.org/usersearch.py?name=Spoirier%7Eenwiki&page=Foundations_of_mathematics&server=enwiki&max=
- Wikipedia:Copyright_violations
- Wikipedia:Close_paraphrasing
- Wikipedia:FAQ/Copyright
- Wikipedia:Text_copyright_violations_101
- Template:Close_paraphrasing
- Template:Copyvio-revdel
Use template {{close paraphrasing}} with params |Foundations of mathematics|source=https://www.britannica.com/science/foundations-of-mathematics |talk=Section name. Place at top of section or article.
Draft for Talk:
Apparent close paraphrasing
It looks to me as if this article has parts which are very close in phrasing and logical flow to the corresponding article on Encyclopedia Britannica. This is the first time I've come across this issue, so I may not have pitched it at the right level (it may be more or less severe than I've judged it to be). The originating edit was some time ago [1]: "Revision as of 19:08, 12 September 2012 Spoirier~enwiki (BIG DEVELOPMENT first step)". There are some further steps (similar edits) afterwards.
Here are a couple of text examples.
1. Britannica reads:
- example from source
The present day article reads:
- example from article
2. Britannica:
- ex
Present article:
- ex
There are other passages that similarly follow quite closely. I've left a message on the originator's talk page — their last edit is dated August 2013, so they may be no longer active.
The WP guidelines suggest an offending article "should be revised to separate it further from its source". (Although this is a slightly different issue, if rewriting proves necessary, I wonder whether the current framing as a historical narrative of crisis and resolution could be better replaced by a more neutrally pitched contemporary view of what we currently understand by mathematical foundations.)
Draft for contributor:
Apparent close paraphrasing
I've recently noticed that the Foundations of mathematics article you contributed to several years ago has parts which are very close in phrasing to the corresponding article on Encyclopedia Britannica. This can be a problem under Wikipedia's copyright policy and its guideline on plagiarism.
I've left further details on the article talk page. I don't know if Britannica and yourself were drawing from a common public source, which might explain the similarities.
Please get in touch if you have any queries. --
Leave for now
No As I'm sure you're aware, although facts are not copyrightable, creative elements of presentation – including both structure and language – are.
No Wikipedia advises that, as a website that is widely read and reused, it takes copyright very seriously to protect the interests of the holders of copyright as well as those of the Wikimedia Foundation and our reusers. Wikipedia's copyright policies require that the content we take from non-free sources, aside from brief and clearly marked quotations, be rewritten from scratch. So that we can be sure it does not constitute a derivative work, this article should be revised to separate it further from its source. The essay Wikipedia:Close paraphrasing contains some suggestions for rewriting that may help avoid these issues. The article Wikipedia:Wikipedia Signpost/2009-04-13/Dispatches also contains some suggestions for reusing material from sources that may be helpful, beginning under "Avoiding plagiarism".
NeilOnWiki (talk) 13:41, 31 July 2021 (UTC)
I wonder if the following interpretation is correct. If so it would help to make this article more accessible, especially to a reader looking for a way to visualise the properties of continued fraction approximations to an irrational α.
The continued fraction approximation irrational number α can be visualised by superimosing an integer grid over a plot of y = αx.
- Vertices as convergents.
- Why convex.
- Picture?
NeilOnWiki (talk) 21:31, 30 June 2021 (UTC)
Can I find some refs? NeilOnWiki (talk) 12:14, 26 November 2021 (UTC)
Consider:
NeilOnWiki (talk) 10:53, 29 November 2021 (UTC)
Question: Can I use this?.. The Lebesgue integration article states that "For a measure theory novice, this construction of the Lebesgue integral makes more intuitive sense when it is compared to the way Riemann sum is used with the definition/construction of the Riemann integral. Simple functions can be used to approximate a measurable function, by partitioning the range into layers." Maybe my intuition is totally out of synch!
- (Copied) If one wants to, it's probably fairly easy to see how an approximation as horizontal slabs can be converted into one expressed as simple functions, either geometrically by dropping some verticals onto the x-axis or perhaps algebraically by the summation by parts described above. This might be needed to reconcile the diagram with the preceding quotation from Lebesgue that "I order the bills and coins according to identical values and then I pay the several heaps one after the other".
(Not copied) This last operation is illustrated in the diagram labelled "Approximating a function by simple functions" at the start of the Via simple functions section, except the text seems to suggest interpreting it the other way round!
Question: Is the article confusing because it juggles two points of view - one working from layers directly (subsequently summed using the improper Riemann integral), the other via simple functions per se? NeilOnWiki (talk) 22:22, 19 January 2022 (UTC)
Reminder Linearity Para 2. NeilOnWiki (talk) 11:47, 18 January 2024 (UTC)
∅
Possible tasks:
- Edit lead.
- Make more accessible to eg. someone who knows basic calculus.
- Consider autodidacticism vs. taught curricula.
See:
- Absolutely integrable function
- Fundamental lemma of the calculus of variations
- Calculus_on_Euclidean_space#Weak_derivatives
NeilOnWiki (talk) 07:55, 16 May 2024 (UTC)
Addenda
Quick drafts
∅
Cited
- ^ Megginson, Robert E. (1998), "Chapter 1", An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3