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).

Erlang (unit)

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)

Pseudo-Euclidean space

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)

Ritz method

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)

Blade element momentum theory

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)

Feed-in tariffs in the United Kingdom

See Feed-in tariffs in the United Kingdom

Out of date and historical - warning box?

NeilOnWiki (talk) 17:39, 16 September 2019 (UTC)

Rough set

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)

Radon–Nikodym theorem

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:
  • σ-finite measure
  • Hahn decomposition theorem
  • Signed measure
  • Freudenthal spectral theorem
• ...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

∅

Intersection (set theory)

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.

$\bigcap _{A\in M\cup \{B\}}A=\left(\bigcap _{A\in M}A\right)\cap B=R(M)\cap B\subseteq R(M).$

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)

$\bigcap _{A\in M}A=\{x:\forall A\in M,x\in A\}.$

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) , 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. This can symbolically be written as

$\bigcap _{A\in M}A\subseteq \bigcup _{A\in M}A.$

Therefore, we can modify the definition slightly to

$\bigcap _{A\in M}A=\left\{x\in \bigcup _{A\in M}A:\forall A\in M,x\in A\right\}.$

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)

Limit superior and limit inferior

esp Definition for filter bases — limit points, "similar to". NeilOnWiki (talk) 10:24, 14 January 2021 (UTC)

Pontryagin duality

eg. Pairing NeilOnWiki (talk) 10:24, 14 January 2021 (UTC)

Definition of the limit of a function

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

$\lim _{x\to c}f(x)=L\iff (\forall \varepsilon >0,\,\exists \ \delta >0,\,\forall x\in D,\,0<|x-c|<\delta \ \Rightarrow \ |f(x)-L|<\varepsilon ).$

Because of the condition $0<|x-c|,$ I call this definition the "punctured definition".

The definition that I have learnt more than 40 years ago, is the "unpunctured definition"

$\lim _{x\to c}f(x)=L\iff (\forall \varepsilon >0,\,\exists \ \delta >0,\,\forall x\in D,\,|x-c|<\delta \ \Rightarrow \ |f(x)-L|<\varepsilon ).$

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 $f\colon \mathbf {Z} \to \mathbf {R}$. This definition means that every real number is a limit of $f$ 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)

Particular values of the Riemann zeta function

 Question: Pole vs. princ value: see Euler–Mascheroni constant

NeilOnWiki (talk) 18:03, 4 May 2021 (UTC)

Foundations of mathematics

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)

Klein polyhedron#Relation to continued fractions

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)

Three Dialogues Between Hylas and Philonous

Can I find some refs? NeilOnWiki (talk) 12:14, 26 November 2021 (UTC)

History of calculus

Consider:

• Lebesgue integration;
• Partial derivative and Symmetry of second derivatives;
• Non-standard analysis.

NeilOnWiki (talk) 10:53, 29 November 2021 (UTC)

Lebesgue integration

 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)

Sign function

Discontinuity at zero

Although the sign function takes the value −1 when $x$ is negative, the ringed point (0, −1) in the plot of $\operatorname {sgn} x$ indicates that this is not the case when $x=0$. Instead, the value jumps abruptly to the solid point at (0, 0) where $\operatorname {sgn}(0)=0$. There is then a similar jump to $\operatorname {sgn}(x)=+1$ when $x$ is positive. Either jump demonstrates visually that the sign function $\operatorname {sgn} x$ is discontinuous at zero, even though it is continuous at any point where $x$ is either positive or negative.

These observations are confirmed by any of the various equivalent formal definitions of continuity in mathematical analysis. A function $f(x)$, such as $\operatorname {sgn}(x),$ is continuous at a point $x=a$ if the value $f(a)$ can be approximated arbitrarily closely by the sequence of values $f(a_{1}),f(a_{2}),f(a_{3}),\dots ,$ where the $a_{n}$ make up any infinite sequence which becomes arbitrarily close to $a$ as $n$ becomes sufficiently large. In the notation of mathematical limits, continuity of $f$ at $a$ requires that $f(a_{n})\to f(a)$ as $n\to \infty$ for any sequence $\left(a_{n}\right)_{n=1}^{\infty }$ for which $a_{n}\to a.$ The arrow symbol can be read to mean approaches, or tends to, and it applies to the sequence as a whole.

This criterion fails for the sign function at $a=0$. For example, we can choose $a_{n}$ to be the sequence $1,{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\dots ,$ which tends towards zero as $n$ increases towards infinity. In this case, $a_{n}\to a$ as required, but $\operatorname {sgn}(a)=0$ and $\operatorname {sgn}(a_{n})=+1$ for each $n,$ so that $\operatorname {sgn}(a_{n})\to 1\neq \operatorname {sgn}(a)$. This counterexample confirms more formally the discontinuity of $\operatorname {sgn} x$ at zero that is visible in the plot.

Despite the sign function having a very simple form, the step change at zero causes difficulties for traditional calculus techniques, which are quite stringent in their requirements. Continuity is a frequent constraint. One solution can be to approximate the sign function by a smooth continuous function; others might involve less stringent approaches that build on classical methods to accommodate larger classes of function.

NeilOnWiki (talk) 16:54, 21 May 2024 (UTC)

Weak derivative

Possible tasks

• Edit lead.
• Make more accessible to eg. someone who knows basic calculus.
• Consider autodidacticism vs. taught curricula.

Misc notes

Permissible domains

I'm wondering if we can make the initial definition for functions on [a, b] more accessible to a wider readership. Unfortunately, there are aspects of the current wording that seem unclear, not least the relationship between this first definition in 1D and its generalisation in higher dimensions.

Do we ordinarily require a<b in the first definition? (The integrals involved do seem to make sense for a and b equal, so we may not need to.)

We don't specify the domain of φ in the 1D case. I imagine it needs to be defined on an open set containing [a, b] if it's to be differentiable there. It would be good to be explicit, even if there's some flexibility (eg. if it were sufficient for it to be infinitely differentiable in the interior of [a, b]).

Why does the definition section generalise from functions on a closed interval in R to functions on an open subset U of Rⁿ? These are two different (and seemingly incompatible) types of domain, even when n=1.

Should we insert an intermediate generalisation applying to functions on a compact subspace K of Rⁿ? (Assuming such a definition would be a recognised one.)

Or would it be better in the first definition to require that u, v and φ all be defined on the same open set U in R? This would bring it into line with the more general Rⁿ definition.

For later

In the latter case, I'm also guessing (perhaps wrongly!) that there may be an equivalent definition of the form that v is a weak derivative of u iff the integral identity holds for any closed sub-interval [a,b] within U in R and any infinitely differentiable φ on U with φ(a)=φ(b)=0.

Also, as far as I can tell, the Lebesgue space L¹([a, b]) is simply the space of Lebesgue-integrable functions on [a, b]. We don't appeal to any other L^p spaces in the article or L¹ spaces of functions on domains of dimension n>1. This being the case, I'd like to reword the first definition so that it's more widely accessible to mathematically literate readers, who might understand calculus but may not think in terms of function spaces or be overly familiar with measure theory.

See...

• Absolutely integrable function
• Fundamental lemma of the calculus of variations
• Calculus on Euclidean space#Weak derivatives

NeilOnWiki (talk) 11:32, 20 May 2024 (UTC)

Addenda

Quick drafts

∅