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1259:. This function satisfies the triangle inequality but is non-symmetric: rising the elevator always costs a lot of energy, but lowering it should cost less, or zero, or maybe we can even obtain energy from that operation. The important thing is that we can't obtain energy from travelling a closed cycle. So the relevant positivity axiom is
2877:, have length 12 and are the shortest path between the two points. The current description doesn't seem to mention anything about the taxicab metric that is actually important. However I'm hesitant to edit it since it's been there for so long without anybody noticing. Is there an important reason why it's like this?
2536:, first paragraph , if y is between z and x on the real number line, then d(x,y)≤max(d(x,z),d(z,y)) seems like the correct version. However due to my naivety with Maths, I have not made any changes to ether page concerning this. Regardless, I believe that one of the two pages is wrong concerning this.
234:
By the way, as one side-note, we would call this a non-negative real-valued function, not a positive definite function. The term positive definite is a linear algebra term for matrices; providing that distinction and link sends the reader off on a very unclearly connected idea. Let's just keep this
3045:
is a metric whenever, for any subset of three points or less, it is the case there is a subset of
Euclidean space of the same size that yields the same pairwise distances." (Your input here would be much appreciated.) And as to #2, you are absolutely right; I will look through my textbooks for the
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As it currently reads in
Knowledge, the distance induced by a quasi-norm is a semi-metric, and a distance induced by a semi-norm is a quasi-metric. This is confusing, and something should be done to fix this. Whatever the solution is, it should be consistent between generalized norms and generalized
1051:
It becomes important when you consider certain subsets of the axioms; for many such subsets there are names. Positive definiteness is even specifically mentioned in the article. Also in some axiomatisations non-negativity is built in even before the first axiom is mentioned, by saying that "a metric
792:
is PlanetMath. PlanetMath does not give a reference at all. The presyllable "hemi" is usually used as a synonym for "semi". The concept itself is often referred to as pseudo-quasi-metric, which is a bit longer but logical once pseudometric and quasimetric have been defined. Google has 852 hits. Some
214:
I think both of you are emphasizing different important ideas. As presently stated, it is NOT mathematically elegant (and actually inaccurate) to claim that all three axioms are responsible for the result that distinct points have positive distances. Here is why: Because the function is defined to
2928:
You need (1) to make clear the relevance of what you are talking about to distance functions, rather than to
Euclidean spaces (a different thing, and not the topic of this article), and (2) to back up all claims in whatever you add by reliable published sources. Also, obviously (3) to say something
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is
Arkhangelskii and Pontryagin: General Topology I. My library doesn't have the book, and it's not on Google books either, therefore I can't check that the strange word "prametric" really appears in the book. Google has 1660 hits for it. Each of the first 100 hits is either an obvious misspelling
176:
The first sentence in the definition of the metric explains about the range being positive, immediately explaining that 'distance' can't be negative. Then, property 1 of a metric says it's positive, which is actually specifying its range again. And then it says that non-negativity follows from the
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are the same. We have two separate articles for these topics. Is the lead correct? If that's the case, then these articles should be merged. Is metric tensor applied more typically in the context of differentiable manifolds, whereas metric is the more general concept, used usually in discrete
995:
A little update: I checked the definition with a book I'm using at my university (not a definitive resource as it is printed locally) and it states that there are three 'axioms of a metric' (identical to axioms 2, 3 and 4 here on wikipedia) and that nonnegativity follows from them (apparently all
224:
That said, what it SEEMS that one of you is trying to say is that if d were defined as a real-valued function without the non-negative range restriction, then all three axioms together would imply the positive criteria. I'm going to make a change to this effect to see if we can't make both
2759:
How does the
Minkowski metric fit in? It is called a metric on Minkowski space, but it does not appear to satisfy any of the definitions in this article. Each of the special flavors (pseudometrics, quasimetrics, metametrics, semimetrics, premetrics) include semi-positive definiteness
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of them are for "m-hemimetric", whatever that is, and I suspect that many are variants for "semimetric". Google
Scholar has 0 hits. Google Books has 2 hits: One for "m-hemi-metric", and one for "quasi-hemi-metric", a combination that makes no sense with our definition.
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The comment you supplied when reverting my edit is confusing to me, which leads me to the belief that my edit to the article was confusing to the reader. I will try for more clarity with my next attempt. Thank you for looking out for the quality of
Knowledge.
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for "parametric", or it's on
Knowledge or a clone. Google Scholar has 94 hits, every single one is an obvious misspelling for "parametric". Google books has, surprise, one relevant hit among 25 misspellings: Ruben Aldrovandi and J. G. Pereira,
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I see two problems here: a missing comma before "which", and ambiguity as to which noun phrase it qualifies. (It is intended to refer back to the "structure" mentioned.) Definitely needs rephrasing to avoid the ambiguity. I have done so. —
822:
Since all combinations of the axioms can be expressed by combinations of pseudo-, semi- and quasi-, there is no reason why
Knowledge should coin a strange new term such as "prametric". (Or help the translator of a Russian book to do so.)
2365:
The word "induce" is used on this and several related pages. Does this have some specific technical meaning? Is it related to the word "induction" in logic? Or is it perhaps a synonym for "implies", or just "results in" or ????
726:. May be his intentions were good, but now the articles create a confusing mess rather than a coherent mathematical discourse. Please someone can initiate a microproject to put some order/system into this topic? `'
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Not logical induction. In a mathematical context it seems to mean "implies the existence of". Or more informally, "produces". Though I don't see this precise meaning in the online dictionaries that I browsed. —
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Apparently d(x,x) = 0 is also required, although it's not stated explicitly. Thus the notion is identical with that of pseudoquasimetric or hemimetric, already discussed further down in its own section.
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I think that splitting was perhaps unfortunate, or could have been done better, as witness the section asking for synchronisation of content. I would suggest that if articles are spilt there should be:
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I think this splitting up is a good idea, but more cross-links are needed between these two pages (and they should refer to precise sections and/or examples, so section titles should be well-thought).
443:
The notion of 'metric induced by a norm' is already used 2 sections before(Examples) so there should be at least a forward reference (maybe better give the definition in a subsection of "examples")
981:
Correct me if I'm wrong but I believe that the first condition in the definition is redundant as it follows from identity of indiscernibles (condition 2) and triangle inequality (condition 4)
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Section linking between articles is a very bad idea, because the links are prone to breaking. If section linking cannot be avoided the two articles should be merged again. What do you mean by
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User:Valandil211 started changing html to inline TeX in the article. I strongly believe that there is no consensus for it, and I'd like to get some support to tell him to stop doing this. --
151:
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Misnomer: The condition that is relaxed in comparison to quasimetrics is positive definiteness, not positivity. Positivity, at least as used in this article, just says that d(x,y) : -->
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I guess the maximum that we need here is a mention that someone has called pseudoquasimetrics non-positive quasimetrics and maybe a link to the book for an example of such a thing.
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To show that mathematicians have used the term quasi-metric to refer either to a lack of symmetry, or to a relaxed triangle inequality, a brief search provides the following papers
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Why keep it if it's redundant? Perhaps it would be better style to keep 2-4 as axiom and remark that nonnegativity follows from them (i. e. is not an axiom but a useful property).
196:
below. my private opinion: non negativity is too fundamental to not mention it (and later the other axioms are removed), ps next time add new sections yat the end of the page :)
2615:
That is a misreading. They are not the same, and should not be merged. In particular, there are many examples of metrics and metric spaces that do not come from a metric tensor. —
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929:. The remaining question is whether to delete them altogether. I think there should be something like a notability criterion for definitions, and I suspect they wouldn't pass.
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A clear indication at the beginning of each article which other article(s) there are, and of what content can be found where, consistent with the principle guiding the split.
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be non-negative, d(x,y) can only be zero or positive. Now, axiom 1 rules-out distinct points being 0, thus forcing their distance to be positive--without axioms 2 & 3.
2324:"In differential geometry, the word "metric" is also used to refer to a structure defined only on a differentiable manifold which is more properly termed a metric tensor"
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are probably from me. I had intended to do a cleanup in the next few days. All your suggestions seem reasonable so go ahead. However I would keep the old section title
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1520:(Remark: Here when I say "energy" I mean 'useful energy', so heat doesn't count. So even if energy is always conserved, useful energy is conserved or diminished.)
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Perhaps some people who actually work in topology and functional analysis, or in related disciplines, can check whether the passages about the two definitions of
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really describe usage in their respective fields. I have also asked the MSC2010 group to clarify what the MSC2000 category "54E25 Semimetric spaces" is about. --
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As a sidenote, I think it is uneccessary to discuss such small modifications on the talk page. The next time you should be more bold and just do the changes.
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1564:. Smyth, M. (1987). "Quasi uniformities: reconciling domains with metric spaces". And I merged the example explained above into the text of the article.
1011:
It's easy to prove 1 from 2-4 (and an easy counterexample with two points shows just 2 and 4 are not enough). But this doesn't mean we need to drop 1. --
2327:
This claims that a differential manifold is more properly termed a metric tensor. I suspect "which is" was intended to refer back to the structure.
785:
So that's a total of two books defining the term. (The book on Google Books doesn't seem to do anything with it except that it derives a topology.)
429:
in "Definition", it is written "function function" - should we just delete the first "function", or was it intended to replace it with "distance" ?
826:
To be absolutely sure, I made a number of searches in publications. Numbers are for search without hypen/with hyphen, e.g. prametric/pra-metric.
2330:
Perhaps: "...used to refer to a structure defined only on a differentiable manifold. Such a structure is more properly termed a metric tensor"
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962:. "Praclosure" has 0 hits in Google Scholar and Google Books, but 167 on Google. All on Knowledge, including clones and translations. --
400:
In the case of this particular split I suppose the principle could be that variants on the concept and properties of the function go in
1581:
Does anybody know how a
Minkowsky space would fit in here? It seems that it drops 1,2, and 4 but I never found a mathematical name.
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coefficent to a specific geometric series. That's what's in my book, and I guess it's needed to ensure convergence of this sum -
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No more overlap between them than is necessary to make the individual articles readable without excessive jumping back and forth.
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772:. Most people are reluctant to remove something just because they have never heard of it. So am I, but this was a special case.
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2869:, and is the unique shortest path.” This is misleading, because the red, yellow and blue paths have the same length in taxicab
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The article mentions offhand, in the section on quasimetrics, that the quasimetric inducing the Sorgenfrey Line may defined as
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Section links need not get broken if you add anchors to the sections, and if people realise the implications of the anchors :(
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1) refers to a property "5" which is not defined ; please correct (and, if possible in 1 or 2 lines, write the proof).
408:. I am afraid I don’t have time at the moment to clean things up on this basis … perhaps later, perhaps someone else?
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I think these results show clearly that it is seriously misleading to direct unsuspecting readers to the articles on
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to the Euclidean metric on the plane: According to the taxicab metric the red, yellow, and blue paths have the same
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An easily formulated principle enabling an editor to decide which content belongs where, perhaps in an HTML comment.
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I suggest that at least "pseudo" and "ultra" should be defined in the same place than "intrinsic" (much more used,
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764:. Most of them should speak for themselves, but perhaps I should explain why I have removed all references to
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spaces? In other words, is there a distinction here, which should be pointed out? If not, why not merge?
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I am not fundamentally opposed, I just don't think it would be a clear improvement to drop the axiom. --
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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675:{\displaystyle d(x,y)=\sum _{n=1}^{\infty }{\frac {1}{2^{n}}}{\frac {p_{n}(x-y)}{1+p_{n}(x-y)}}}
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The picture right on the top of the page has this description: “An illustration comparing the
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at zero cost by doing nothing). The most similar concept described in this article is that of
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that is meaningful and correct mathematically. None of 1,2,3 was evident in what you added. —
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three). I will try to research this further but I have limited access to sources in English.
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The example is excessive compared to the other equally or more important generalisations.
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The added value (#1 and #3) was supposed to be an easier to read version of: a function
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Fixed point theorems in generalizing spaces of quasi-metric family and applications
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I also agree that there is no consensus, and I strongly prefer the inline TeX.
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1508:. By the way, I find the term quite descriptive, think of it: quASIMETRIC :-P.
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seem sort of arbitrary and bewildering; does anyone have a source for them?
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I think this new material is not appropriate, at least in the present form:
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Quasi-Metric Spaces, Quasi-Metric Hyperspaces and Uniform Local Compactness
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I agree that there is no consensus, and I don't agree with the change. --
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439:"Notes" (I would prefer "Comments") - here the last phrase (2,3,5 =: -->
2873:. The important thing is that with taxicab metric, all the four lines,
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The real number line is a metric space, but not an ultrametric space. —
722:, and maybe something else, have evolved independently after mathMarti
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is a metric (aka a distance function) if and only if for every subset
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The only frequently used terms for generalized metrics appear to be:
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as saying a non-negative function. Dr. Belnap 15:49, 7 May 2021 (UTC)
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2836:(12). According to the Euclidean metric, the green path has length
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be the space of possible states of a certain machine (for example,
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is the usual Euclidean distance function. Maybe I can say that "
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See my simultaneous contribution to the section on the split.
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2064:{\displaystyle {\sqrt {(t_{1}-t_{0})^{2}-(x_{1}-x_{0})^{2}}}}
319:. "Equivalent metrics" are now defined, they define the same
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be the infimum energy needed to take the machine from state
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source. (Your input here would also be much appreciated.) —
2801:). But the Minkowski metric is not semi-positive definite.
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Terminology is inconsistent with semi-norms and quasi-norms
1797:{\displaystyle t_{1}-t_{0}\geq {\sqrt {(x_{1}-x_{0})^{2}}}}
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Euclidean metric as the motivation for metrics generally
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A new approach to function spaces on quasi-metric spaces
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but that this 1 may be replaced with infinity or with
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The taxicab×Euclidean figure has a misleading caption
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Fixed Point Results in Dislocated Quasi Metric Spaces
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1603:Notice that if you can go from an event
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177:others. Sounds a bit superfluous to me.
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2406:= 0 for all x,y. This actually holds.
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1094:Consider the following example: Let
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1968:{\displaystyle e_{1}=(t_{1},x_{1})}
1909:{\displaystyle e_{0}=(t_{0},x_{0})}
1714:{\displaystyle e_{1}=(t_{1},x_{1})}
1655:{\displaystyle e_{0}=(t_{0},x_{0})}
460:I recently split this article from
49:It is of interest to the following
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1157:{\displaystyle X=\{0,\dots ,n-1\}}
950:PS: I was drawn into this because
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404:while properties of spaces go in
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2361:"induce" -- what does that mean?
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2528:Definition of Ultrametric
2071:. Separation satisfies a
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1501:{\displaystyle d(x,x)=0}
1334:{\displaystyle d(x,x)=0}
1084:11:51, 14 May 2008 (UTC)
1039:06:41, 14 May 2008 (UTC)
1021:23:38, 13 May 2008 (UTC)
1006:20:52, 13 May 2008 (UTC)
991:20:36, 13 May 2008 (UTC)
457:12:59, 19 Apr 2005 (UTC)
395:to the related articles.
341:13:28, 19 Apr 2005 (UTC)
304:12:27, 8 Apr 2005 (UTC)
152:project's priority scale
1382:because we can go from
436:), i.e. in "Definition"
109:WikiProject Mathematics
2863:
2825:
2795:
2745:
2700:
2507:On quasi-metric spaces
2483:Weighted quasi-metrics
2241:
2215:
2214:{\displaystyle s(y,x)}
2180:
2179:{\displaystyle s(x,y)}
2145:
2065:
1969:
1910:
1847:
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1715:
1656:
1558:
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1461:
1426:, but it does require
1416:
1396:
1376:
1335:
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1213:
1212:{\displaystyle d(x,y)}
1178:
1158:
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349:same should mean equal
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1804:), then the interval
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724:decided to split them
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712:Metric (mathematics)
706:Content syncronizing
546:
402:Metric (mathematics)
263:metric (mathematics)
248:metric (mathematics)
132:mathematics articles
18:Talk:Prametric space
2824:This is the figure.
2240:{\displaystyle x=y}
956:praclosure operator
952:preclosure operator
752:Generalized metrics
284:normed vector space
225:distinctions clear.
194:#Axioms of a metric
2979:such that for all
2968:injective function
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977:Axioms of a metric
958:and had a link to
954:was misspelled as
685:so I changed that
672:
280:norm (mathematics)
101:Mathematics portal
45:content assessment
2851:
2611:
2599:comment added by
2560:
2543:comment added by
2059:
1792:
1587:comment added by
1415:{\displaystyle x}
1395:{\displaystyle x}
1252:{\displaystyle y}
1232:{\displaystyle x}
1177:{\displaystyle n}
1107:{\displaystyle X}
919:
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325:topological space
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16:(Redirected from
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2966:there exists an
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2803:Klaas van Aarsen
2800:
2798:
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2755:Minkowski Metric
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323:, i.e. the same
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1116:
1115:
1096:
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960:prametric space
754:
708:
693:
644:
637:
611:
610:
596:
544:
543:
530:
516:
508:
497:
485:
483:remove fraction
464:so most of the
426:
269:. Reasons were
255:
174:
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99:
92:
72:
23:
22:
15:
12:
11:
5:
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2942:
2941:
2931:David Eppstein
2899:David Eppstein
2893:
2890:
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2830:taxicab metric
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2814:
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2632:
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2628:
2627:
2617:David Eppstein
2601:70.247.173.205
2581:
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2566:David Eppstein
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2013:
2007:
2003:
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1986:
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1942:
1938:
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1930:
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1779:
1775:
1771:
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1731:
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1697:
1692:
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1672:
1651:
1646:
1642:
1638:
1633:
1629:
1625:
1622:
1617:
1613:
1589:76.193.219.196
1579:
1578:
1577:
1576:
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1544:
1541:
1538:
1535:
1532:
1497:
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1371:
1368:
1365:
1362:
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1327:
1324:
1321:
1318:
1315:
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1309:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1248:
1228:
1208:
1205:
1202:
1199:
1196:
1193:
1173:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1103:
1091:
1090:
1089:
1088:
1087:
1086:
1072:
1071:satisfying..."
1068:
1052:is a function
1044:
1043:
1042:
1041:
1024:
1023:
978:
975:
917:
916:
913:
910:
906:
905:
902:
899:
895:
894:
891:
888:
884:
883:
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872:
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858:
855:
851:
850:
847:
844:
840:
839:
836:
835:Math. Reviews
833:
820:
819:
813:
804:
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753:
750:
749:
748:
707:
704:
689:
683:
682:
668:
665:
662:
659:
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635:
632:
629:
626:
623:
618:
614:
603:
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583:
580:
577:
573:
569:
566:
563:
560:
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551:
537:
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535:
534:
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512:
504:
495:
484:
481:
445:
444:
441:
437:
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389:
386:
383:
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328:
309:
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240:
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219:
218:
217:
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209:
208:
173:
170:
168:
164:
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144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
37:
24:
14:
13:
10:
9:
6:
4:
3:
2:
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3085:
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3077:
3076:
3074:
3067:
3066:
3062:
3058:
3054:
3051:
3044:
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3034:
3027:
3023:
3019:
3015:
3010:
3006:
3002:
2998:
2994:
2987:
2983:
2977:
2973:
2969:
2952:
2948:
2940:
2936:
2932:
2927:
2926:
2925:
2924:
2920:
2916:
2912:
2909:
2900:
2891:
2889:
2888:
2884:
2880:
2876:
2872:
2871:and Euclidean
2856:
2853:
2848:
2843:
2835:
2831:
2822:
2815:
2813:
2812:
2808:
2804:
2788:
2785:
2779:
2776:
2773:
2767:
2754:
2752:
2733:
2730:
2727:
2720:
2716:
2713:
2688:
2685:
2682:
2675:
2671:
2668:
2657:
2653:
2649:
2645:
2641:
2638:
2637:
2636:
2630:
2626:
2622:
2618:
2614:
2613:
2612:
2610:
2606:
2602:
2598:
2591:
2590:metric tensor
2587:
2579:
2575:
2571:
2567:
2563:
2562:
2561:
2558:
2554:
2550:
2546:
2542:
2535:
2527:
2525:
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2520:
2516:
2508:
2505:
2502:
2499:
2496:
2493:
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2478:
2475:
2472:
2469:
2466:
2463:
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2455:
2452:
2449:
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2404:
2403:
2402:
2396:
2392:
2389:
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2337:
2333:
2328:
2325:
2319:
2315:
2311:
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2301:
2297:
2293:
2289:
2288:
2287:
2286:
2282:
2278:
2274:
2270:
2262:
2258:
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2250:
2234:
2231:
2228:
2205:
2202:
2199:
2193:
2170:
2167:
2164:
2158:
2135:
2132:
2129:
2123:
2120:
2114:
2111:
2108:
2102:
2099:
2093:
2090:
2087:
2081:
2074:
2054:
2044:
2040:
2036:
2031:
2027:
2020:
2015:
2005:
2001:
1997:
1992:
1988:
1957:
1953:
1949:
1944:
1940:
1933:
1928:
1924:
1898:
1894:
1890:
1885:
1881:
1874:
1869:
1865:
1856:
1835:
1831:
1827:
1822:
1818:
1811:
1787:
1777:
1773:
1769:
1764:
1760:
1751:
1746:
1742:
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1733:
1729:
1703:
1699:
1695:
1690:
1686:
1679:
1674:
1670:
1644:
1640:
1636:
1631:
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1615:
1611:
1602:
1601:
1600:
1598:
1594:
1590:
1586:
1575:
1571:
1567:
1551:
1548:
1542:
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1536:
1530:
1522:
1521:
1519:
1515:
1511:
1495:
1492:
1486:
1483:
1480:
1474:
1454:
1451:
1445:
1442:
1439:
1433:
1425:
1409:
1389:
1369:
1366:
1360:
1357:
1354:
1348:
1328:
1325:
1319:
1316:
1313:
1307:
1287:
1284:
1278:
1275:
1272:
1266:
1246:
1226:
1203:
1200:
1197:
1191:
1184:floors). Let
1171:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1124:
1121:
1101:
1093:
1092:
1085:
1081:
1077:
1073:
1067:
1063:
1059:
1055:
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1049:
1048:
1047:
1046:
1045:
1040:
1036:
1032:
1028:
1027:
1026:
1025:
1022:
1018:
1014:
1010:
1009:
1008:
1007:
1003:
999:
993:
992:
988:
984:
976:
974:
973:
969:
965:
961:
957:
953:
948:
947:
943:
939:
935:
930:
928:
924:
914:
911:
908:
907:
903:
900:
897:
896:
892:
889:
887:pseudometric
886:
885:
881:
878:
875:
874:
870:
867:
864:
863:
859:
856:
853:
852:
848:
845:
842:
841:
838:Zentralblatt
837:
834:
831:
830:
827:
824:
817:
814:
812:
808:
805:
803:
800:
799:
798:
791:
787:
784:
779:
775:
774:
773:
771:
767:
763:
759:
751:
747:
743:
739:
735:
734:
733:
732:
729:
725:
721:
717:
713:
710:The articles
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703:
701:
697:
692:
688:
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660:
657:
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645:
641:
638:
630:
627:
624:
616:
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489:
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2431:metrics.
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923:prametric
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3061:contribs
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1585:unsigned
912:138/273
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150:on the
2974:: S →
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242:Split
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