User talk:LokiClock - Knowledge (XXG)

3459:, start with a normal 2-D vector space with a Euclidean metric. Start with the normal orthogonormal x–y basis, and express some fixed vector in terms of that basis. Then choose another basis that varies in the obvious ways: increase the length of the basis vectors, and find the coefficients required to express the fixed vector. Find the dual basis: those (co)vectors that when dotted with the basis in every combination produce the identity matrix. Use it to find the coefficients of the fixed vector. Change the angle of one of the basis vectors. See how the covector basis changes, and how the fixed vector's coefficients change for the new basis. If you're feeling energetic, play with three dimensions, though you'll get the idea from 2-D. Then play with a space with a 3178:

matched by the same rotation, in the same direction, of the dual basis. Any collective scaling of the basis is matched by an inverse scaling of the dual basis. Any distortion of the basis (e.g. a scaling on one axis) is matched by the inverse distortion (an inverse scaling on the same axis), so if the angle between two basis vectors is reduced, the angle between duals increases. As to overlap between the basis and its dual, this occurs with an othonormal basis and its dual in a Euclidian space. It never occurs for an indefinite or a negative definite metric. In these cases, you can still get the dual being identical to the basis except for the appropriate number of its vectors being of the opposite sign. It is usual (at least in physics) to broaden the term

2343:

Multiplying this by the unit vector will produce the unit again (p=1), but so it will for any covector corresponding to a vector with a length inverse to the cosine of the angle between it and the unit vector. Now I'm assuming that the failure of an inner product to support vector-covector identification on its own boils down to the lack of definition of length - Without an explicit map between vectors and covectors, i.e. a metric, I can't be sure that any vector corresponds to a covector, and thus I can't raise or lower an index.

2078:

And for any metric, the mapping of two vectors to a scalar is equivalent to defining a covector for the scalar, because in every basis the components have to satisfy the inner product (cancel) through regular matrix multiplication. If we have the vector in one basis, we can find the covector, but we don't need to find the covector every time, because the inverse relationship of the transformation law lets us know what form that covector will take if we forget about it while we manipulate the basis of the vector. ᛭

721: 2396:. Maybe I'm not interpreting the coefficient in the transformation law properly, because I see this as conflicting with the idea that tensors are independent of their choice of basis, the very fact I'm trying to illustrate with my proposed parallel. Or is that why you're pointing out that raising and lowering an index is linear? That it's a correspondence between vector and covector, but not the notion of covector-vector duality I'm looking for? ᛭ 3834: 470: 3247:−1 other vectors you choose to build the basis from. The component decomposition in terms of a basis is the same as linear decomposition of any linear system: you have a set of synthesis functions and a corresponding basis of analysis functions. Just like any transform: the analysis functions are the covector basis and the synthesis functions are the vectors. Anyhow, this seems to be rather a digression. 4128: 3257:

coefficients is immaterial. In a metric space, whether this is called covariant or contravariant becomes moot; the terms now only have significance with respect to what you label as your "vector" basis. You could just as easily use the vectors of your covector basis as your vectors with no effect. The only importance of having two bases of this nature then is to express the metric of the space.

1184:". Besides scholarly normalization, you encounter a specific normalized orthography, Standard Normalized Spelling, which is not flexible to the needs of individual texts like an arbitrary normalized spelling, and does not live up to modern standards of Old Norse normalization. I prefer ǿ myself, chiefly for consistency and connection with The First Grammatical Treatise orthography. ᛭ 3365:−1)-D space. The covectors are simply at the intersection of all the constraints for that covector. This means that the dual for an additional vector does not really make sense – what constraints would apply? The only sensible interpretation would be as though it were yet another basis vector, and the result is overconstrained: it has no solution. 298: 3863:. However, you changed the symbol of "left syntactic relation" (2nd def.), while the clarification request complained about "right syntactic equivalence" (1st def.) and "syntactic congruence" (3rd def.) looking the same. Therefore, in the new version, they still look the same. Maybe you intended to change the 1st or 3rd def.'s symbol? - 2795:, the metric is like the transition map between them, making them into a manifold where change of covariant basis eventually overlaps with change of contravariant basis. I don't know if they really overlap, but I'm trying to make a logical continuity between change of basis inside each space and change of basis between spaces. 3177:

I'll try to sketch typical transformations of the basis in a metric space. Imagine you have a vector basis and its covector dual. Being in a metric space allows us to refer to rotation, angles and length. Also, we can picture both the bases in the same space. Any collective rotation of the basis is

3134:

Yes, in Einstein notation the scalars with indices are the coefficients (or as everyone but me calls them them, components) for a specific basis. When you are dealing with numerical values, you have to make an actual choice of basis, but when you are dealing symbolically the actual choice can remain

1685:

Because a space of one-forms is a vector space, the dual to the all-covariant tensor should be the same as looking at the all-covariant tensor as an all-contravariant tensor in the dual space - each covariant index is contravariant relative to the dual space. So, if you represented each contravariant

1123:

To use the system, you can simply edit the page as you normally would, but you should also mark the latest revision as "reviewed" if you have looked at it to ensure it isn't problematic. Edits should generally be accepted if you wouldn't undo them in normal editing: they don't have obvious vandalism,

3333:

I followed for treating V×V as V×V after the metric, but not that the product of those two spaces is treated as if no complexity was added by the product. You allow a higher order of object, (1,1) tensors, but the (1,0) & (0,1) objects aren't orthogonal sets. They're redundant images of the same

2071:

Okay, so you can take that unique map, and take a covector and multiply it by a vector twice. The first application of the vector will cancel the stretching of the components performed by the change of basis, and the second will, in all, turn the covector into a vector. However, because we don't yet

3965:

gave differing pronunciations, and I'd rather not mix and match the pronunciations you guys offered. Since Nora hasn't been around for a couple years, I was hoping you could show me in this table how you'd render these names and patronymics in IPA. That would give me a simple and consistent list to

3040:

The easy questions first: Isolating the coefficients of a tensor with respect to a specific basis can be done via the dot product with each dual basis vector (this can be done for any basis, for a tensor of any order, with any mix of covariant and contravariant factor spaces). Anyhow, it should be

2716:

is replaced by a linear transformation (a matrix, and hence its inverse for transforming the coefficients). There is nothing magical about a dual basis any more than there is for an orthonormal basis: any basis for the dual space would do, regardless of what basis we use for the vector space. But

2077:

To identify them based on the scalar they produce by combination, we have to cancel out the effect their lengths have on the Kronecker delta, which in the case where the inner product is the dot product means dividing the result by the two vectors' lengths and only identifying them if they equal 1.

3622:

By the way, I don't suppose it's surprising that a person who (a) describes himself as liking to learn and think about languages should also (b) be intrigued by isomorphism. At least that's the way it appears to this correspondent who loves learning and thinking about languages and is intriuged by

1691:

Going on this I imagine raising and lowering indices as looking at pieces of an same object being pushed through a door between the collective vector and dual spaces, and when all the pieces are to one side or the other it looks the same from that side of the door (the two products of all non-dual

3291:

Right, like when they change basis and none of the vectors change relative angle or distance, the dual basis sees no inverse effect. Taking the dual transmits contextual information, so it wouldn't be well-defined for taking the dual of any single object. So if I have two vectors, is there an n-2

3210:

depending on the basis, but once used as coefficients for their respective basis, the objects should be the same again. Plus, I'm not sure if this fits with a tensor space as a product space, unless the product with the dual space is itself once they have a metric, not just that the space and its

1217:

Please accept my apology. I realise now that my editing behaviour in this case was highly questionable. I did not intend to start a flame war, and I can only blame this on my lack of Wikipedian experience (though I've had an account for several years, I've never been a prolific contributor). I'll

1127:

The "reviewers" property does not obligate you to do any additional work, and if you like you can simply ignore it. The expectation is that many users will have this property, so that they can review pending revisions in the course of normal editing. However, if you explicitly want to decline the

3266:

I'm a little lost by your comment about a product space; the tensor space of a given order is not closed under any product, if that us what you're talking about. The order of the tensor product is the sum of the orders; contraction reduces the order by 2. So the two products of vectors are the

1103:

system that is currently being tried. This system loosens page protection by allowing anonymous users to make "pending" changes which don't become "live" until they're "reviewed". However, logged-in users always see the very latest version of each page with no delay. A good explanation of the

3256:

I'm not too sure what you mean by active and passive transformations. As far as I'm concerned, changes between bases is not doing anything to the object you are transforming the components of: you are still representing the same object; what basis you choose to do so in terms of with a set of

3139:

and the fact that usually there is no need to be clear whether this is being used, and you can even mix them freely if desired. They even look identical, except for a subtle indicator such as the use of Latin vs. Greek indices. The down side is potential confusion if you do not know which is

1753:

and then applying a rotation will show the contravariant component moving against the rotation, and the covariant component moving in advance of the rotation (or is it equal?), but when you raise the index again it will be the same as if the rotation was applied with both indices contravariant

3173:

multiplied (via tensor product) with the term, and duplication of an abstract index implies contraction, not summation. Because the notation is so well-behaved, one generally does not have to worry whether an equation is as per Eintein (and hence dealing with scalars), or abstract (and hence

2342:

p is the vector I expect from combination with the corresponding covector, then I've found my covector. I'm saying my fallacy was that multiple combinations of vector and covector can produce the same scalar. Take for example the dot product over a unit vector and its corresponding covector.

428:

Well, I'm probably going to the University library here fairly soon, so I'll take another look at the Haugen book. However, since you're the one who added the name Thorodd to the article, the general burden for providing documentation on the Thorodd hypothesis would appear to fall on you...

1163:

to the transcription list? My authority for stating that this is a modern, scholarly way of representing /ø:/ in Old Norse is Terje Spurkland: "Innføring i norrønt språk", Universitetsforlaget, 9th edition (2007). This is the standard textbook in Old Norse at the University of Oslo. From a

3205:

If there's no sense of a vector in a basis corresponding to a covector in its dual basis, how can we say that changing to the dual basis is a passive transformation? I can understand if the co/vector is essentially untyped in a metric space, and only has covariant or contravariant

3216:

The basis overlap comment wasn't about an overlap between the basis and its dual basis, but an overlap in the set of bases for the spaces once the spaces are unified, so that change from covariant to contravariant basis is the same as some strictly covariant change of basis. ᛭

3130:

This feels a little odd, each basis vector having its specific counterpart covector in the dual basis, yet there is no such thing as duality of standalone vectors in this sense. Also not to be confused with a myriad independent senses in which the term "dual" is used.

3507:

If you want to undo the edit you can click "View history", then click on "00:22 5 December 2011, then click on "Edit" and then on "Save page". (Unless I have misunderstood what you wanted or the wiki software has been changed in some way I am not aware of.) - Haukur

2072:

know the lengths of the two vectors, the cancellation of the change of basis coefficients is not a property of a specific combination of a covector and vector, but will work for any of them - that they cancel does not identify the vector with the covector.

1198:

You see also at Old Norse orthography a "Standard Normalized spelling" column which perhaps should be accompanied by "First Grammatical Treatise orthography" and "General normalized spelling" columns for a fuller account of discrete orthographic norms. ᛭

584: 319:

located above the edit window. This will automatically insert a signature with your username or IP address and the time you posted the comment. This information is useful because other editors will be able to tell who said what, and when. Thank you.

4140:

is open from Monday, 00:00, 21 November through Sunday, 23:59, 4 December to all unblocked users who have registered an account before Wednesday, 00:00, 28 October 2016 and have made at least 150 mainspace edits before Sunday, 00:00, 1 November 2016.

3305:

If you have enough context to take the dual of the basis, why couldn't you piggyback the vector or covector onto the basis when you take the dual? If the operation's fully determined for a basis, it should be fully determined for a vector in a known

2733:

that is induced by the metric. It is so natural once we have the metric, we can treat the two spaces as the same vector space. In this sense the process becomes passive: we are simply choosing which basis we are expressing our tensor in terms of:

2337:

p is then vector. I'm attempting to convert the vector to a covector, change the basis, and convert the changed covector to a vector. My hypothesis is that the vector counterpart to the changed covector is the changed vector. I was assuming that if

3319:

I don't see what the metric is doing that allows you to still use the covariance and contravariance that specifies which space contributed an object, while treating the spaces as indistinct by making the phenomenon of contravariance and covariance

3454:

It is not the object that is covariant or contravariant; it is the basis and the coefficients. Although, without a metric, there seems to be a definite difference in what you can do with objects from the vector or covector space. To learn about

1686:

index by a vector, the covariant version would be that same set of vectors in the dual space. And vice versa - the level set representation of each covariant component is the same after taking the dual tensor and observing it in the dual space.

1683: 1284:~ perhaps we should insert the IPA transcriptions there and provide a link?. In his original article on the system Turrell V. Wylie did not detail the sounds of the letters - though he did mistakenly call the system he outlined Tibetan 3382:

The covariance and contravariance, when you have a metric, is still mathematically convenient, even though it is no longer necessary. It saves having to remember when to negate the square of a component. This convenience applies in

1512:

I see you've recently made changes to collapse(topology). If t is a face of s, t has already two cofaces if t and s are distinct. Should the definition read t is a free face os s if t and s are the only cofaces of t in the complex?

2724:

I think the answer to your question is that there is no one-to-one vector–covector duality; the duality is only between the vector spaces as a whole. The raising and lowering of indices is entirely related to a natural mapping

113:

What do you mean? That the distinction is only made in the spelling and no longer in the language? Also, there was a medial distinction between /f/ and /v/ ('sævar' vs. 'sofa') but there wasn't any final distinction, was there?

3360:

Each vector in the vector basis provides a one-D constraint on each of the dual vectors, producing a "flat" hyperslice oc covectors. The mapping must be either to 1 or to 0. So the first vector constrains each covector to an

1218:

revert my edit. I'd also like to thank you for your outstanding work on the Old Norse article, it's good that Anglophones take an interest in the classical form of the Nordic languages, as we sadly neglect it ourselves.

1111:

If there are "pending" (unreviewed) edits for a page, they will be apparent in a page's history screen; you do not have to go looking for them. There is, however, a list of all articles with changes awaiting review at

4168: 3914:. It has the authority to enact binding solutions for disputes between editors, primarily related to serious behavioural issues that the community has been unable to resolve. This includes the ability to impose 3292:

dimensional subspace that can be the dual of the set of vectors, or maybe it reduces each vector's corresponding covector subspace separately? Do we know the shape the subspace will have for some set of vectors?

3419:

Where can I learn to take a dual basis? Useful is one thing, but I don't see how it still exists. The covariant and contravariant objects are the same, yet they can still be distinguished by that property. ᛭

1240:

It's no big deal. Thanks for being swell and reasonable. Also I'd like to say that the tables on the Old Norse page are a bit old and misrepresentative to various degrees. I'm trying to make replacements on

2460:). Without the bold, we are referring to the tensor's coefficients for a specific basis. And for the time being, the only indexed tensors are our basis vectors and covectors, where we mean that there are 3199:

Okay, I just out about the swapping of upper and lower index between abstract index notation and Einstein notation, so the basis decomposition formulae are more consistent with my understanding of linear

1846: 376:

Well, Einar Haugen conspicuously refrained from endorsing any Thorodd theory, so I strongly doubted that it could be the clear mainstream scholarly consensus unless there was some recent discovery...

4152:. It has the authority to impose binding solutions to disputes between editors, primarily for serious conduct disputes the community has been unable to resolve. This includes the authority to impose 1444:

s in *fefall and *sat respectively. I also changed the separate links of causative weak verb to a section link, which gives exact forms for reconstructions and the umlauting suffix. Note that the

1751: 1528:

t is not a coface of itself, because the faces of an object are 1-lower-dimensional objects. If s is a triangular coface of t, t is an edge. If s is a tetrahedron, t is a triangle. ᛭

218:

Sure, if you want to slice it that way. Note that if I've commented on your talk page I'll be watching it, you don't need to specifically notify me of replies over on my talk page.

4209: 4202: 3182:

to allow these cases, where the basis vectors are orthogonal but the sqaure of these vectors may be in the set {+1, 0, −1}, thus allowing an "orthonormal" basis for any metric. —

2492:

entries. You will notice that the only time the order of a product in this notation is significant is in the tensor product. So I would put (being a little pedantic for now):

2755:

when we have a metric seem to be kept separate for formal reasons, but indices are raised and lowered without thinking in general, usually considering it as the same tensor. —

3475:. Here there is no orthonormal basis, only orthogonal bases in which one basis vector squares to +1 and the other to −1. Reading about the respective aspects will help. — 1334:

I removed it because the "Old Norse alphabet" doesn't exist. But I didn't take into account the consonants, so I'll put it back up until the consonant tables are created. ᛭

1080:

G'day, I've noticed that you're quite active on Old Norse articles, so I thought you might be interested in this template. Feel free to add, remove, or rearrange things.

1043: 3235:

Don't misunderstand me; I was trying to make the distinction between dual vectors in a basis and no dual for an arbitrary standalone vector that is not part of a set of

163:

All right, I would phrase that as "the distinction between medial /v/ and medial /f/ disappeared, though the distinction is made in normalized spelling". But note that,

1563: 1319:

idea of the consonants and the Latin graphemes that were used to represent them. This is not covered anywhere else in the article – only vowels are covered. Thanks,

1124:

personal attacks, etc. If an edit is problematic, you can fix it by editing or undoing it, just like normal. You are permitted to mark your own changes as reviewed.

2617:

is doing is what you'd expect from this notation with normal vectors: you are performing a projection (and a scaling), thus losing most of the information about

2820:

Now, I still have some issues of notational clarity. You're saying that in Einstein notation, choosing an index is choosing a basis? So the vector is by itself

1560:

My understanding is that a tensor with all the indices raised is contravariant and with all the indices lowered is covariant, and that the two tensors are dual

2488:. Use juxtaposition only to denote the scalar multiplication. And stick to the Einstein summation convention: repeated indices give an explicit summation if 2250:, which is one acting on the other (or the contracted tensor product) followed by scalar multiplication. Apply this with the basis and its dual, and you get ( 1409:

is already fronted, what back vowel got fronted? Maybe just a little more detail would clarify it for me. Also, could you put in some more detail about the

3243:−1)-dimensional subspace of the covector space that can be the covector to a given vector; which specific covector it turns out to be is determined by the 1067: 3812:

I will probably not be able to contribute significantly to those articles. I was just reading random JSTOR previews and noticed the statement I added. ᛭

1116:. Because there are so few pages in the trial so far, the latter list is almost always empty. The list of all pages in the pending review system is at 50:

Yes, the introduction seems to be an improvement. I'm still not entirely sure I know what it is, but I'm less unclear about how unclear I am now. ;-)- (

2096:

I'm sorry, I'm having difficulty following that. I'll try to illustrate the process that can be achieved, which is to get a basis and the dual of that

1029: 2835:

are components of the vector in a chosen co- or contravariant basis? Let me try and build on one of your formulas: We've chosen a metric, so that my

3456: 1449: 192:, /v/, so that /v/ was confined to being a medial allophone of /f/. I failed to note that it was a compound word, but you get my point, anyway. 3174:

representing the tensors themselves). I would not be surprized if many physicists are not clear on the distinction between the two notations.

2333:, which supports this notion of the inner product as a product of a covector and vector, not immediately one between two vectors or covectors. 2029:

Very important to understanding this: This is all without any reference to a metric: this all happens without defining the length of a vector.

137:

still as /ˈfɑː/. /f/ only occurs initially, and is otherwise /v/, so it's an initial vs. medio-final allophonic relationship, like that of ð.

4182: 3939: 3847: 3515: 520:

Huh? Sorry, but those comments are old, and I'm really not sure which comment you´re referring to. But I cannot see any claim from me that

483: 3972: 4208:

Because of your inactivity, you have been removed from the list. If you would like to resubscribe, you can do so at any time by visiting

2145:

specific points in the interval of the polynomial. By solving a linear equation, we can find the weight for each point in each covector

1397:

took a suffix which caused the /ɔ/ (or a different vowel in the past tense?) to front to /ɛ/, after which the suffix disappeared. But in

3015:? I may be able to show the components' correspondence but to demonstrate this graphically I'd have to choose the basis (repeatedly). ᛭ 1514: 2452:

We need to be careful with notation. Let's indicate a tensor in bold here, and use indices to indicate literal indexing (i.e. we will

3566: 3551: 3387:

and particularly in curved manifolds, where it is impossible to choose coordinates so that the metric takes a simple form globally. —

1965:. So there is not a linear relationship, but more like an inverse relationship. Raising and lowering indices is a linear relationship. 2195:. The coefficients (or "components" as they are rather unfortunately called) are just that: coefficients. A column of coefficients 2032:

You cannot have more than two of each index in a term, so you need to introduce an index with a different name in your last equation.

560:

only refers to those dialects. Of course you may think of a completely different comment, in which case my reply is pure nonsense ;)

3789:

article. Perhaps you'd be interested in improving it or know someone else who is. Similar pages that could use some attention are

248:

Alrighty, then. If you have any suggestions on how I could word that better, please let me know, or just clarify the text yourself.

2791:

Okay, beautiful. Thank you very much for walking me through this! I'm thinking, if the sets of bases for each space are like two

1280:(IPA) system. The IPA transcription of the sounds of the isolated Tibetan consonants more properly belongs in the article on the 316: 283: 356:

http://en.wikipedia.org/search/?title=First_Grammatical_Treatise&action=historysubmit&diff=328491616&oldid=327953691

4178: 3935: 2281:, which is simply mapping the covector basis back onto the original vector basis, the reverse map of getting the dual basis. — 2141:. Now imagine a covector: anything that linearly maps these vectors (the polynomials) onto scalars, perhaps a weighted sum of 1060: 451: 4161: 3953:

Hi LokiClock. A long time ago I asked for help in rendering a few Old Norse names into IPA and you kindly gave your thoughts (

2422:

My question boils down to, is raising and lowering an index an active transformation which I've mistaken for a passive one? ᛭

1757: 4108: 3840: 3833: 506: 476: 469: 3565:

I have now responded, but I will copy the discussion to the article talk page, which is a more appropriate place for it.--

3481: 3393: 3273: 3188: 2761: 2356:

Now, it seems like the idea that they still correspond after change of basis is flawed by the inverse transformation - if

2287: 2182:-dimensional vector space. Am I being clear about the vectors and covectors being abstract vector spaces? The vectors in 2112:

of the dual basis covectors. Think of the vector space as an abstract mathematical object; all you know is that it is an

2060: 3041:

clear that you get different scalar coefficients depending on whether you are dotting with the unprimed or primed basis:

634:

when they already have a subject-specific stub tag. And when you're adding stub tags, please put them at the end, as per

524:

wasn't used about the Danish language as it was spoken then. Icelandic court documents from the 10th century referred to

339:

I didn't remove any references, I merely moved the speculation about Þorodd out of the first paragraph of the article...

1861:

You need to distinguish between finding the dual of a vector, and raising and lowering indices. They are not the same.

944: 3926:

describes the Committee's roles and responsibilities in greater detail. If you wish to participate, you are welcome to

1881:(or ℝ if you prefer). This definition alone does not allow pinpointing a specific covector for any particular vector: 1100: 4173: 3931: 2464:

actual tensors; this is the only time bold and indexing should be combined here. Denote the inner product with a dot

1556:

I have some assertions based on my present understanding of tensors, and if they're wrong, could someone explain why?

1288:. Can we just put the IPA in the article on Tibetan script and then remove it in the Wylie transliteration article? 3588:

Hey, I really like your addition, distinguishing between "we share all properties" and "our structures share all of

3868: 1053: 926: 399: 4104: 1705:

Since the object itself isn't altered by a change of basis, only its matrix representation, lowering a component

1440:. They were not derived from "sat" or "fell" directly, but use the same ablauts of the original past tenses, the 1708: 500: 2133:) – the superscript here being a power, not an index – we can express any polynomial in the vector space using 2100:: a set of vectors and a set of covectors that together satisfy the duality requirement. The point is that in 4149: 3519: 3628: 3597: 3136: 2457: 2050:) and what that allows one to do, for example mapping vectors onto covectors (raising and lowering indices). 1518: 951: 565: 3511: 2843:

are intrinsically the same object, aliases used only for visual pairing with the covector and vector forms.

2721:

arises from its ability to represent the most general possible linear projection of a vector onto a scalar.

2378:(the covariant change of basis is a scaling of all basis vectors by k), then the covector corresponding to 306: 4095: 4085: 4075: 4065: 4055: 4045: 4035: 4025: 4015: 4005: 3995: 3570: 3555: 1113: 653:

Sorry about that, I didn't see. Also, I didn't know that (obviously) about stubs at the end, so thanks. ᛭

598:

I award LokiClock this barnstar for his enthusiastic contributions to articles on the Old Norse language.

107: 103: 99: 25: 3927: 302: 3970:

who regurgitated some of these. The last two are Old Norse forms of Gaelic names (the latter appears in

3864: 3751: 2215:

does. Now change any one of your basis vectors, and every one of the dual basis covectors might change.

1312: 1266: 1242: 1181: 1105: 826: 798: 358:), so if you don't know where that theory comes from, then maybe it should be removed from the article. 3954: 974: 956: 845: 681:

may result in it being deleted on Jun 18 2010, if there is no sourced content in it at that time. --

280: 59: 4136: 4118: 3903: 3894: 2000:, thus allowing us to define vectors to map covectors to the same scalar as the reversed operation: 496: 2330: 1479: 1426: 1324: 1223: 1170: 1117: 1085: 883: 4145: 3923: 3907: 2154:

so that it will produce exactly the corresponsding coefficient of the polynomial it is acting on:

635: 141:

would be pronounced /ˈsæː ˌwɑɾ/ before the merger, but afterwards would become like an endingless

3976: 3967: 3624: 3593: 1015: 1002: 868: 704: 686: 603: 561: 459: 434: 407: 381: 363: 344: 223: 172: 119: 84: 3592:

properties." That's a very nice way to explain the difference between equality and isomorphism.—

2174:. We can now express any covector as a linear combination of these weight combinations we call 1678:{\displaystyle T^{ijk}\in V\otimes V\otimes V=(T_{ijk}\in V^{*}\otimes V^{*}\otimes V^{*})^{*}.} 2717:

just like orthonormal bases, there is no generality lost and it is convenient. The utility of

1453: 1359: 1315:? Although the orthography of Old Norse was not completely consistent, the table still gave a 989: 765: 752: 325: 4157: 3919: 3911: 4223: 3860: 3747: 3607:

Thank you very much! I think it's what makes isomorphism deep, so I'm happy to hear that. ᛭

3460: 2792: 2305:

Yes, I am thinking of the vector space as arbitrary and abstract. Let me try more carefully.

1128:"reviewer" property, you may ask any administrator to remove it for you at any time. — Carl 1108:. The system is only being used for pages that would otherwise be protected from editing. 1099:

I have added the "reviewers" property to your user account. This property is related to the

1020: 984: 862: 4160:, editing restrictions, and other measures needed to maintain our editing environment. The 4153: 3922:, editing restrictions, and other measures needed to maintain our editing environment. The 3915: 3781:

Hi LokiClock. I noticed you are one of few active editors to have meaningfully edited the

167:

is a bit different - it's a compound word so the 'f' is pronounced /f/ in any time period.

3962: 3802: 3794: 3384: 3140:

intended, and there are times when you must be specific. So you could consistently write

1929: 1374: 1293: 1273: 979: 811: 277: 51: 36: 631: 4219: 3958: 3879: 3813: 3786: 3762: 3743: 3608: 3531: 3421: 3335: 3218: 3016: 2423: 2397: 2079: 1849: 1693: 1529: 1489: 1475: 1457: 1422: 1335: 1320: 1281: 1272:

IMO Putting Wylie and IPA together like this tends to confuse the difference between a

1246: 1219: 1200: 1185: 1166: 1081: 890: 816: 758: 734: 654: 643: 249: 193: 146: 532:. I'm perfectly aware that the language spoken in the north-germanic areas was called 73:

I love the IPA and use it. The issue is its appropriate use in appropriate contexts.

3790: 2036: 1277: 1147: 1135: 1007: 746: 682: 624: 599: 455: 430: 403: 377: 359: 340: 219: 168: 115: 80: 3742:. Ironically, two products with different order have exactly opposite signs, (i.e. 3662:

is an arbitrary abelian group with at least one non-zero element of order ≠ 2. Let

3476: 3388: 3268: 3183: 2756: 2282: 2055: 1355: 678: 354:

By the way, you were the one who first added the name "Þorodd" to the article (see

321: 4205:, but you haven't made any edits to the English Knowledge (XXG) in over 6 months. 4127: 2120:

over the interval . Once we have identified a basis, in this example it could be

76:

So I must not ignore other editors, but they may ignore me. How does that work?

3782: 3582: 1549: 908: 780: 770: 583: 313:( ~~~~ ) at the end of your comment. You may also click on the signature button 3798: 2218:

When you say "multiply it by a vector twice", I imagine you mean for covector

1448:

in *fefall, and thus fell, does not come from umlaut, but reduplication (see:

1289: 915: 32: 556:

and whether it can be used about Old Norwegian and Old Icelandic if the term

838: 833: 741: 712: 639: 3530:

Ah, thank you. Sometimes I see that warning and I forget it's possible. ᛭

189:

No, there is no medial /f/, except in compounds. There's medial <f: -->

4201:

You're receiving this notification because you were previously listed at

1421:

originally /man/ and not /mæn/, so that /a/ got fronted to /ɛ/? Thanks.

1143: 1131: 4164:

describes the Committee's roles and responsibilities in greater detail.

2035:

Once you have this distinction, you can look at introducing a metric (a

674: 297: 133:, for example, would have been pronounced /ˈvɑvl/ after the merger, and 4227: 4186: 4112: 3943: 3910:

is the panel of editors responsible for conducting the Knowledge (XXG)

3887: 3872: 3821: 3806: 3770: 3755: 3632: 3616: 3601: 3574: 3559: 3539: 3523: 3486: 3429: 3398: 3343: 3278: 3226: 3193: 3024: 2766: 2431: 2405: 2292: 2087: 2065: 1537: 1522: 1497: 1483: 1465: 1430: 1363: 1343: 1328: 1297: 1254: 1227: 1208: 1193: 1174: 1152: 1089: 997: 690: 662: 647: 607: 569: 510: 463: 438: 411: 385: 367: 348: 329: 315: 286: 257: 227: 201: 176: 154: 123: 88: 63: 40: 3966:

work with for articles relating to the 11th-century to 13th-century

3267:

scalar and order-2 tensors. The contraction requires the metric. —

3239:

vectors that consitute a basis. For a single vector you have an (

1180:

Note that the column in the table was renamed after that edit to "

804: 310: 2116:-dimensional vector space, for example the polynomials of order 900: 875: 305:

and Knowledge (XXG) pages that have open discussion, you should

79:

By the way, the correct idiom is "free rein," not "free reign."

4167:

If you wish to participate in the 2016 election, please review

3164:(the abstract index notation) may be interpreted as an implied 1984:. However, there is a unique (and hence natural) bijective map 3761:

Thank you. There was a hastily read definition behind that. ᛭

552:). As far as I can see I'm only discussing the correct use of 191:, /w/, merged with the sound represented by medial <f: --> 3850:

at any time by removing the {{Talkback}} or {{Tb}} template.

3832: 2685:

There is nothing magical here - it is simply expressing the

486:

at any time by removing the {{Talkback}} or {{Tb}} template.

468: 3638:"All automorphisms of an Abelian group commute" --LokiClock 2605:

Your description of the process is accurate enough. What (

1980:

to the scalars, and thus not technically the same space as

1552:, since it does not relate to the article or its editing. 4210:

Knowledge (XXG):WikiProject History/Outreach/Participants

4203:

Knowledge (XXG):WikiProject History/Outreach/Participants

2628:

may be more obvious when the notation is used carefully:

1841:{\displaystyle R_{j}^{i}T^{ij}=g^{jj}R_{j}^{i}T_{j}^{i}.} 301:

Hello. In case you didn't know, when you add content to

4148:

is the panel of editors responsible for conducting the

3785:

article. I've recently made some major changes to the

3642: 1377:, I still don't understand the umlaut progression from 1308: 1245:, but I haven't had time to do more than the vowels. ᛭ 355: 2108:

linearly independent vectors before you can determine

1760: 1711: 1566: 1164:

typographical point of view, I too prefer <œ: -->

2026:). So we simply define a vector as a co-covector. 1840: 1745: 1677: 3902:You appear to be eligible to vote in the current 2624:The transformation process and the invariance of 1873:is defined as a linear map from the vector space 1450:wikt:Category:Proto-Germanic class 7 strong verbs 673:This is a courtesy note to let you know that the 3135:unspecified. This is complicated by the use of 677:section you added is still empty, and that the 1932:. You will notice that of we find a new basis 1885:, only a dual vector space. Given a basis for 1550:Talk:Tensor#Understanding check on tensor type 1061: 398:Einar Haugen's book is already listed on the 145:, before and after pronounced /ˈsæː ˌvɑɾ i/. 8: 3839:Hello, LokiClock. You have new messages at 2178:. The polynomials could have been any other 1968:Taking this further, the dual of the dual, ( 1162:Why did you revert my addition of <ǿ: --> 475:Hello, LokiClock. You have new messages at 1746:{\displaystyle T^{ij}\rightarrow T_{j}^{i}} 1068: 1054: 699: 1829: 1824: 1814: 1809: 1796: 1780: 1770: 1765: 1759: 1737: 1732: 1716: 1710: 1666: 1656: 1643: 1630: 1611: 1571: 1565: 581: 3980: 3457:Covariance and contravariance of vectors 2186:are linear combinations of the abstract 1898:, we can find a corresponding basis for 516:Reply: Old talk: "Danish tongue" comment 3777:Bach, Cotton, Lanczos, Schouten tensors 711: 4218:Message delivered to you with love by 3861:Syntactic monoid#Syntactic equivalence 2480:) and the outer (tensor) product with 1452:). Yes, man was not front before the 1375:Germanic umlaut#Morphological effects 1044:WikiProject Norse history and culture 499:. Periods go inside the math tags. 190:. The sound represented by <v: --> 7: 4137:2016 Arbitration Committee elections 2331:inner product space#Related products 1165:, but that doesn't change the fact. 129:Yes, that is precisely what I mean. 4150:Knowledge (XXG) arbitration process 3746:of these is zero, not commutator). 1548:I've copied this request here from 2712:. In the general case the scalar 2199:does not constitute an element of 1953:, then we have for the duals that 1544:Understanding check on tensor type 14: 3928:review the candidates' statements 1140:12:33, 18 June 2010 (UTC) — Carl 4194:You have been pruned from a list 4134:Hello, LokiClock. Voting in the 4126: 3855:Syntactic monoid (clarification) 3841:The Great Redirector's talk page 2693:in terms of two different bases 719: 582: 314: 296: 110:to become merely etymological." 3878:I absolutely did, thank you. ᛭ 3550:I'll respond on my talk page.-- 2495:The inner product V × V → K of 1856:) 22:50, 19 December 2011 (UTC) 1700:) 22:50, 19 December 2011 (UTC) 1095:I have marked you as a reviewer 452:Talk:First Grammatical Treatise 4113:00:31, 17 September 2016 (UTC) 3934:. For the Election committee, 3904:Arbitration Committee election 3895:ArbCom elections are now open! 1725: 1663: 1604: 528:and Snorrí also used the term 1: 4187:22:08, 21 November 2016 (UTC) 3961:). The thing is that you and 3944:16:33, 23 November 2015 (UTC) 3633:18:42, 17 February 2012 (UTC) 3617:15:35, 17 February 2012 (UTC) 3602:15:17, 17 February 2012 (UTC) 3487:15:58, 27 December 2011 (UTC) 3430:12:06, 26 December 2011 (UTC) 3399:10:50, 26 December 2011 (UTC) 3344:08:14, 26 December 2011 (UTC) 3279:05:04, 25 December 2011 (UTC) 3227:23:54, 22 December 2011 (UTC) 3194:07:31, 22 December 2011 (UTC) 3025:01:41, 22 December 2011 (UTC) 2767:07:01, 21 December 2011 (UTC) 2432:23:46, 20 December 2011 (UTC) 2406:23:12, 20 December 2011 (UTC) 2293:21:16, 20 December 2011 (UTC) 2088:19:38, 20 December 2011 (UTC) 2066:05:51, 20 December 2011 (UTC) 1436:I changed the description to 1354:I responded on my talk page. 511:02:07, 31 December 2009 (UTC) 464:23:57, 28 December 2009 (UTC) 439:14:45, 28 December 2009 (UTC) 412:21:41, 27 December 2009 (UTC) 386:17:29, 27 December 2009 (UTC) 368:03:59, 27 December 2009 (UTC) 349:03:56, 27 December 2009 (UTC) 287:14:51, 30 November 2009 (UTC) 258:20:48, 29 November 2009 (UTC) 228:20:42, 29 November 2009 (UTC) 202:20:03, 29 November 2009 (UTC) 177:11:16, 29 November 2009 (UTC) 155:00:48, 29 November 2009 (UTC) 124:23:51, 28 November 2009 (UTC) 89:23:10, 10 November 2009 (UTC) 64:03:20, 4 September 2009 (UTC) 4222::) | Is this wrong? Contact 3888:02:55, 9 February 2014 (UTC) 3873:15:46, 8 February 2014 (UTC) 3859:Hi! Thanks for your edit of 3822:01:34, 20 October 2012 (UTC) 3807:23:41, 13 October 2012 (UTC) 3575:14:23, 24 January 2012 (UTC) 3560:13:55, 24 January 2012 (UTC) 3540:12:54, 24 January 2012 (UTC) 3524:12:49, 24 January 2012 (UTC) 1869:, the dual (covector) space 1538:23:15, 3 December 2011 (UTC) 1523:19:17, 3 December 2011 (UTC) 1344:16:02, 18 October 2010 (UTC) 1329:06:15, 18 October 2010 (UTC) 1182:Standard Normalized Spelling 638:, not at the start. Thanks. 608:20:24, 7 February 2010 (UTC) 570:21:45, 2 February 2010 (UTC) 330:09:04, 1 December 2009 (UTC) 4171:and submit your choices on 4092:Bjaðmunjo Mýrjartaksdóttir 3973:Ágrip af Nóregskonungasǫgum 3930:and submit your choices on 2318:The inner product V×V-: --> 1992:that allows us to identify 1298:10:11, 29 August 2010 (UTC) 679:discussion on the talk page 4243: 4179:MediaWiki message delivery 4169:the candidates' statements 3936:MediaWiki message delivery 1393:I think I understand that 1158:Old Norse orthography edit 927:First Grammatical Treatise 663:22:49, 21 April 2010 (UTC) 648:22:11, 21 April 2010 (UTC) 477:Sławomir Biały's talk page 400:First Grammatical Treatise 335:First Grammatical Treatise 41:17:21, 7 August 2009 (UTC) 3949:Old Norse IPA help needed 3771:00:11, 8 March 2012 (UTC) 3756:16:46, 7 March 2012 (UTC) 1889:, say the set of vectors 1153:13:29, 18 June 2010 (UTC) 1090:05:31, 17 June 2010 (UTC) 630:tags to articles such as 588: 4228:18:00, 1 July 2024 (UTC) 1498:02:49, 4 July 2011 (UTC) 1484:00:13, 4 July 2011 (UTC) 1466:00:56, 3 July 2011 (UTC) 1431:23:40, 2 July 2011 (UTC) 1364:10:16, 17 May 2011 (UTC) 1255:04:22, 9 July 2010 (UTC) 1228:21:14, 8 July 2010 (UTC) 1209:02:50, 8 July 2010 (UTC) 1194:02:46, 8 July 2010 (UTC) 1175:16:31, 6 July 2010 (UTC) 1114:Special:OldReviewedPages 691:15:28, 4 June 2010 (UTC) 253: 197: 150: 98:"and the distinction of 4072:Affrica Guðrøðardóttir 4032:Rǫgnvaldr Rǫgnvaldsson 3137:abstract index notation 2458:abstract index notation 1883:there is no dual vector 1865:Given the vector space 4052:Ragnhildr Óláfsdóttir 3837: 1859: 1842: 1747: 1679: 1438:from a past tense form 473: 26:Talk:Cupola_(geometry) 4146:Arbitration Committee 4119:ArbCom Elections 2016 3908:Arbitration Committee 3836: 1976:is a linear map from 1843: 1748: 1680: 1554: 1313:Old Norse orthography 1303:Old Norse orthography 1276:system (Wylie) and a 1267:Wylie transliteration 1243:Old Norse orthography 592:The Original Barnstar 472: 24:I anwsered to you on 4105:Brianann MacAmhlaidh 4022:Guðrøðr Guðrøðarson 3581:Your enhancement to 2104:dimensions you need 1758: 1709: 1564: 1508:topological collapse 4012:Haraldr Haraldsson 3912:arbitration process 2329:is a scalar p. See 1834: 1819: 1775: 1742: 1417:progression -- was 1307:Why did you remove 1118:Special:StablePages 1104:system is given in 945:Proto-Indo-European 94:Merely etymological 4162:arbitration policy 4121:: Voting now open! 3968:Kings of the Isles 3924:arbitration policy 3848:remove this notice 3838: 3211:dual are the same. 3160:, where the Latin 1838: 1820: 1805: 1761: 1743: 1728: 1675: 990:Greenlandic Norse 869:alliterative verse 536:by it's speakers ( 484:remove this notice 474: 4101: 4100: 4042:Magnús Magnússon 3514:comment added by 3485: 3397: 3277: 3192: 3001:How do I isolate 2765: 2472:or as a function 2291: 2064: 1454:Great Vowel Shift 1265:IPA removal from 1151: 1139: 1078: 1077: 753:Greenlandic Norse 620:Please don't add 613: 612: 540:being synomym to 292:Your recent edits 4234: 4130: 4097: 4087: 4077: 4067: 4057: 4047: 4037: 4027: 4017: 4007: 3997: 3981: 3865:Jochen Burghardt 3851: 3741: 3721: 3701: 3681: 3661: 3657: 3526: 3479: 3461:Minkowski metric 3391: 3271: 3186: 2759: 2285: 2058: 1847: 1845: 1844: 1839: 1833: 1828: 1818: 1813: 1804: 1803: 1788: 1787: 1774: 1769: 1752: 1750: 1749: 1744: 1741: 1736: 1724: 1723: 1684: 1682: 1681: 1676: 1671: 1670: 1661: 1660: 1648: 1647: 1635: 1634: 1622: 1621: 1582: 1581: 1350:Old Norse tables 1141: 1129: 1070: 1063: 1056: 1032:Old Norse origin 1030:English words of 1011: 993: 872: 723: 722: 700: 696:Old Norse topics 675:C++0x#Criticisms 629: 623: 586: 579: 578: 544:as in Icelandic 503: 495:Please refer to 487: 318: 300: 109: 106:for medio-final 105: 101: 4242: 4241: 4237: 4236: 4235: 4233: 4232: 4231: 4224:my bot operator 4196: 4191: 4190: 4174:the voting page 4131: 4123: 4002:Óláfr Óláfsson 3963:User:Nora lives 3951: 3932:the voting page 3898: 3857: 3852: 3845: 3830: 3795:Schouten tensor 3779: 3723: 3703: 3683: 3663: 3659: 3645: 3640: 3586: 3548: 3509: 3505: 3484: 3396: 3385:Minkowski space 3276: 3191: 3172: 3159: 3125: 3116: 3108: 3099: 3086: 3077: 3065: 3056: 3014: 2973: 2956: 2943: 2928: 2914: 2900: 2890: 2881: 2833: 2764: 2742: 2711: 2701: 2679: 2665: 2645: 2599: 2587: 2579: 2563: 2554: 2539: 2530: 2395: 2388: 2369: 2362: 2328: 2290: 2280: 2271: 2262: 2214: 2194: 2169: 2153: 2128: 2063: 2021: 2012: 1952: 1940: 1930:Kronecker delta 1927: 1918: 1897: 1792: 1776: 1756: 1755: 1712: 1707: 1706: 1662: 1652: 1639: 1626: 1607: 1567: 1562: 1561: 1546: 1510: 1371: 1352: 1305: 1274:transliteration 1270: 1160: 1101:Pending changes 1097: 1074: 1038: 1037: 1034: 1009: 991: 970: 962: 961: 940: 932: 931: 923: 922: 898: 897: 873: 866: 859: 851: 850: 823: 822: 812:Younger Futhark 794: 786: 785: 777: 776: 756: 751: 731: 720: 698: 671: 627: 621: 618: 577: 518: 501: 493: 491:Manual of style 488: 481: 337: 309:by typing four 307:sign your posts 294: 274: 96: 71: 48: 31:See you later! 19: 12: 11: 5: 4240: 4238: 4195: 4192: 4132: 4125: 4124: 4122: 4116: 4099: 4098: 4093: 4089: 4088: 4083: 4079: 4078: 4073: 4069: 4068: 4063: 4059: 4058: 4053: 4049: 4048: 4043: 4039: 4038: 4033: 4029: 4028: 4023: 4019: 4018: 4013: 4009: 4008: 4003: 3999: 3998: 3993: 3989: 3988: 3985: 3950: 3947: 3901: 3897: 3892: 3891: 3890: 3856: 3853: 3844: 3831: 3829: 3826: 3825: 3824: 3787:Lanczos tensor 3778: 3775: 3774: 3773: 3744:anticommutator 3639: 3636: 3620: 3619: 3585: 3579: 3578: 3577: 3547: 3544: 3543: 3542: 3516:157.157.183.55 3504: 3501: 3500: 3499: 3498: 3497: 3496: 3495: 3494: 3493: 3492: 3491: 3490: 3489: 3480: 3441: 3440: 3439: 3438: 3437: 3436: 3435: 3434: 3433: 3432: 3408: 3407: 3406: 3405: 3404: 3403: 3402: 3401: 3392: 3373: 3372: 3371: 3370: 3369: 3368: 3367: 3366: 3351: 3350: 3349: 3348: 3347: 3346: 3326: 3325: 3324: 3323: 3322: 3321: 3312: 3311: 3310: 3309: 3308: 3307: 3298: 3297: 3296: 3295: 3294: 3293: 3284: 3283: 3282: 3281: 3272: 3261: 3260: 3259: 3258: 3251: 3250: 3249: 3248: 3230: 3229: 3213: 3212: 3202: 3201: 3187: 3168: 3155: 3128: 3127: 3121: 3112: 3104: 3095: 3082: 3073: 3061: 3052: 3038: 3037: 3036: 3035: 3034: 3033: 3032: 3031: 3030: 3029: 3028: 3027: 3010: 2988: 2987: 2986: 2985: 2984: 2983: 2982: 2981: 2980: 2979: 2978: 2977: 2976: 2975: 2969: 2952: 2945: 2939: 2924: 2919: 2910: 2896: 2888: 2877: 2855: 2854: 2853: 2852: 2851: 2850: 2849: 2848: 2847: 2846: 2845: 2844: 2831: 2807: 2806: 2805: 2804: 2803: 2802: 2801: 2800: 2799: 2798: 2797: 2796: 2778: 2777: 2776: 2775: 2774: 2773: 2772: 2771: 2770: 2769: 2760: 2747:. The spaces 2738: 2722: 2707: 2697: 2683: 2682: 2681: 2675: 2661: 2641: 2622: 2603: 2602: 2601: 2595: 2583: 2575: 2559: 2550: 2535: 2526: 2441: 2440: 2439: 2438: 2437: 2436: 2435: 2434: 2413: 2412: 2411: 2410: 2409: 2408: 2393: 2386: 2367: 2360: 2349: 2348: 2347: 2346: 2345: 2344: 2326: 2311: 2310: 2309: 2308: 2307: 2306: 2298: 2297: 2296: 2295: 2286: 2276: 2267: 2258: 2216: 2210: 2190: 2165: 2149: 2124: 2091: 2090: 2074: 2073: 2059: 2052: 2051: 2033: 2030: 2027: 2017: 2008: 1966: 1948: 1936: 1923: 1914: 1893: 1858: 1857: 1837: 1832: 1827: 1823: 1817: 1812: 1808: 1802: 1799: 1795: 1791: 1786: 1783: 1779: 1773: 1768: 1764: 1740: 1735: 1731: 1727: 1722: 1719: 1715: 1702: 1701: 1688: 1687: 1674: 1669: 1665: 1659: 1655: 1651: 1646: 1642: 1638: 1633: 1629: 1625: 1620: 1617: 1614: 1610: 1606: 1603: 1600: 1597: 1594: 1591: 1588: 1585: 1580: 1577: 1574: 1570: 1545: 1542: 1541: 1540: 1509: 1506: 1505: 1504: 1503: 1502: 1501: 1500: 1488:Sure thing. ᛭ 1469: 1468: 1373:Hi, regarding 1370: 1369:set/sit umlaut 1367: 1351: 1348: 1347: 1346: 1304: 1301: 1282:Tibetan script 1269: 1263: 1262: 1261: 1260: 1259: 1258: 1257: 1233: 1232: 1231: 1230: 1212: 1211: 1196: 1159: 1156: 1096: 1093: 1076: 1075: 1073: 1072: 1065: 1058: 1050: 1047: 1046: 1040: 1039: 1036: 1035: 1031: 1028: 1025: 1024: 1023: 1018: 1013: 1005: 1000: 995: 987: 982: 977: 971: 968: 967: 964: 963: 960: 959: 954: 952:Proto-Germanic 948: 947: 941: 938: 937: 934: 933: 930: 929: 921: 920: 912: 903: 899: 896: 895: 887: 878: 874: 865: 861: 860: 857: 856: 853: 852: 849: 848: 842: 841: 836: 830: 829: 827:Latin alphabet 821: 820: 814: 807: 805:Runic alphabet 803: 802: 801: 795: 792: 791: 788: 787: 784: 783: 775: 774: 768: 761: 759:Old East Norse 757: 750: 749: 744: 737: 735:Old West Norse 733: 732: 729: 728: 725: 724: 716: 715: 709: 708: 697: 694: 670: 667: 666: 665: 617: 614: 611: 610: 595: 594: 589: 587: 576: 573: 517: 514: 502:Sławomir Biały 492: 489: 480: 467: 448: 447: 446: 445: 444: 443: 442: 441: 419: 418: 417: 416: 415: 414: 402:article page. 391: 390: 389: 388: 371: 370: 336: 333: 293: 290: 273: 270: 269: 268: 267: 266: 265: 264: 263: 262: 261: 260: 237: 236: 235: 234: 233: 232: 231: 230: 209: 208: 207: 206: 205: 204: 182: 181: 180: 179: 158: 157: 95: 92: 70: 67: 47: 44: 18: 15: 13: 10: 9: 6: 4: 3: 2: 4239: 4230: 4229: 4226:. | Sent at 4225: 4221: 4217: 4213: 4211: 4206: 4204: 4200: 4199:Hi LokiClock! 4193: 4189: 4188: 4184: 4180: 4176: 4175: 4170: 4165: 4163: 4159: 4155: 4151: 4147: 4142: 4139: 4138: 4129: 4120: 4117: 4115: 4114: 4110: 4106: 4094: 4091: 4090: 4084: 4081: 4080: 4074: 4071: 4070: 4064: 4062:Óspakr-Hákon 4061: 4060: 4054: 4051: 4050: 4044: 4041: 4040: 4034: 4031: 4030: 4024: 4021: 4020: 4014: 4011: 4010: 4004: 4001: 4000: 3994: 3991: 3990: 3986: 3983: 3982: 3979: 3977: 3975: 3974: 3969: 3964: 3960: 3956: 3948: 3946: 3945: 3941: 3937: 3933: 3929: 3925: 3921: 3917: 3913: 3909: 3905: 3896: 3893: 3889: 3885: 3881: 3877: 3876: 3875: 3874: 3870: 3866: 3862: 3854: 3849: 3842: 3835: 3827: 3823: 3819: 3815: 3811: 3810: 3809: 3808: 3804: 3800: 3796: 3792: 3791:Cotton tensor 3788: 3784: 3776: 3772: 3768: 3764: 3760: 3759: 3758: 3757: 3753: 3749: 3745: 3739: 3735: 3731: 3727: 3719: 3715: 3711: 3707: 3699: 3695: 3691: 3687: 3679: 3675: 3671: 3667: 3656: 3652: 3648: 3643: 3637: 3635: 3634: 3630: 3626: 3625:PaulTanenbaum 3623:isomorphism.— 3618: 3614: 3610: 3606: 3605: 3604: 3603: 3599: 3595: 3594:PaulTanenbaum 3591: 3584: 3580: 3576: 3572: 3568: 3564: 3563: 3562: 3561: 3557: 3553: 3545: 3541: 3537: 3533: 3529: 3528: 3527: 3525: 3521: 3517: 3513: 3502: 3488: 3483: 3478: 3474: 3470: 3466: 3462: 3458: 3453: 3452: 3451: 3450: 3449: 3448: 3447: 3446: 3445: 3444: 3443: 3442: 3431: 3427: 3423: 3418: 3417: 3416: 3415: 3414: 3413: 3412: 3411: 3410: 3409: 3400: 3395: 3390: 3386: 3381: 3380: 3379: 3378: 3377: 3376: 3375: 3374: 3364: 3359: 3358: 3357: 3356: 3355: 3354: 3353: 3352: 3345: 3341: 3337: 3332: 3331: 3330: 3329: 3328: 3327: 3318: 3317: 3316: 3315: 3314: 3313: 3304: 3303: 3302: 3301: 3300: 3299: 3290: 3289: 3288: 3287: 3286: 3285: 3280: 3275: 3270: 3265: 3264: 3263: 3262: 3255: 3254: 3253: 3252: 3246: 3242: 3238: 3234: 3233: 3232: 3231: 3228: 3224: 3220: 3215: 3214: 3209: 3204: 3203: 3198: 3197: 3196: 3195: 3190: 3185: 3181: 3175: 3171: 3167: 3163: 3158: 3154: 3151: 3147: 3143: 3138: 3132: 3124: 3120: 3115: 3111: 3107: 3103: 3098: 3094: 3090: 3085: 3081: 3076: 3072: 3068: 3064: 3060: 3055: 3051: 3047: 3044: 3043: 3042: 3026: 3022: 3018: 3013: 3008: 3004: 3000: 2999: 2998: 2997: 2996: 2995: 2994: 2993: 2992: 2991: 2990: 2989: 2972: 2967: 2963: 2959: 2955: 2950: 2946: 2942: 2938: 2935: 2931: 2927: 2923: 2920: 2917: 2913: 2908: 2904: 2899: 2895: 2891: 2884: 2880: 2876: 2872: 2869: 2868: 2867: 2866: 2865: 2864: 2863: 2862: 2861: 2860: 2859: 2858: 2857: 2856: 2842: 2838: 2834: 2827: 2823: 2819: 2818: 2817: 2816: 2815: 2814: 2813: 2812: 2811: 2810: 2809: 2808: 2794: 2790: 2789: 2788: 2787: 2786: 2785: 2784: 2783: 2782: 2781: 2780: 2779: 2768: 2763: 2758: 2754: 2750: 2746: 2741: 2737: 2732: 2728: 2723: 2720: 2715: 2710: 2705: 2700: 2696: 2692: 2688: 2684: 2678: 2673: 2669: 2664: 2660: 2657: 2653: 2649: 2644: 2640: 2637: 2633: 2630: 2629: 2627: 2623: 2620: 2616: 2612: 2608: 2604: 2598: 2594: 2591: 2586: 2582: 2578: 2574: 2571: 2567: 2562: 2558: 2553: 2549: 2546: 2542: 2538: 2534: 2529: 2525: 2522: 2518: 2514: 2510: 2506: 2502: 2498: 2494: 2493: 2491: 2487: 2483: 2479: 2475: 2471: 2467: 2463: 2459: 2455: 2451: 2450: 2449: 2448: 2447: 2446: 2445: 2444: 2443: 2442: 2433: 2429: 2425: 2421: 2420: 2419: 2418: 2417: 2416: 2415: 2414: 2407: 2403: 2399: 2392: 2385: 2381: 2377: 2373: 2366: 2359: 2355: 2354: 2353: 2352: 2351: 2350: 2341: 2336: 2332: 2325: 2322: 2317: 2316: 2315: 2314: 2313: 2312: 2304: 2303: 2302: 2301: 2300: 2299: 2294: 2289: 2284: 2279: 2275: 2270: 2266: 2261: 2257: 2253: 2249: 2245: 2241: 2237: 2233: 2229: 2225: 2221: 2217: 2213: 2209: 2206: 2202: 2198: 2193: 2189: 2185: 2181: 2177: 2173: 2168: 2164: 2161: 2157: 2152: 2148: 2144: 2140: 2137:coefficients 2136: 2132: 2127: 2123: 2119: 2115: 2111: 2107: 2103: 2099: 2095: 2094: 2093: 2092: 2089: 2085: 2081: 2076: 2075: 2070: 2069: 2068: 2067: 2062: 2057: 2049: 2045: 2041: 2038: 2037:bilinear form 2034: 2031: 2028: 2025: 2020: 2016: 2011: 2007: 2003: 1999: 1995: 1991: 1987: 1983: 1979: 1975: 1971: 1967: 1964: 1960: 1956: 1951: 1947: 1944: 1939: 1935: 1931: 1926: 1922: 1917: 1913: 1909: 1905: 1902:that we call 1901: 1896: 1892: 1888: 1884: 1880: 1877:onto scalars 1876: 1872: 1868: 1864: 1863: 1862: 1855: 1851: 1835: 1830: 1825: 1821: 1815: 1810: 1806: 1800: 1797: 1793: 1789: 1784: 1781: 1777: 1771: 1766: 1762: 1738: 1733: 1729: 1720: 1717: 1713: 1704: 1703: 1699: 1695: 1690: 1689: 1672: 1667: 1657: 1653: 1649: 1644: 1640: 1636: 1631: 1627: 1623: 1618: 1615: 1612: 1608: 1601: 1598: 1595: 1592: 1589: 1586: 1583: 1578: 1575: 1572: 1568: 1559: 1558: 1557: 1553: 1551: 1543: 1539: 1535: 1531: 1527: 1526: 1525: 1524: 1520: 1516: 1515:132.236.54.92 1507: 1499: 1495: 1491: 1487: 1486: 1485: 1481: 1477: 1473: 1472: 1471: 1470: 1467: 1463: 1459: 1455: 1451: 1447: 1443: 1439: 1435: 1434: 1433: 1432: 1428: 1424: 1420: 1416: 1412: 1408: 1404: 1400: 1396: 1392: 1388: 1384: 1380: 1376: 1368: 1366: 1365: 1361: 1357: 1349: 1345: 1341: 1337: 1333: 1332: 1331: 1330: 1326: 1322: 1318: 1314: 1310: 1302: 1300: 1299: 1295: 1291: 1287: 1286:transcription 1283: 1279: 1278:transcription 1275: 1268: 1264: 1256: 1252: 1248: 1244: 1239: 1238: 1237: 1236: 1235: 1234: 1229: 1225: 1221: 1216: 1215: 1214: 1213: 1210: 1206: 1202: 1197: 1195: 1191: 1187: 1183: 1179: 1178: 1177: 1176: 1172: 1168: 1157: 1155: 1154: 1149: 1145: 1137: 1133: 1125: 1121: 1119: 1115: 1109: 1107: 1102: 1094: 1092: 1091: 1087: 1083: 1071: 1066: 1064: 1059: 1057: 1052: 1051: 1049: 1048: 1045: 1042: 1041: 1033: 1027: 1026: 1022: 1019: 1017: 1014: 1012: 1006: 1004: 1001: 999: 996: 994: 988: 986: 983: 981: 978: 976: 973: 972: 966: 965: 958: 955: 953: 950: 949: 946: 943: 942: 936: 935: 928: 925: 924: 918: 917: 913: 911: 910: 905: 904: 902: 893: 892: 888: 886: 885: 884:of Icelanders 880: 879: 877: 870: 864: 855: 854: 847: 844: 843: 840: 837: 835: 832: 831: 828: 825: 824: 818: 815: 813: 809: 808: 806: 800: 797: 796: 790: 789: 782: 779: 778: 772: 769: 767: 763: 762: 760: 754: 748: 747:Old Norwegian 745: 743: 742:Old Icelandic 739: 738: 736: 727: 726: 718: 717: 714: 710: 706: 702: 701: 695: 693: 692: 688: 684: 680: 676: 668: 664: 660: 656: 652: 651: 650: 649: 645: 641: 637: 633: 626: 615: 609: 605: 601: 597: 596: 593: 590: 585: 580: 574: 572: 571: 567: 563: 562:Dylansmrjones 559: 555: 554:danish tongue 551: 547: 543: 539: 535: 531: 527: 523: 515: 513: 512: 508: 504: 498: 490: 485: 478: 471: 466: 465: 461: 457: 453: 440: 436: 432: 427: 426: 425: 424: 423: 422: 421: 420: 413: 409: 405: 401: 397: 396: 395: 394: 393: 392: 387: 383: 379: 375: 374: 373: 372: 369: 365: 361: 357: 353: 352: 351: 350: 346: 342: 334: 332: 331: 327: 323: 317: 312: 308: 304: 299: 291: 289: 288: 285: 282: 279: 271: 259: 255: 251: 247: 246: 245: 244: 243: 242: 241: 240: 239: 238: 229: 225: 221: 217: 216: 215: 214: 213: 212: 211: 210: 203: 199: 195: 188: 187: 186: 185: 184: 183: 178: 174: 170: 166: 162: 161: 160: 159: 156: 152: 148: 144: 140: 136: 132: 128: 127: 126: 125: 121: 117: 111: 93: 91: 90: 86: 82: 77: 74: 68: 66: 65: 61: 57: 53: 45: 43: 42: 38: 34: 29: 27: 22: 16: 4215: 4214: 4207: 4198: 4197: 4172: 4166: 4143: 4135: 4133: 4102: 3971: 3952: 3899: 3883: 3858: 3817: 3780: 3766: 3737: 3733: 3729: 3725: 3717: 3713: 3709: 3705: 3697: 3693: 3689: 3685: 3677: 3673: 3669: 3665: 3654: 3650: 3646: 3641: 3621: 3612: 3589: 3587: 3567:91.148.159.4 3552:91.148.159.4 3549: 3546:On Old Norse 3535: 3510:— Preceding 3506: 3503:Undo request 3472: 3468: 3464: 3425: 3362: 3339: 3244: 3240: 3236: 3222: 3207: 3200:combination. 3179: 3176: 3169: 3165: 3161: 3156: 3152: 3149: 3145: 3141: 3133: 3129: 3122: 3118: 3113: 3109: 3105: 3101: 3096: 3092: 3088: 3083: 3079: 3074: 3070: 3066: 3062: 3058: 3053: 3049: 3045: 3039: 3020: 3011: 3006: 3002: 2970: 2965: 2961: 2957: 2953: 2948: 2940: 2936: 2933: 2929: 2925: 2921: 2915: 2911: 2906: 2902: 2897: 2893: 2886: 2882: 2878: 2874: 2870: 2840: 2836: 2829: 2825: 2821: 2752: 2748: 2744: 2739: 2735: 2730: 2726: 2718: 2713: 2708: 2703: 2698: 2694: 2690: 2686: 2676: 2671: 2667: 2662: 2658: 2655: 2651: 2647: 2642: 2638: 2635: 2631: 2625: 2618: 2614: 2610: 2606: 2596: 2592: 2589: 2584: 2580: 2576: 2572: 2569: 2565: 2560: 2556: 2551: 2547: 2544: 2540: 2536: 2532: 2527: 2523: 2520: 2516: 2512: 2508: 2504: 2503:is a scalar 2500: 2496: 2489: 2485: 2481: 2477: 2473: 2469: 2465: 2461: 2453: 2427: 2401: 2390: 2383: 2379: 2375: 2371: 2364: 2357: 2339: 2334: 2323: 2320: 2277: 2273: 2268: 2264: 2259: 2255: 2251: 2247: 2243: 2239: 2235: 2231: 2227: 2223: 2219: 2211: 2207: 2204: 2200: 2196: 2191: 2187: 2183: 2179: 2175: 2171: 2166: 2162: 2159: 2155: 2150: 2146: 2142: 2138: 2134: 2130: 2125: 2121: 2117: 2113: 2109: 2105: 2101: 2097: 2083: 2053: 2047: 2043: 2039: 2023: 2018: 2014: 2009: 2005: 2001: 1997: 1993: 1989: 1985: 1981: 1977: 1973: 1969: 1962: 1958: 1954: 1949: 1945: 1942: 1937: 1933: 1924: 1920: 1915: 1911: 1907: 1903: 1899: 1894: 1890: 1886: 1882: 1878: 1874: 1870: 1866: 1860: 1853: 1697: 1555: 1547: 1533: 1511: 1493: 1461: 1445: 1441: 1437: 1418: 1414: 1410: 1406: 1402: 1398: 1394: 1390: 1386: 1382: 1378: 1372: 1353: 1339: 1316: 1306: 1285: 1271: 1250: 1204: 1189: 1161: 1126: 1122: 1110: 1098: 1079: 914: 907: 889: 882: 672: 658: 619: 591: 557: 553: 549: 545: 541: 537: 533: 529: 525: 521: 519: 494: 449: 338: 295: 275: 164: 142: 138: 134: 130: 112: 97: 78: 75: 72: 69:Jagermeister 55: 49: 46:Tensor field 30: 23: 20: 17:Hypercupolae 3783:Bach tensor 3748:Incnis Mrsi 3644:False. Let 3583:isomorphism 3180:orthonormal 2222:and vector 1692:spaces). ᛭ 975:Dalecarlian 969:Descendants 957:Proto-Norse 909:Poetic Edda 799:Orthography 781:Old Gutnish 771:Old Swedish 530:dansk tunga 526:dansk tunga 522:dansk tunga 4216:Thank you! 4158:topic bans 3920:topic bans 3208:components 1906:such that 1309:this table 1290:Chris Fynn 1106:this image 916:Prose Edda 858:Literature 846:Morphology 766:Old Danish 497:WP:MOSMATH 303:talk pages 278:Briangotts 104:<f: --> 100:<v: --> 56:Wolfkeeper 4220:Yapperbot 4154:site bans 3916:site bans 3880:LokiClock 3814:LokiClock 3763:LokiClock 3609:LokiClock 3532:LokiClock 3422:LokiClock 3336:LokiClock 3219:LokiClock 3017:LokiClock 2424:LokiClock 2398:LokiClock 2080:LokiClock 1850:LokiClock 1694:LokiClock 1530:LokiClock 1490:LokiClock 1476:Duoduoduo 1474:Thanks! 1458:LokiClock 1423:Duoduoduo 1336:LokiClock 1321:Hayden120 1247:LokiClock 1220:Devanatha 1201:LokiClock 1186:LokiClock 1167:Devanatha 1082:Hayden120 1016:Norwegian 1010:(extinct) 1003:Icelandic 992:(extinct) 939:Ancestors 891:Legendary 839:Phonology 713:Old Norse 655:LokiClock 636:WP:LAYOUT 632:Málaháttr 558:Old Norse 548:, Danish 284:(Contrib) 272:Hnefatafl 250:LokiClock 194:LokiClock 147:LokiClock 3992:Lǫgmaðr 3846:You can 3828:talkback 3512:unsigned 3320:shallow. 2382:is (1/k) 1405:, since 817:Medieval 730:Dialects 705:a series 703:Part of 683:Sacolcor 616:Stubbing 575:Barnstar 550:tungemål 546:tungumál 542:language 482:You can 456:AnonMoos 431:AnonMoos 404:AnonMoos 378:AnonMoos 360:AnonMoos 341:AnonMoos 276:Thanks! 81:Wahrmund 4082:Bjaðǫk 3702:. Then 3477:Quondum 3389:Quondum 3334:set. ᛭ 3269:Quondum 3184:Quondum 3005:′ and 2757:Quondum 2389:, not k 2370:k, and 2283:Quondum 2056:Quondum 1356:Benwing 1317:general 1021:Swedish 998:Gutnish 985:Faroese 834:Grammar 322:SineBot 3906:. The 3732:) = (− 3722:, but 3692:) = (− 3658:where 3306:basis. 2824:, and 2793:charts 2689:thing 2374:=(1/k) 2203:, but 1928:, the 1513:Thanks 980:Danish 863:Poetry 600:Haukur 538:tongue 534:danish 311:tildes 281:(Talk) 220:Haukur 169:Haukur 165:sæfari 143:sæfari 116:Haukur 3984:Name 3799:Teply 3724:ψ∘φ ( 3712:) = ( 3704:φ∘ψ ( 3672:) = ( 3590:their 2543:v) = 2319:K of 2098:basis 1996:with 1957:= (1/ 1385:. In 1311:from 1008:Norn 876:Sagas 669:C++0x 139:Sævar 102:from 33:Padex 4183:talk 4144:The 4109:talk 3987:IPA 3959:here 3957:and 3955:here 3940:talk 3884:talk 3869:talk 3818:talk 3803:talk 3793:and 3767:talk 3752:talk 3682:and 3629:talk 3613:talk 3598:talk 3571:talk 3556:talk 3536:talk 3520:talk 3426:talk 3340:talk 3223:talk 3100:) = 3021:talk 2960:′ = 2905:) = 2839:and 2828:and 2751:and 2702:and 2687:same 2666:) = 2568:) = 2499:and 2456:use 2428:talk 2402:talk 2170:) = 2084:talk 1919:) = 1854:talk 1698:talk 1534:talk 1519:talk 1494:talk 1480:talk 1462:talk 1456:. ᛭ 1427:talk 1395:fall 1391:fell 1387:fall 1360:talk 1340:talk 1325:talk 1294:talk 1251:talk 1224:talk 1205:talk 1190:talk 1171:talk 1148:talk 1136:talk 1086:talk 901:Edda 687:talk 659:talk 644:talk 640:PamD 625:stub 604:talk 566:talk 507:talk 460:talk 450:See 435:talk 408:talk 382:talk 364:talk 345:talk 326:talk 254:talk 224:talk 198:talk 173:talk 151:talk 131:Vafl 120:talk 85:talk 60:Talk 52:User 37:talk 21:Hi! 4096:non 4086:non 4076:non 4066:non 4056:non 4046:non 4036:non 4026:non 4016:non 4006:non 3996:non 3978:). 3900:Hi, 3716:, − 3684:ψ ( 3664:φ ( 3057:= ( 2885:= ( 2743:or 2646:= ( 2531:)⋅( 2519:= ( 2454:not 2238:= ( 2226:, ( 2129:= ( 2110:any 1419:man 1415:men 1411:man 1407:sit 1403:set 1399:sit 1389:to 1383:set 1381:to 1379:sit 1144:CBM 1132:CBM 1120:. 793:Use 108:/v/ 4212:. 4185:) 4177:. 4156:, 4111:) 4103:-- 3942:) 3918:, 3886:) 3871:) 3820:) 3805:) 3797:. 3769:) 3754:) 3740:) 3736:, 3728:, 3720:) 3708:, 3696:, 3688:, 3676:, 3668:, 3653:⊕ 3649:= 3631:) 3615:) 3600:) 3573:) 3558:) 3538:) 3522:) 3473:∆y 3471:− 3469:∆x 3467:= 3465:∆s 3463:: 3428:) 3342:) 3225:) 3148:= 3144:= 3117:= 3078:= 3069:)⋅ 3023:) 3009:′ 2964:′ 2951:′ 2947:∴ 2918:′. 2909:′ 2892:)( 2887:kc 2873:= 2729:→ 2670:′ 2654:)( 2634:= 2588:= 2511:= 2507:: 2430:) 2404:) 2380:v' 2372:v' 2358:v' 2327:ij 2272:= 2263:)) 2246:)) 2234:)) 2086:) 2054:— 2013:)= 1972:)= 1941:= 1848:᛭ 1726:→ 1668:∗ 1658:∗ 1650:⊗ 1645:∗ 1637:⊗ 1632:∗ 1624:∈ 1596:⊗ 1590:⊗ 1584:∈ 1536:) 1521:) 1496:) 1482:) 1464:) 1446:fe 1429:) 1413:→ 1401:→ 1362:) 1342:) 1327:) 1296:) 1253:) 1226:) 1207:) 1192:) 1173:) 1146:· 1134:· 1088:) 707:on 689:) 661:) 646:) 628:}} 622:{{ 606:) 568:) 509:) 462:) 454:. 437:) 410:) 384:) 366:) 347:) 328:) 320:-- 256:) 226:) 200:) 175:) 153:) 135:fá 122:) 87:) 62:) 54:) 39:) 28:! 4181:( 4107:( 3938:( 3882:( 3867:( 3843:. 3816:( 3801:( 3765:( 3750:( 3738:x 3734:y 3730:y 3726:x 3718:x 3714:y 3710:y 3706:x 3700:) 3698:y 3694:x 3690:y 3686:x 3680:) 3678:x 3674:y 3670:y 3666:x 3660:T 3655:T 3651:T 3647:A 3627:( 3611:( 3596:( 3569:( 3554:( 3534:( 3518:( 3482:c 3424:( 3394:c 3363:n 3361:( 3338:( 3274:c 3245:n 3241:n 3237:n 3221:( 3189:c 3170:b 3166:e 3162:b 3157:β 3153:e 3150:v 3146:v 3142:v 3126:. 3123:i 3119:c 3114:i 3110:δ 3106:j 3102:c 3097:i 3093:e 3091:⋅ 3089:ε 3087:( 3084:j 3080:c 3075:i 3071:e 3067:ε 3063:j 3059:c 3054:i 3050:e 3048:⋅ 3046:c 3019:( 3012:j 3007:c 3003:v 2974:. 2971:j 2968:′ 2966:e 2962:v 2958:ε 2954:j 2949:c 2944:. 2941:i 2937:e 2934:v 2932:= 2930:ε 2926:i 2922:c 2916:ε 2912:j 2907:c 2903:k 2901:/ 2898:i 2894:ε 2889:i 2883:ε 2879:i 2875:c 2871:c 2841:v 2837:c 2832:i 2830:v 2826:v 2822:v 2762:c 2753:V 2749:V 2745:ε 2740:i 2736:e 2731:V 2727:V 2719:V 2714:k 2709:j 2706:′ 2704:e 2699:i 2695:e 2691:v 2680:. 2677:j 2674:′ 2672:e 2668:v 2663:i 2659:e 2656:k 2652:k 2650:/ 2648:v 2643:i 2639:e 2636:v 2632:v 2626:v 2621:. 2619:c 2615:v 2613:) 2611:c 2609:⋅ 2607:v 2600:. 2597:i 2593:c 2590:v 2585:i 2581:δ 2577:j 2573:c 2570:v 2566:ε 2564:⋅ 2561:i 2557:e 2555:( 2552:j 2548:c 2545:v 2541:ε 2537:j 2533:c 2528:i 2524:e 2521:v 2517:c 2515:⋅ 2513:v 2509:p 2505:p 2501:c 2497:v 2490:n 2486:v 2484:⊗ 2482:c 2478:v 2476:( 2474:c 2470:v 2468:⋅ 2466:c 2462:n 2426:( 2400:( 2394:i 2391:v 2387:i 2384:v 2376:v 2368:i 2365:v 2363:= 2361:i 2340:v 2335:v 2324:c 2321:v 2288:c 2278:i 2274:e 2269:j 2265:e 2260:i 2256:e 2254:( 2252:ε 2248:v 2244:v 2242:( 2240:c 2236:v 2232:c 2230:( 2228:v 2224:v 2220:c 2212:i 2208:e 2205:a 2201:V 2197:a 2192:i 2188:e 2184:V 2180:n 2176:ε 2172:a 2167:i 2163:e 2160:a 2158:( 2156:ε 2151:i 2147:ε 2143:n 2139:a 2135:n 2131:x 2126:i 2122:e 2118:n 2114:n 2106:n 2102:n 2082:( 2061:c 2048:K 2046:→ 2044:V 2042:× 2040:V 2024:e 2022:( 2019:j 2015:e 2010:j 2006:e 2004:( 2002:e 1998:V 1994:V 1990:V 1988:→ 1986:V 1982:V 1978:V 1974:V 1970:V 1963:e 1961:) 1959:k 1955:f 1950:i 1946:e 1943:k 1938:i 1934:f 1925:j 1921:δ 1916:j 1912:e 1910:( 1908:e 1904:e 1900:V 1895:i 1891:e 1887:V 1879:K 1875:V 1871:V 1867:V 1852:( 1836:. 1831:i 1826:j 1822:T 1816:i 1811:j 1807:R 1801:j 1798:j 1794:g 1790:= 1785:j 1782:i 1778:T 1772:i 1767:j 1763:R 1739:i 1734:j 1730:T 1721:j 1718:i 1714:T 1696:( 1673:. 1664:) 1654:V 1641:V 1628:V 1619:k 1616:j 1613:i 1609:T 1605:( 1602:= 1599:V 1593:V 1587:V 1579:k 1576:j 1573:i 1569:T 1532:( 1517:( 1492:( 1478:( 1460:( 1442:a 1425:( 1358:( 1338:( 1323:( 1292:( 1249:( 1222:( 1203:( 1188:( 1169:( 1150:) 1142:( 1138:) 1130:( 1084:( 1069:e 1062:t 1055:v 919:) 906:( 894:) 881:( 871:) 867:( 819:) 810:( 773:) 764:( 755:) 740:( 685:( 657:( 642:( 602:( 564:( 505:( 479:. 458:( 433:( 406:( 380:( 362:( 343:( 324:( 252:( 222:( 196:( 171:( 149:( 118:( 83:( 58:( 35:(

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index