[math-fun] What is a "k-vector"?
On the GA list http://groups.google.com/group/geometric_algebra/ I recently claimed that the term "k-vector" (as in 2-vector, 3-vector) routinely denotes a vector with dimension k. This was subsequently queried, whereupon I was obliged to admit my inability to quote any actual example from the literature. Can anybody clarify the situation --- or better still, provide some relevant citations? Fred Lunnon
Quoting Fred lunnon <fred.lunnon@gmail.com>:
Can anybody clarify the situation --- or better still, provide some relevant citations?
The phrase is commkon enough, but I don't have a reference handy. -hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
On Sun, Mar 14, 2010 at 9:56 AM, <mcintosh@servidor.unam.mx> wrote:
Quoting Fred lunnon <fred.lunnon@gmail.com>:
Can anybody clarify the situation --- or better still, provide some relevant citations?
The phrase is commkon enough, but I don't have a reference handy.
-hvm
How about wikipedia? http://en.wikipedia.org/wiki/Multivector
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Good reference, but that points out an ambiguity. The wikipedia article in question uses k-vector for the wedge product of k vectors of any fixed dimension. Fred uses it for a vector in a vector space of dimension k. I think that both terms are in use. Victor On Sun, Mar 14, 2010 at 2:25 PM, Mike Stay <metaweta@gmail.com> wrote:
On Sun, Mar 14, 2010 at 9:56 AM, <mcintosh@servidor.unam.mx> wrote:
Quoting Fred lunnon <fred.lunnon@gmail.com>:
Can anybody clarify the situation --- or better still, provide some relevant citations?
The phrase is commkon enough, but I don't have a reference handy.
-hvm
How about wikipedia? http://en.wikipedia.org/wiki/Multivector
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Indeed, the point I had made at the GA site was that the Clifford / Grassmann algebra usage should be deprecated, partly on these grounds. But now I'm beginning to wonder if said ambiguity resulted only from my confusing "k-vector" with "k-tuple", etc? WFL On 3/14/10, Victor Miller <victorsmiller@gmail.com> wrote:
Good reference, but that points out an ambiguity. The wikipedia article in question uses k-vector for the wedge product of k vectors of any fixed dimension. Fred uses it for a vector in a vector space of dimension k. I think that both terms are in use.
Victor
On Sun, Mar 14, 2010 at 2:25 PM, Mike Stay <metaweta@gmail.com> wrote:
On Sun, Mar 14, 2010 at 9:56 AM, <mcintosh@servidor.unam.mx> wrote:
Quoting Fred lunnon <fred.lunnon@gmail.com>:
Can anybody clarify the situation --- or better still, provide some relevant citations?
The phrase is commkon enough, but I don't have a reference handy.
-hvm
How about wikipedia? http://en.wikipedia.org/wiki/Multivector
------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
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I somewhat agree. There is already the term k-tensor, and the wedge product of k-vectors is essentially like a k-tensor (strictly speaking Lambda^k V is a quotient of \otimes^V ). Victor On Sun, Mar 14, 2010 at 7:29 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Indeed, the point I had made at the GA site was that the Clifford / Grassmann algebra usage should be deprecated, partly on these grounds. But now I'm beginning to wonder if said ambiguity resulted only from my confusing "k-vector" with "k-tuple", etc?
WFL
On 3/14/10, Victor Miller <victorsmiller@gmail.com> wrote:
Good reference, but that points out an ambiguity. The wikipedia article in question uses k-vector for the wedge product of k vectors of any fixed dimension. Fred uses it for a vector in a vector space of dimension k. I think that both terms are in use.
Victor
On Sun, Mar 14, 2010 at 2:25 PM, Mike Stay <metaweta@gmail.com> wrote: > On Sun, Mar 14, 2010 at 9:56 AM, <mcintosh@servidor.unam.mx> wrote: >> Quoting Fred lunnon <fred.lunnon@gmail.com>: >>> >>> Can anybody clarify the situation --- or better still, provide >>> some relevant citations? >> >> The phrase is commkon enough, but I don't have a reference handy. >> >> -hvm > > How about wikipedia? > http://en.wikipedia.org/wiki/Multivector > >> >> ------------------------------------------------- >> www.correo.unam.mx >> UNAMonos Comunicándonos >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun@mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > > -- > Mike Stay - metaweta@gmail.com > http://math.ucr.edu/~mike > http://reperiendi.wordpress.com > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
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I think this should have read "the wedge product of k vectors" ? I've been trying to coin a replacement for the "k-vector part" notation currently in use in GA for the terms of grade (polynomial degree) k within a given multivector --- on top of the (potential) ambiguity, it is simply too unwieldy when reading an expression aloud, or programming a computer! My current private neologism is "grator" or "k-grator" [contraction of "grade(d) vector"]. WFL On 3/15/10, Victor Miller <victorsmiller@gmail.com> wrote:
I somewhat agree. There is already the term k-tensor, and the wedge product of k-vectors is essentially like a k-tensor (strictly speaking Lambda^k V is a quotient of \otimes^V ).
Victor
On Sun, Mar 14, 2010 at 7:29 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
Indeed, the point I had made at the GA site was that the Clifford / Grassmann algebra usage should be deprecated, partly on these grounds. But now I'm beginning to wonder if said ambiguity resulted only from my confusing "k-vector" with "k-tuple", etc?
WFL
On 3/14/10, Victor Miller <victorsmiller@gmail.com> wrote:
Good reference, but that points out an ambiguity. The wikipedia article in question uses k-vector for the wedge product of k vectors of any fixed dimension. Fred uses it for a vector in a vector space of dimension k. I think that both terms are in use.
Victor
On Sun, Mar 14, 2010 at 2:25 PM, Mike Stay <metaweta@gmail.com> wrote:
On Sun, Mar 14, 2010 at 9:56 AM, <mcintosh@servidor.unam.mx> wrote:
Quoting Fred lunnon <fred.lunnon@gmail.com>:
Can anybody clarify the situation --- or better still, provide some relevant citations?
The phrase is commkon enough, but I don't have a reference handy.
-hvm
How about wikipedia? http://en.wikipedia.org/wiki/Multivector
------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
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On 3/14/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
Indeed, the point I had made at the GA site was that the Clifford / Grassmann algebra usage should be deprecated, partly on these grounds. But now I'm beginning to wonder if said ambiguity resulted only from my confusing "k-vector" with "k-tuple", etc?
Searching on "n-vector" turned up the following on the first page http://mathworld.wolfram.com/n-Vector.html << n-Vector An n-dimensional vector, i.e., a vector (x_1, x_2, ..., x_n) with n components. >> http://webpages.dcu.ie/~oriordae/matrix-problems.pdf << Q1: Let x be an n-vector and A a n x n matrix. ... >> http://ptp.ipap.jp/link?PTP/49/1451/ << Critical Behavior of the Anisotropic Classical n-Vector Model ... The critical exponents of the anisotropic classical model with n components are investigated in the 1/n-expansion. >> http://kr.cs.ait.ac.th/~radok/math/mat5/algebra12.htm << Let (y_1, ··· , y_n) (4) be the vectors of this space. The vectors (y1, ··· , yn, 0) (5) form a vector space X' consisting of (n + 1)-vectors. n-vectors (4) are independent if and only if the corresponding vectors (5) are independent, ... >> In contrast --- but in contexts which are more remote from mainstream linear algebra --- http://www.farcaster.com/papers/sm-thesis/node24.html uses "n-vector" to mean a set of n vectors. http://ptp.ipap.jp/link?PTP/49/1451/ << Critical Behavior of the Anisotropic Classical n-Vector Model ... The critical exponents of the anisotropic classical model with n components are investigated in the 1/n-expansion. >> This and several more concern a percolation model in which matrices in O(n) --- effectively Cl(n,0,0) versors --- are associated with nodes of a lattice. No doubt searching on "m-vector", "3-vector" would find plenty more. It's noteworthy that the Hestenes(?) "k-vector" usage is absent from these pages! Fred Lunnon
This URL has a page from Hermann Weyl's 1939 book "The classical groups: their invariants and representations," in which k-vector is used to describe an n-tuple with entries from a number field k http://books.google.com/books?id=zmzKSP2xTtYC&pg=PA18&lpg=PA18&dq=k-vector+h... On Mon, Mar 15, 2010 at 5:16 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 3/14/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
Indeed, the point I had made at the GA site was that the Clifford / Grassmann algebra usage should be deprecated, partly on these grounds. But now I'm beginning to wonder if said ambiguity resulted only from my confusing "k-vector" with "k-tuple", etc?
Searching on "n-vector" turned up the following on the first page
http://mathworld.wolfram.com/n-Vector.html << n-Vector An n-dimensional vector, i.e., a vector (x_1, x_2, ..., x_n) with n components. >>
http://webpages.dcu.ie/~oriordae/matrix-problems.pdf << Q1: Let x be an n-vector and A a n x n matrix. ... >>
http://ptp.ipap.jp/link?PTP/49/1451/ << Critical Behavior of the Anisotropic Classical n-Vector Model ... The critical exponents of the anisotropic classical model with n components are investigated in the 1/n-expansion. >>
http://kr.cs.ait.ac.th/~radok/math/mat5/algebra12.htm << Let (y_1, ··· , y_n) (4) be the vectors of this space. The vectors (y1, ··· , yn, 0) (5) form a vector space X' consisting of (n + 1)-vectors. n-vectors (4) are independent if and only if the corresponding vectors (5) are independent, ... >>
In contrast --- but in contexts which are more remote from mainstream linear algebra --- http://www.farcaster.com/papers/sm-thesis/node24.html uses "n-vector" to mean a set of n vectors.
http://ptp.ipap.jp/link?PTP/49/1451/ << Critical Behavior of the Anisotropic Classical n-Vector Model ... The critical exponents of the anisotropic classical model with n components are investigated in the 1/n-expansion. >> This and several more concern a percolation model in which matrices in O(n) --- effectively Cl(n,0,0) versors --- are associated with nodes of a lattice.
No doubt searching on "m-vector", "3-vector" would find plenty more. It's noteworthy that the Hestenes(?) "k-vector" usage is absent from these pages!
Fred Lunnon
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Nice one --- and what's more, not unrelated to Geometric Algebra! One way and another, "k-vector" begins to look fairly flaky notation in any context ... WFL On 3/16/10, Thane Plambeck <tplambeck@gmail.com> wrote:
This URL has a page from Hermann Weyl's 1939 book "The classical groups: their invariants and representations," in which k-vector is used to describe an n-tuple with entries from a number field k
http://books.google.com/books?id=zmzKSP2xTtYC&pg=PA18&lpg=PA18&dq=k-vector+h...
On Mon, Mar 15, 2010 at 5:16 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 3/14/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
Indeed, the point I had made at the GA site was that the Clifford / Grassmann algebra usage should be deprecated, partly on these grounds. But now I'm beginning to wonder if said ambiguity resulted only from my confusing "k-vector" with "k-tuple", etc?
Searching on "n-vector" turned up the following on the first page
http://mathworld.wolfram.com/n-Vector.html << n-Vector An n-dimensional vector, i.e., a vector (x_1, x_2, ..., x_n) with n components. >>
http://webpages.dcu.ie/~oriordae/matrix-problems.pdf << Q1: Let x be an n-vector and A a n x n matrix. ... >>
http://ptp.ipap.jp/link?PTP/49/1451/ << Critical Behavior of the Anisotropic Classical n-Vector Model ... The critical exponents of the anisotropic classical model with n components are investigated in the 1/n-expansion. >>
http://kr.cs.ait.ac.th/~radok/math/mat5/algebra12.htm << Let (y_1, ··· , y_n) (4) be the vectors of this space. The vectors (y1, ··· , yn, 0) (5) form a vector space X' consisting of (n + 1)-vectors. n-vectors (4) are independent if and only if the corresponding vectors (5) are independent, ... >>
In contrast --- but in contexts which are more remote from mainstream linear algebra --- http://www.farcaster.com/papers/sm-thesis/node24.html uses "n-vector" to mean a set of n vectors.
http://ptp.ipap.jp/link?PTP/49/1451/ << Critical Behavior of the Anisotropic Classical n-Vector Model ... The critical exponents of the anisotropic classical model with n components are investigated in the 1/n-expansion. >> This and several more concern a percolation model in which matrices in O(n) --- effectively Cl(n,0,0) versors --- are associated with nodes of a lattice.
No doubt searching on "m-vector", "3-vector" would find plenty more. It's noteworthy that the Hestenes(?) "k-vector" usage is absent from these pages!
Fred Lunnon
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Quoting Fred lunnon <fred.lunnon@gmail.com>:
Nice one --- and what's more, not unrelated to Geometric Algebra!
One way and another, "k-vector" begins to look fairly flaky notation in any context ... WFL
Quite possibly that depends on whether one wants to regard it as a formal definition, or just the way some mathematicians tend to say certain things. -hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
And watch out when you use this term among physicists, who will think you are referring to the momentum vector of an electron moving in a crystal. In that setting, momentum is conserved modulo the "momentum-lattice" associated with the crystal lattice (the dual lattice scaled by Planck's constant). In other words, the momentum equivalence classes of electrons in crystals are universally identified by their "k-vector". Veit On Mar 16, 2010, at 2:48 AM, mcintosh@servidor.unam.mx wrote:
Quoting Fred lunnon <fred.lunnon@gmail.com>:
Nice one --- and what's more, not unrelated to Geometric Algebra!
One way and another, "k-vector" begins to look fairly flaky notation in any context ... WFL
Quite possibly that depends on whether one wants to regard it as a formal definition, or just the way some mathematicians tend to say certain things.
-hvm
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participants (6)
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Fred lunnon -
mcintosh＠servidor.unam.mx -
Mike Stay -
Thane Plambeck -
Veit Elser -
Victor Miller