Received: from 173-13-139-235-sfba.hfc.comcastbusiness.net ([173.13.139.235]:33091 helo=jukni.digitalkingdom.org) by stodi.digitalkingdom.org with smtp (Exim 4.80.1) (envelope-from ) id 1Xo5pO-0003ya-Dp; Mon, 10 Nov 2014 23:25:31 -0800 Received: by jukni.digitalkingdom.org (sSMTP sendmail emulation); Mon, 10 Nov 2014 23:25:25 -0800 From: "Apache" Date: Mon, 10 Nov 2014 23:25:25 -0800 To: webmaster@lojban.org, gleki.is.my.name@gmail.com Subject: [jvsw] Definition Edited At Word aigne -- By gleki Bcc: jbovlaste-admin@lojban.org Message-ID: <5461b9e5.HffkwoO0/mRJ0P+I%webmaster@lojban.org> User-Agent: Heirloom mailx 12.5 7/5/10 MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Spam-Score: 0.5 (/) X-Spam_score: 0.5 X-Spam_score_int: 5 X-Spam_bar: / X-Spam-Report: Spam detection software, running on the system "stodi.digitalkingdom.org", has NOT identified this incoming email as spam. The original message has been attached to this so you can view it or label similar future email. If you have any questions, see @@CONTACT_ADDRESS@@ for details. Content preview: In jbovlaste, the user gleki has edited a definition of "aigne" in the language "English". Differences: 5,5c5,5 < For any eigenvector v in generalized eigenspace x3 of linear transformation x2 for eigenvalue x1, where I is the identity matrix/transformation that works/makes sense in the context, the following equation is satisfied: $(((x_2)-(x_1)I)^{x_4})v = 0$. When the argument of x4 is 1, the generalized eigenspace x3 is simply a strict/simple/basic eigenspace; this is the typical (and probable cultural default) meaning of this word. x4 will typically be restricted to integer values k > 0. x2 should always be specified (at least implicitly by context), for an eigenvalue does not mean much without the linear transformation being known. However, since one usually knows the said linear transformation, and since the basic underlying relationship of this word is "eigen-ness", the eigenvalue is given the primary terbri (x1). When filling x3 and/or x4, x2 and x1 (in that order of importance) should already be (at least contextually implicitly) specified. x3 is the set of all eigenvectors of linear transformation x2, endowed with all of the typical operations of the vector space at hand. The default includes the zero vector (else the x3 eigenspace is not actually a vector space); normally in the context of mathematics, the zero vector is not considered to be an eigenvector, but by this definition it is included. Thus, a Lojban mathematician would consider the zero vector to be an (automatic) eigenvector of the given (in fact, any) linear transformation (particularly ones represented by a square matrix in a given basis). This is the logically most basic definition, but is contrary to typical mathematical culture. This word implies neither nondegeneracy nor degeneracy of eigenspace x3. In other words there may or may not be more than one linearly independent vector in the eigenspace x3 for a given eigenvalue x1 of linear transformation x2. x3 is the unique generalized eigenspace of x2 for given values of x1 and x4. x1 is not necessarily the unique eigenvalue of linear transformation x2, nor is its multiplicity necessarily [...] Content analysis details: (0.5 points, 5.0 required) pts rule name description ---- ---------------------- -------------------------------------------------- 0.0 URIBL_BLOCKED ADMINISTRATOR NOTICE: The query to URIBL was blocked. See http://wiki.apache.org/spamassassin/DnsBlocklists#dnsbl-block for more information. [URIs: lojban.org] 1.4 RCVD_IN_BRBL_LASTEXT RBL: No description available. [173.13.139.235 listed in bb.barracudacentral.org] -1.9 BAYES_00 BODY: Bayes spam probability is 0 to 1% [score: 0.0000] 1.0 RDNS_DYNAMIC Delivered to internal network by host with dynamic-looking rDNS In jbovlaste, the user gleki has edited a definition of "aigne" in the language "English". Differences: 5,5c5,5 < =09=09For any eigenvector v in generalized eigenspace x3 of linear tr= ansformation x2 for eigenvalue x1, where I is the identity matrix/trans= formation that works/makes sense in the context, the following equation= is satisfied: $(((x_2)-(x_1)I)^{x_4})v =3D 0$. When the argument of x= 4 is 1, the generalized eigenspace x3 is simply a strict/simple/basic e= igenspace; this is the typical (and probable cultural default) meaning = of this word. x4 will typically be restricted to integer values k > 0. = x2 should always be specified (at least implicitly by context), for an = eigenvalue does not mean much without the linear transformation being k= nown. However, since one usually knows the said linear transformation,= and since the basic underlying relationship of this word is "eigen-nes= s", the eigenvalue is given the primary terbri (x1). When filling x3 an= d/or x4, x2 and x1 (in that order of importance) should already be (at = least contextually implicitly) specified. x3 is the set of all eigenvec= tors of linear transformation x2, endowed with all of the typical opera= tions of the vector space at hand. The default includes the zero vecto= r (else the x3 eigenspace is not actually a vector space); normally in = the context of mathematics, the zero vector is not considered to be an = eigenvector, but by this definition it is included. Thus, a Lojban mat= hematician would consider the zero vector to be an (automatic) eigenvec= tor of the given (in fact, any) linear transformation (particularly one= s represented by a square matrix in a given basis). This is the logica= lly most basic definition, but is contrary to typical mathematical cult= ure. This word implies neither nondegeneracy nor degeneracy of eigenspa= ce x3. In other words there may or may not be more than one linearly i= ndependent vector in the eigenspace x3 for a given eigenvalue x1 of lin= ear transformation x2. x3 is the unique generalized eigenspace of x2 fo= r given values of x1 and x4. x1 is not necessarily the unique eigenvalu= e of linear transformation x2, nor is its multiplicity necessarily 1 fo= r the same. Beware when converting the terbri structure of this word. = In fact, the set of all eigenvalues for a given linear transformation = x2 will include scalar zero (0); therefore, any linear transformation w= ith a nontrivial set of eigenvalues will have at least two eigenvalues = that may fill in terbri x1 of this word. The 'eigenvalue' of zero for = a proper/nice linear transformation will produce an 'eigenspace' that i= s equivalent to the entire vector space at hand. If x3 is specified by = a set of vectors, the span of that set should fully yield the entire ei= genspace of the linear transformation x2 associated with eigenvalue x1,= however the set may be redundant (linearly dependent); the zero vector= is automatically included in any vector space. A multidimensional eige= nspace (that is to say a vector space of eigenvectors with dimension st= rictly greater than 1) for fixed eigenvalue and linear transformation (= and generalization exponent) is degenerate by definition. The algebraic= multiplicity x5 of the eigenvalue does not entail degeneracy (of eigen= space) if greater than 1; it is the integer number of occurrences of a = given eigenvalue x1 in the multiset of eigenvalues (spectrum) of the gi= ven linear transformation/square matrix x2. In other words, the charact= eristic polynomial can be factored into linear polynomial primes (with = root x1) which are exponentiated to the power x5 (the multiplicity; not= ably, not x4). For x4 > x5, the eigenspace is trivial. x2 may not be di= agonalizable. The scalar zero (0) is a naturally permissible argument o= f x1 (unlike some cultural mathematical definitions in English). Eigens= paces retain the operations and properties endowing the vectorspaces to= which they belong (as subspaces). Thus, an eigenspace is more than a = set of objects: it is a set of vectors such that that set is endowed wi= th vectorspace operators and properties. Thus {klesi} alone is insuffi= cient. But the set underlying eigenspace x3 is a type of {klesi}, with= the property of being closed under linear transformation x2 (up to sca= lar multiplication). The vector space and basis being used are not spec= ified by this word. Use this word as a seltau in constructions such as = "eigenket", "eigenstate", etc. (In such cases, {te aigne} is recommend= ed for the typical English usages of such terms. Use {zei} in lujvo for= med by these constructs. The term "eigenvector" may be rendered as "cm= ima be le te .aigne"). See also: {gei'ai}, {klesi}, {daigno}. --- > =09=09For any eigenvector v in generalized eigenspace $x_3$ of linear= transformation $x_2$ for eigenvalue $x_1$, where I is the identity mat= rix/transformation that works/makes sense in the context, the following= equation is satisfied: $((x_2 - x_1 * I)^{x_4})v =3D 0$. When the argu= ment of $x_4$ is 1, the generalized eigenspace $x_3$ is simply a strict= /simple/basic eigenspace; this is the typical (and probable cultural de= fault) meaning of this word. $x_4$ will typically be restricted to inte= ger values k > 0. $x_2$ should always be specified (at least implicitly= by context), for an eigenvalue does not mean much without the linear t= ransformation being known. However, since one usually knows the said li= near transformation, and since the basic underlying relationship of thi= s word is "eigen-ness", the eigenvalue is given the primary terbri ($x_= 1$). When filling $x_3$ and/or $x_4$, $x_2$ and $x_1$ (in that order of= importance) should already be (at least contextually implicitly) speci= fied. $x_3$ is the set of all eigenvectors of linear transformation $x_= 2$, endowed with all of the typical operations of the vector space at h= and. The default includes the zero vector (else the $x_3$ eigenspace is= not actually a vector space); normally in the context of mathematics, = the zero vector is not considered to be an eigenvector, but by this def= inition it is included. Thus, a Lojban mathematician would consider the= zero vector to be an (automatic) eigenvector of the given (in fact, an= y) linear transformation (particularly ones represented by a square mat= rix in a given basis). This is the logically most basic definition, but= is contrary to typical mathematical culture. This word implies neither= nondegeneracy nor degeneracy of eigenspace $x_3$. In other words there= may or may not be more than one linearly independent vector in the eig= enspace $x_3$ for a given eigenvalue $x_1$ of linear transformation $x_= 2$. $x_3$ is the unique generalized eigenspace of $x_2$ for given value= s of $x_1$ and $x_4$. $x_1$ is not necessarily the unique eigenvalue of= linear transformation $x_2$, nor is its multiplicity necessarily 1 for= the same. Beware when converting the terbri structure of this word. In= fact, the set of all eigenvalues for a given linear transformation $x_= 2$ will include scalar zero (0); therefore, any linear transformation w= ith a nontrivial set of eigenvalues will have at least two eigenvalues = that may fill in terbri $x_1$ of this word. The 'eigenvalue' of zero fo= r a proper/nice linear transformation will produce an 'eigenspace' that= is equivalent to the entire vector space at hand. If $x_3$ is specifie= d by a set of vectors, the span of that set should fully yield the enti= re eigenspace of the linear transformation $x_2$ associated with eigenv= alue $x_1$, however the set may be redundant (linearly dependent); the = zero vector is automatically included in any vector space. A multidimen= sional eigenspace (that is to say a vector space of eigenvectors with d= imension strictly greater than 1) for fixed eigenvalue and linear trans= formation (and generalization exponent) is degenerate by definition. Th= e algebraic multiplicity $x_5$ of the eigenvalue does not entail degene= racy (of eigenspace) if greater than 1; it is the integer number of occ= urrences of a given eigenvalue $x_1$ in the multiset of eigenvalues (sp= ectrum) of the given linear transformation/square matrix $x_2$. In othe= r words, the characteristic polynomial can be factored into linear poly= nomial primes (with root $x_1$) which are exponentiated to the power $x= _5$ (the multiplicity; notably, not $x_4$). For $x_4$ > $x_5$, the eige= nspace is trivial. $x_2$ may not be diagonalizable. The scalar zero (0)= is a naturally permissible argument of $x_1$ (unlike some cultural mat= hematical definitions in English). Eigenspaces retain the operations an= d properties endowing the vectorspaces to which they belong (as subspac= es). Thus, an eigenspace is more than a set of objects: it is a set of = vectors such that that set is endowed with vectorspace operators and pr= operties. Thus {klesi} alone is insufficient. But the set underlying ei= genspace $x_3$ is a type of {klesi}, with the property of being closed = under linear transformation $x_2$ (up to scalar multiplication). The ve= ctor space and basis being used are not specified by this word. Use thi= s word as a seltau in constructions such as "eigenket", "eigenstate", e= tc. (In such cases, {te} {aigne} is recommended for the typical English= usages of such terms. Use {zei} in lujvo formed by these constructs. T= he term "eigenvector" may be rendered as {cmima} {be} {le} {te} {aigne}= ). See also {gei'ai}, {klesi}, {daigno} 12,12d11 < =09=09Word: eigenspace (generalized), In Sense: mathematical; linear = transformation, vector space; generalized according to equation aforeme= ntioned \n13a13,13 \n> =09=09Word: eigenspace (generalized), In Sense: mathematical; linea= r transformation, vector space; generalized according to equation afore= mentioned Old Data: =09Definition: =09=09$x_1$ is an eigenvalue (or zero) of linear transformation/square = matrix $x_2$, associated with/'owning' all vectors in generalized eigen= space $x_3$ (implies neither nondegeneracy nor degeneracy; default incl= udes the zero vector) with 'eigenspace-generalization' power/exponent $= x_4$ (typically and probably by cultural default will be 1), with algeb= raic multiplicity (of eigenvalue) $x_5$ =09Notes: =09=09For any eigenvector v in generalized eigenspace x3 of linear tran= sformation x2 for eigenvalue x1, where I is the identity matrix/transfo= rmation that works/makes sense in the context, the following equation i= s satisfied: $(((x_2)-(x_1)I)^{x_4})v =3D 0$. When the argument of x4 = is 1, the generalized eigenspace x3 is simply a strict/simple/basic eig= enspace; this is the typical (and probable cultural default) meaning of= this word. x4 will typically be restricted to integer values k > 0. x2= should always be specified (at least implicitly by context), for an ei= genvalue does not mean much without the linear transformation being kno= wn. However, since one usually knows the said linear transformation, a= nd since the basic underlying relationship of this word is "eigen-ness"= , the eigenvalue is given the primary terbri (x1). When filling x3 and/= or x4, x2 and x1 (in that order of importance) should already be (at le= ast contextually implicitly) specified. x3 is the set of all eigenvecto= rs of linear transformation x2, endowed with all of the typical operati= ons of the vector space at hand. The default includes the zero vector = (else the x3 eigenspace is not actually a vector space); normally in th= e context of mathematics, the zero vector is not considered to be an ei= genvector, but by this definition it is included. Thus, a Lojban mathe= matician would consider the zero vector to be an (automatic) eigenvecto= r of the given (in fact, any) linear transformation (particularly ones = represented by a square matrix in a given basis). This is the logicall= y most basic definition, but is contrary to typical mathematical cultur= e. This word implies neither nondegeneracy nor degeneracy of eigenspace= x3. In other words there may or may not be more than one linearly ind= ependent vector in the eigenspace x3 for a given eigenvalue x1 of linea= r transformation x2. x3 is the unique generalized eigenspace of x2 for = given values of x1 and x4. x1 is not necessarily the unique eigenvalue = of linear transformation x2, nor is its multiplicity necessarily 1 for = the same. Beware when converting the terbri structure of this word. I= n fact, the set of all eigenvalues for a given linear transformation x2= will include scalar zero (0); therefore, any linear transformation wit= h a nontrivial set of eigenvalues will have at least two eigenvalues th= at may fill in terbri x1 of this word. The 'eigenvalue' of zero for a = proper/nice linear transformation will produce an 'eigenspace' that is = equivalent to the entire vector space at hand. If x3 is specified by a = set of vectors, the span of that set should fully yield the entire eige= nspace of the linear transformation x2 associated with eigenvalue x1, h= owever the set may be redundant (linearly dependent); the zero vector i= s automatically included in any vector space. A multidimensional eigens= pace (that is to say a vector space of eigenvectors with dimension stri= ctly greater than 1) for fixed eigenvalue and linear transformation (an= d generalization exponent) is degenerate by definition. The algebraic m= ultiplicity x5 of the eigenvalue does not entail degeneracy (of eigensp= ace) if greater than 1; it is the integer number of occurrences of a gi= ven eigenvalue x1 in the multiset of eigenvalues (spectrum) of the give= n linear transformation/square matrix x2. In other words, the character= istic polynomial can be factored into linear polynomial primes (with ro= ot x1) which are exponentiated to the power x5 (the multiplicity; notab= ly, not x4). For x4 > x5, the eigenspace is trivial. x2 may not be diag= onalizable. The scalar zero (0) is a naturally permissible argument of = x1 (unlike some cultural mathematical definitions in English). Eigenspa= ces retain the operations and properties endowing the vectorspaces to w= hich they belong (as subspaces). Thus, an eigenspace is more than a se= t of objects: it is a set of vectors such that that set is endowed with= vectorspace operators and properties. Thus {klesi} alone is insuffici= ent. But the set underlying eigenspace x3 is a type of {klesi}, with t= he property of being closed under linear transformation x2 (up to scala= r multiplication). The vector space and basis being used are not specif= ied by this word. Use this word as a seltau in constructions such as "e= igenket", "eigenstate", etc. (In such cases, {te aigne} is recommended= for the typical English usages of such terms. Use {zei} in lujvo forme= d by these constructs. The term "eigenvector" may be rendered as "cmim= a be le te .aigne"). See also: {gei'ai}, {klesi}, {daigno}. =09Jargon: =09=09 =09Gloss Keywords: =09=09Word: eigen-, In Sense: prefix; mathematical/physical =09=09Word: eigenspace (generalized), In Sense: mathematical; linear tr= ansformation, vector space; generalized according to equation aforement= ioned =09=09Word: eigenspace-generalization exponent, In Sense: mathematical =09=09Word: eigenvalue, In Sense: mathematical; of a square matrix/line= ar transformation =09=09Word: eigenvector, In Sense: mathematical; linear transformation/= square matrix =09=09Word: multiplicity (algebraic) of eigenvalue, In Sense: mathemati= cal; degree of linear terms in characteristic polynomial of the linear = transformation/square matrix; useful for Jordan canonical form computat= ions; algebraic mulitplicity =09=09Word: self-preserving vector under mapping/transformation, In Sen= se: mathematical; (perfect preservation not implied: dilation/contracti= on by scalar, including by scalar zero (0), allowed) =09Place Keywords: New Data: =09Definition: =09=09$x_1$ is an eigenvalue (or zero) of linear transformation/square = matrix $x_2$, associated with/'owning' all vectors in generalized eigen= space $x_3$ (implies neither nondegeneracy nor degeneracy; default incl= udes the zero vector) with 'eigenspace-generalization' power/exponent $= x_4$ (typically and probably by cultural default will be 1), with algeb= raic multiplicity (of eigenvalue) $x_5$ =09Notes: =09=09For any eigenvector v in generalized eigenspace $x_3$ of linear t= ransformation $x_2$ for eigenvalue $x_1$, where I is the identity matri= x/transformation that works/makes sense in the context, the following e= quation is satisfied: $((x_2 - x_1 * I)^{x_4})v =3D 0$. When the argume= nt of $x_4$ is 1, the generalized eigenspace $x_3$ is simply a strict/s= imple/basic eigenspace; this is the typical (and probable cultural defa= ult) meaning of this word. $x_4$ will typically be restricted to intege= r values k > 0. $x_2$ should always be specified (at least implicitly b= y context), for an eigenvalue does not mean much without the linear tra= nsformation being known. However, since one usually knows the said line= ar transformation, and since the basic underlying relationship of this = word is "eigen-ness", the eigenvalue is given the primary terbri ($x_1$= ). When filling $x_3$ and/or $x_4$, $x_2$ and $x_1$ (in that order of i= mportance) should already be (at least contextually implicitly) specifi= ed. $x_3$ is the set of all eigenvectors of linear transformation $x_2$= , endowed with all of the typical operations of the vector space at han= d. The default includes the zero vector (else the $x_3$ eigenspace is n= ot actually a vector space); normally in the context of mathematics, th= e zero vector is not considered to be an eigenvector, but by this defin= ition it is included. Thus, a Lojban mathematician would consider the z= ero vector to be an (automatic) eigenvector of the given (in fact, any)= linear transformation (particularly ones represented by a square matri= x in a given basis). This is the logically most basic definition, but i= s contrary to typical mathematical culture. This word implies neither n= ondegeneracy nor degeneracy of eigenspace $x_3$. In other words there m= ay or may not be more than one linearly independent vector in the eigen= space $x_3$ for a given eigenvalue $x_1$ of linear transformation $x_2$= . $x_3$ is the unique generalized eigenspace of $x_2$ for given values = of $x_1$ and $x_4$. $x_1$ is not necessarily the unique eigenvalue of l= inear transformation $x_2$, nor is its multiplicity necessarily 1 for t= he same. Beware when converting the terbri structure of this word. In f= act, the set of all eigenvalues for a given linear transformation $x_2$= will include scalar zero (0); therefore, any linear transformation wit= h a nontrivial set of eigenvalues will have at least two eigenvalues th= at may fill in terbri $x_1$ of this word. The 'eigenvalue' of zero for = a proper/nice linear transformation will produce an 'eigenspace' that i= s equivalent to the entire vector space at hand. If $x_3$ is specified = by a set of vectors, the span of that set should fully yield the entire= eigenspace of the linear transformation $x_2$ associated with eigenval= ue $x_1$, however the set may be redundant (linearly dependent); the ze= ro vector is automatically included in any vector space. A multidimensi= onal eigenspace (that is to say a vector space of eigenvectors with dim= ension strictly greater than 1) for fixed eigenvalue and linear transfo= rmation (and generalization exponent) is degenerate by definition. The = algebraic multiplicity $x_5$ of the eigenvalue does not entail degenera= cy (of eigenspace) if greater than 1; it is the integer number of occur= rences of a given eigenvalue $x_1$ in the multiset of eigenvalues (spec= trum) of the given linear transformation/square matrix $x_2$. In other = words, the characteristic polynomial can be factored into linear polyno= mial primes (with root $x_1$) which are exponentiated to the power $x_5= $ (the multiplicity; notably, not $x_4$). For $x_4$ > $x_5$, the eigens= pace is trivial. $x_2$ may not be diagonalizable. The scalar zero (0) i= s a naturally permissible argument of $x_1$ (unlike some cultural mathe= matical definitions in English). Eigenspaces retain the operations and = properties endowing the vectorspaces to which they belong (as subspaces= ). Thus, an eigenspace is more than a set of objects: it is a set of ve= ctors such that that set is endowed with vectorspace operators and prop= erties. Thus {klesi} alone is insufficient. But the set underlying eige= nspace $x_3$ is a type of {klesi}, with the property of being closed un= der linear transformation $x_2$ (up to scalar multiplication). The vect= or space and basis being used are not specified by this word. Use this = word as a seltau in constructions such as "eigenket", "eigenstate", etc= . (In such cases, {te} {aigne} is recommended for the typical English u= sages of such terms. Use {zei} in lujvo formed by these constructs. The= term "eigenvector" may be rendered as {cmima} {be} {le} {te} {aigne}).= See also {gei'ai}, {klesi}, {daigno} =09Jargon: =09=09 =09Gloss Keywords: =09=09Word: eigen-, In Sense: prefix; mathematical/physical =09=09Word: eigenspace-generalization exponent, In Sense: mathematical =09=09Word: eigenspace (generalized), In Sense: mathematical; linear tr= ansformation, vector space; generalized according to equation aforement= ioned =09=09Word: eigenvalue, In Sense: mathematical; of a square matrix/line= ar transformation =09=09Word: eigenvector, In Sense: mathematical; linear transformation/= square matrix =09=09Word: multiplicity (algebraic) of eigenvalue, In Sense: mathemati= cal; degree of linear terms in characteristic polynomial of the linear = transformation/square matrix; useful for Jordan canonical form computat= ions; algebraic mulitplicity =09=09Word: self-preserving vector under mapping/transformation, In Sen= se: mathematical; (perfect preservation not implied: dilation/contracti= on by scalar, including by scalar zero (0), allowed) =09Place Keywords: You can go to to see it.