tcanapu
fu'ivla x_{1} (node/vertex/station) is backward of/along from x_{2} in oriented graph x_{3} (graph with orientation) using oriented edge path x_{4} (ordered sequence of ordered pairs; oriented edges). The path from x_{2} to x_{1} along the (nowunoriented version of the) edges used in x_{4} is counter/against/upstream of the orientation of x_{3} (and/or given by the oriented version of x_{4}). Orientation is given from/by x3, so only an edge need be submitted if x3 is fully specified; if x3 is not fully specified, x4 can take the burden of specifying the orientation and subgraph of particular interest (namely, the two vertices x1 and x2, intervening vertices along the path, and the orientation of the given edges connecting them). Note that on an unoriented edge or along a cycle, x1 and x2 might be able to exchange places and/or be equal one another. x4 is an ordered sequence of ordered pairs; the first entry of each pair is the origin node, the second pair is the destination node; the path should probably be connected (so that the destination node of one pair is the origin node of the next, except possibly if it is the last such pair). x1 is not necessarily last (id est: backward adjacent of/from) x2, but it can be. Useful for pages, webpages, family relationships, utterances, etc. See also: grafu, tcanaba, tcanaca


vlalikei
fu'ivla x_{1} [mass/sequence] plays the Lojban word chaining game (vlalinkei) with ruleset x_{2} and winner x_{3} with resulting sentence x_{4} against world champion x_{6} for fabulous cash prize x_{7} and endorsement deal(s) x_{8} groupies x_{9} (except they probably go earlier), played at time x_{10} at location x_{11} and honorific title x_{12} breaking record(s) x_{13} with mindless spectators x_{14} taking time x_{15} [amount] containing most frequently used word x_{16} (zo) and not using perfectly good words x_{17} (zo) displaying new strategy/trick x_{18} supervised by x_{19} with referee x_{20} and used message transmission system x_{21} time limit per move x_{22} shortest move of the game x_{23} broadcast on TV network(s) x_{24} with Neilson ratings x_{25} supplanting previously most watched show x_{26} winning new fans x_{27} who formerly played x_{28} which is inferior for reasons x_{29} by standard x_{30} with banned words x_{31} with words winning additional points x_{32} with climax of suspense x_{33} and best comeback x_{34}


xipfne
obsolete fu'ivla x_{1} emits light characteristic of/is of the color that is described by/(as) arising from/associated with the approximately twentyone centimeter wavelength, hydrogen hyperfine proton/electron spinflip transition (from parallel to antiparallel configuration, id est: from the higher energy state to the lower energy state), electromagnetic radiation In analogy to xunre and the like. See also: skari, blabi, xekri, kandi, carmi, nukni, narju, rozgu, zirpu, pelxu, xunre, cidro, lektoni, protoni, dikca, maksi, guska'u, gusni, cradi. This color is a subset/element (depending on interpretation/usage) of the colors associated with light in the radio and microwave regions of the electromagnetic spectrum. Speaker determines how close "approximately twentyone centimeters" is to exactly twentyone centimeters. Technically can only be used for the coloration of light that would arise from the hyperfine spinflip transition of a hydrogen atom (without any neutrons) in which the protonelectron quantum spin configuration abruptly changes from parallel to antiparallel in the 1s groundstate. But other isotopes and/or hydrogenic (id est: singleelectron) atoms can be referenced by semantic broadening; note that in any case, this word always refers to the color of the light emitted from such an object. This is a color, not a description of the process, conditions under which the light of this color is emitted, etc., nor is it the light itself nor wavelength of the light. However, this term probably will come up in all such descriptions. For example, "hyperfine spinflip transition" might be rendered {xipfne binxo}. Technically, any object that emits photons with wavelengths of approximately twentyone centimeters will be of the color xipfne, regardless of why such emission is occurring (id est: it need not be due to hydrogen hyperfine, proton/electron spinflip transitions). Usage in such a case is perfectly acceptable. However, in practice, such occasions/contexts will be rare (essentially completely absent except in theory) and the only common usage will be in the context of hydrogen hyperfine proton/electron spinflip transitions (as in astronomy, chemistry, or quantum mechanics). This is an electromagnetic (id est: light, photon) color. It is physical (being derived from the properties of the wavelength of the emitted and received photon(s)), but is interpreted by some instrument (such as a telescope/camera system, an animal's optical system, etc.). Usage need not be technical.


aigne
fu'ivla x_{1} is an eigenvalue (or zero) of linear transformation/square matrix x_{2}, associated with/'owning' all vectors in generalized eigenspace x_{3} (implies neither nondegeneracy nor degeneracy; default includes the zero vector) with 'eigenspacegeneralization' power/exponent x_{4} (typically and probably by cultural default will be 1), with algebraic multiplicity (of eigenvalue) x_{5} For any eigenvector v in generalized eigenspace x_{3} of linear transformation x_{2} for eigenvalue x_{1}, 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 x_{4} is 1, the generalized eigenspace x_{3} is simply a strict/simple/basic eigenspace; this is the typical (and probable cultural default) meaning of this word. x_{4} will typically be restricted to integer values k > 0. x_{2} 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 "eigenness", 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) specified. 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 hand. 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 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 x_{3}. In other words there may or may not be more than one linearly independent vector in the eigenspace 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 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 with 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 for a proper/nice linear transformation will produce an 'eigenspace' that is 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 eigenvalue x_{1}, however the set may be redundant (linearly dependent); the zero vector is automatically included in any vector space. A multidimensional eigenspace (that is to say a vector space of eigenvectors with dimension strictly greater than 1) for fixed eigenvalue and linear transformation (and generalization exponent) is degenerate by definition. The algebraic multiplicity x_{5} of the eigenvalue does not entail degeneracy (of eigenspace) if greater than 1; it is the integer number of occurrences of a given eigenvalue x_{1} in the multiset of eigenvalues (spectrum) of the given linear transformation/square matrix x_{2}. In other words, the characteristic polynomial can be factored into linear polynomial 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 eigenspace is trivial. x_{2} may not be diagonalizable. The scalar zero (0) is a naturally permissible argument of x_{1} (unlike some cultural mathematical 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 vectors such that that set is endowed with vectorspace operators and properties. Thus klesi alone is insufficient. But the set underlying eigenspace x_{3} is a type of klesi, with the property of being closed under linear transformation x_{2} (up to scalar multiplication). The vector 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 usages 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


daigno
fu'ivla x_{1} (ordered list) is a sampling of entries of matrix/tensor x_{2} in which exactly one entry is sampled from each row and/or column (etc.) between entries x_{3} (list; default: the largest 'square'/'hypercubic' sampling possible in the entire tensor starting with the first entry, see notes) inclusively following selection procedure/rule/function/order x_{4} (default: diagonally, see notes), where the tensor/matrix is expressed in basis/under conditions x_{5} Entries of the list in x3 need not actually be sampled; the entries listed are merely to name the minimal and maximal indices between which the sampling may be drawn. Thus, the indices/labels specified are included in the range of sampling; id est: if the matrix entries listed belong to the ith row and jth column and the (i+n)th row and (j+m)th column respectively (for positive integers i,j,n,m), then the sampling will be conducted in all rows of number between (and including) i and i+n (yielding n+1 sampled rows) and in all columns of number between (and including) j and j+m (yielding m+1 sampled columns). The default diagonal sampling procedure for x4 is as follows: The first sampled entry has the minimum allowed (as specified in x3) indices. All latter sampled entries (by default) have indices of the immediately previous sampled entry each augmented by 1. (Which is to say that if the kth sampled entry has indices (x,y,...), in that order, then the (k+1)th sampled entry has indices (x+1,y+1,...), in that order and where each subsequent index would be the respective index of the kth sampled entry augmented by 1). The process terminates generally whenever exactly one entry is sampled from each of the rows, each of the columns, etc. of the tensor. In the default, the process terminates when at least one of the indices of a sampled entry of the tensor is as large as possible in the range specified by x3. Thus, in order to reconcile the general and the default termination conditions, the range specified by x3 must be compatible with both; id est: it must be a rdimensional hypercube of entries, so to speak, where r is the rank of tensor x2. The default for sampling range x3 is between and including the entry in the first row and first column (etc.) and the entry in the last row and last column (etc.) for an rdimensional hypercube tensor (meaning that each row, column, etc. of the tensor has exactly the same number of entries as the others). Generally, the default range begins with the entry of indices each minimal in the tensor (called 'the first entry') and extends to include ("draw") the maximal rdimensional hypercube of entries in the tensor with one vertex on the first entry; in other words, if the minimum of the set of maximal indices in the tensor is g, then the sampling range is every row between the first and the gth, every column between the first and the gth, etc. Generally, the sampling range must be an rdimensional orthotope of some positive size (that is to say: including at least one entry) no larger than the tensor itself, but with the freedom to place at most r of its vertices among the entries thereof; if the default sampling procedure x4 is being used, then the rdimensional orthotope must be an rdimensional hypercube. Generalizes to any tensor, but is only interesting for tensors of rank at least 1. Any mention of geometric terminology (such as mention of diagonals, orthotopes, etc.) in the definition or notes of this word should be interpreted cautiously and is not necessarily good Lojbanic practice; such terminology should not necessarily be emulated in practicing Lojbanic thought or speech. Not for use for geometric diagonals (such as between vertices); confer: digno.
