On the construction of staggered linear bases

A staggered linear basis is a specific basis for a polynomial ideal generating the ideal as a linear vector space. There are some algorithms in the literature to compute these bases and we show that they do not work properly. In this paper, due to the strong connection between Gröbner bases and staggered linear bases and by applying a signature-based structure, we present a new and correct algorithm to compute staggered linear bases. Since the output of this algorithm also remains a Gröbner basis for its input ideal, we conclude the paper by discussing the performance of this algorithm compared with the newest signature-based algorithm to compute Gröbner bases.

Paleochronic reversion in Psophocarpus: Dynamics II, rhizogeny on floral anatomic fields

Paleochronic reversion is confirmed in Psophocarpus as a basic floral ground state. That state can expand to include dynamics T (g,,h) ) of axial expansion (AE) as a permutation (T X ) phase beginning as phyllotactic floral phylloid (T Phyld ) and/or axial decompression (T Axl ) manifest as linear elongation (T Long ) and/or rotation (T Rtn ) and/or latitudinal (TS Lat ) expansion. Organ regions present a continuum as a vector space LT Axl of floral axial transformation. A generative phase of meristem activity (T (Rz, SAM, Infl) ) can follow. Experiments with 49 phylloid and/or phyllome paleochronically reverted flowers presented varying degrees of phyllotactic permutation involving development of a pericladial stalk (PCL) and/or inter-bracts stem (IBS) and/or activated pedicel (PdcL) and/or gynophore (Gnf), Cupule-like (Gnf)/Cupl-Lk) elongation. A meristem generative function included rhizogeny as root site generation (RSG) at the calyx (Cl), PCL, bracts (Bt), IBS, PdcL and/or Gnf/Cupl-Lk regions manifest as eigenvector functions as RSG whose density of generation was associated with permutation of the ground state. A continuum of pedicel to calyx regions constitutes a subset [PdcL,Cl] of a linear vector space [Bt,Crpl]=T X → [PdcL,Crpl]=LT Axl whose extension is defined within the space: ∑ F PdcL + F Bt (1,,z) ± S IBS (1,,x) + F PCL (1,,w) + F Cl + F Gyncm ± S Gnf ± S Cupl-Lk = L T Axl . The vector space LT Axl transforms to a master vector field (F [c,..,d] ) of generated Euclidian eigenvectors so that: LT Axl → F [c,..,d]

Generation of Basis Vectors for Magnetic Structures and Displacement Modes

Increasing attention is being focused on the use of symmetry-adapted functions to describe magnetic structures, structural distortions, and incommensurate crystallography. Though the calculation of such functions is well developed, significant difficulties can arise such as the generation of too many or too few basis functions to minimally span the linear vector space. We present an elegant solution to these difficulties using the concept of basis sets and discuss previous work in this area using this concept. Further, we highlight the significance of unitary irreducible representations in this method and provide the first validation that the irreducible representations of the crystallographic space groups tabulated by Kovalev are unitary.

Isomorphic Operators and Functional Equations for the Skew-Circulant Algebra

The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra ofn×ncomplex skew-circulant matrices are displayed in this paper.

Linear vector space derivation of new expressions for the pseudo inverse of rectangular matrices

In this paper, a family of simple formulas for the calculation of the pseudo inverse of a rectangular matrix of less than maximum rank is derived using linear vector space methods. The principal result is that the pseudo inverse A+ of a matrix A can be calculated as A+ = Q(PTAQ)−1PT, where P and Q are rectangular matrices whose r columns are vectors that form a basis for the spaces spanned by the columns and rows, respectively, of matrix A. This leaves the user the liberty to choose the basis to take into consideration other questions such as amount of work needed and ill-conditioning of the matrix that has to be inverted. The formulas are particularized for rectangular matrices that have maximum rank and for the trivial case in which the original matrix is non-singular. Illustrative numerical examples are worked out for several choices of basis vectors and the results are compared with those provided by the program Mathematica through its function PseudoInverse[A].