2,520 research outputs found
Parallels between control PDE's (Partial Differential Equations) and systems of ODE's (Ordinary Differential Equations)
System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation
Canonical coordinates for partial differential equations
Necessary and sufficient conditions are found under which operators of the form Sigma(m, j=1) X(2)sub j + X sub 0 can be made constant coefficient. In addition, necessary and sufficient conditions are derived which classify those linear partial differential operators that can be moved to the Kolmogorov type
Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields
Let K be a finite field with q elements and let X be a subset of a projective
space P^{s-1}, over the field K, which is parameterized by Laurent monomials.
Let I(X) be the vanishing ideal of X. Some of the main contributions of this
paper are in determining the structure of I(X) and some of their invariants. It
is shown that I(X) is a lattice ideal. We introduce the notion of a
parameterized code arising from X and present algebraic methods to compute and
study its dimension, length and minimum distance. For a parameterized code
arising from a connected graph we are able to compute its length and to make
our results more precise. If the graph is non-bipartite, we show an upper bound
for the minimum distance. We also study the underlying geometric structure of
X.Comment: Finite Fields Appl., to appea
Cohen-Macaulay graphs and face vectors of flag complexes
We introduce a construction on a flag complex that, by means of modifying the
associated graph, generates a new flag complex whose -factor is the face
vector of the original complex. This construction yields a vertex-decomposable,
hence Cohen-Macaulay, complex. From this we get a (non-numerical)
characterisation of the face vectors of flag complexes and deduce also that the
face vector of a flag complex is the -vector of some vertex-decomposable
flag complex. We conjecture that the converse of the latter is true and prove
this, by means of an explicit construction, for -vectors of Cohen-Macaulay
flag complexes arising from bipartite graphs. We also give several new
characterisations of bipartite graphs with Cohen-Macaulay or Buchsbaum
independence complexes.Comment: 14 pages, 3 figures; major updat
Natural polymer based composite scaffolds for tissue engineering applications
The fabrication, characterization, and bio-assessment of two types of perspective tissue engineering (TE) scaffolds are presented. Principally derived of biopolymers, both types of scaffolds generally followed porous scaffold methodologies for synthesis. Differentiating the two scaffold varieties was chiefly driven by crosslinking attainment, where crosslinking is argued to add structural stability and aid in regulating biodegradability rates in TE scaffolds. Microwave irradiation via conventional microwave was one method used to prospectively crosslink cornstarch to chitosan and sodium alginate. Triethyl orthoformate, was used to prospectively crosslink collagen and chitosan. After the scaffolds were “crosslinked” they were subjected to freeze drying techniques in order to exploit the sublimation of ice crystals frozen within the scaffolds, to produce a porous-permeable microstructure, vital for promoting cellular processes. Osteoblast MC3T3 cells and fibroblast cells were used for the bio-assessment to suggest the scaffolds as viable candidates for tissue engineering applications for bone and skin regeneration programs
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