2,697 research outputs found
Multiplicity Preserving Triangular Set Decomposition of Two Polynomials
In this paper, a multiplicity preserving triangular set decomposition
algorithm is proposed for a system of two polynomials. The algorithm decomposes
the variety defined by the polynomial system into unmixed components
represented by triangular sets, which may have negative multiplicities. In the
bivariate case, we give a complete algorithm to decompose the system into
multiplicity preserving triangular sets with positive multiplicities. We also
analyze the complexity of the algorithm in the bivariate case. We implement our
algorithm and show the effectiveness of the method with extensive experiments.Comment: 18 page
The Howe dual pair in Hermitean Clifford analysis
Clifford analysis offers a higher dimensional function theory studying the null solutions of the rotation invariant, vector valued, first order Dirac operator â. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure J on Euclidean space and a corresponding second Dirac operator âJ , leading to the system of equations âf = 0 = âJf, expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group. In this paper we show that
this choice of equations is fully justified. Indeed, constructing the Howe dual for the action of the unitary group on the space of all spinor valued polynomials, the generators of the resulting Lie superalgebra reveal the natural set of equations to be considered in thiscontext, which exactly coincide with the chosen ones
Rook placements and Jordan forms of upper-triangular nilpotent matrices
The set of n by n upper-triangular nilpotent matrices with entries in a
finite field F_q has Jordan canonical forms indexed by partitions lambda of n.
We present a combinatorial formula for computing the number F_\lambda(q) of
matrices of Jordan type lambda as a weighted sum over standard Young tableaux.
We also study a connection between these matrices and non-attacking rook
placements, which leads to a refinement of the formula for F_\lambda(q).Comment: 25 pages, 6 figure
Polynomial Interrupt Timed Automata
Interrupt Timed Automata (ITA) form a subclass of stopwatch automata where
reachability and some variants of timed model checking are decidable even in
presence of parameters. They are well suited to model and analyze real-time
operating systems. Here we extend ITA with polynomial guards and updates,
leading to the class of polynomial ITA (PolITA). We prove the decidability of
the reachability and model checking of a timed version of CTL by an adaptation
of the cylindrical decomposition method for the first-order theory of reals.
Compared to previous approaches, our procedure handles parameters and clocks in
a unified way. Moreover, we show that PolITA are incomparable with stopwatch
automata. Finally additional features are introduced while preserving
decidability
Nilpotent normal form for divergence-free vector fields and volume-preserving maps
We study the normal forms for incompressible flows and maps in the
neighborhood of an equilibrium or fixed point with a triple eigenvalue. We
prove that when a divergence free vector field in has nilpotent
linearization with maximal Jordan block then, to arbitrary degree, coordinates
can be chosen so that the nonlinear terms occur as a single function of two
variables in the third component. The analogue for volume-preserving
diffeomorphisms gives an optimal normal form in which the truncation of the
normal form at any degree gives an exactly volume-preserving map whose inverse
is also polynomial inverse with the same degree.Comment: laTeX, 20 pages, 1 figur
Nilpotent normal form for divergence-free vector fields and volume-preserving maps
We study the normal forms for incompressible flows and maps in the
neighborhood of an equilibrium or fixed point with a triple eigenvalue. We
prove that when a divergence free vector field in has nilpotent
linearization with maximal Jordan block then, to arbitrary degree, coordinates
can be chosen so that the nonlinear terms occur as a single function of two
variables in the third component. The analogue for volume-preserving
diffeomorphisms gives an optimal normal form in which the truncation of the
normal form at any degree gives an exactly volume-preserving map whose inverse
is also polynomial inverse with the same degree.Comment: laTeX, 20 pages, 1 figur
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