3,450 research outputs found
The Differential Counting Polynomial
The aim of this paper is a quantitative analysis of the solution set of a
system of polynomial nonlinear differential equations, both in the ordinary and
partial case. Therefore, we introduce the differential counting polynomial, a
common generalization of the dimension polynomial and the (algebraic) counting
polynomial. Under mild additional asumptions, the differential counting
polynomial decides whether a given set of solutions of a system of differential
equations is the complete set of solutions
The Differential Dimension Polynomial for Characterizable Differential Ideals
We generalize the differential dimension polynomial from prime differential
ideals to characterizable differential ideals. Its computation is algorithmic,
its degree and leading coefficient remain differential birational invariants,
and it decides equality of characterizable differential ideals contained in
each other
On the Ext-computability of Serre quotient categories
To develop a constructive description of in categories of
coherent sheaves over certain schemes, we establish a binatural isomorphism
between the -groups in Serre quotient categories
and a direct limit of -groups in the
ambient Abelian category . For the isomorphism
follows if the thick subcategory is
localizing. For the higher extension groups we need further assumptions on
. With these categories in mind we cannot assume
to have enough projectives or injectives and
therefore use Yoneda's description of .Comment: updated bibliography and deleted remaining occurrences of "maximally
Thomas decompositions of parametric nonlinear control systems
This paper presents an algorithmic method to study structural properties of
nonlinear control systems in dependence of parameters. The result consists of a
description of parameter configurations which cause different control-theoretic
behaviour of the system (in terms of observability, flatness, etc.). The
constructive symbolic method is based on the differential Thomas decomposition
into disjoint simple systems, in particular its elimination properties
On monads of exact reflective localizations of Abelian categories
In this paper we define Gabriel monads as the idempotent monads associated to
exact reflective localizations in Abelian categories and characterize them by a
simple set of properties. The coimage of a Gabriel monad is a Serre quotient
category. The Gabriel monad induces an equivalence between its coimage and its
image, the localizing subcategory of local objects.Comment: fixed Prop. 2.10, updated bibliograph
Satisfiability Games for Branching-Time Logics
The satisfiability problem for branching-time temporal logics like CTL*, CTL
and CTL+ has important applications in program specification and verification.
Their computational complexities are known: CTL* and CTL+ are complete for
doubly exponential time, CTL is complete for single exponential time. Some
decision procedures for these logics are known; they use tree automata,
tableaux or axiom systems. In this paper we present a uniform game-theoretic
framework for the satisfiability problem of these branching-time temporal
logics. We define satisfiability games for the full branching-time temporal
logic CTL* using a high-level definition of winning condition that captures the
essence of well-foundedness of least fixpoint unfoldings. These winning
conditions form formal languages of \omega-words. We analyse which kinds of
deterministic {\omega}-automata are needed in which case in order to recognise
these languages. We then obtain a reduction to the problem of solving parity or
B\"uchi games. The worst-case complexity of the obtained algorithms matches the
known lower bounds for these logics. This approach provides a uniform, yet
complexity-theoretically optimal treatment of satisfiability for branching-time
temporal logics. It separates the use of temporal logic machinery from the use
of automata thus preserving a syntactical relationship between the input
formula and the object that represents satisfiability, i.e. a winning strategy
in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner
closure of the input formula only. Last but not least, the games presented here
come with an attempt at providing tool support for the satisfiability problem
of complex branching-time logics like CTL* and CTL+
3D billiards: visualization of regular structures and trapping of chaotic trajectories
The dynamics in three-dimensional billiards leads, using a Poincar\'e
section, to a four-dimensional map which is challenging to visualize. By means
of the recently introduced 3D phase-space slices an intuitive representation of
the organization of the mixed phase space with regular and chaotic dynamics is
obtained. Of particular interest for applications are constraints to classical
transport between different regions of phase space which manifest in the
statistics of Poincar\'e recurrence times. For a 3D paraboloid billiard we
observe a slow power-law decay caused by long-trapped trajectories which we
analyze in phase space and in frequency space. Consistent with previous results
for 4D maps we find that: (i) Trapping takes place close to regular structures
outside the Arnold web. (ii) Trapping is not due to a generalized
island-around-island hierarchy. (iii) The dynamics of sticky orbits is governed
by resonance channels which extend far into the chaotic sea. We find clear
signatures of partial transport barriers. Moreover, we visualize the geometry
of stochastic layers in resonance channels explored by sticky orbits.Comment: 20 pages, 11 figures. For videos of 3D phase-space slices and
time-resolved animations see http://www.comp-phys.tu-dresden.de/supp
Algorithmic Thomas Decomposition of Algebraic and Differential Systems
In this paper, we consider systems of algebraic and non-linear partial
differential equations and inequations. We decompose these systems into
so-called simple subsystems and thereby partition the set of solutions. For
algebraic systems, simplicity means triangularity, square-freeness and
non-vanishing initials. Differential simplicity extends algebraic simplicity
with involutivity. We build upon the constructive ideas of J. M. Thomas and
develop them into a new algorithm for disjoint decomposition. The given paper
is a revised version of a previous paper and includes the proofs of correctness
and termination of our decomposition algorithm. In addition, we illustrate the
algorithm with further instructive examples and describe its Maple
implementation together with an experimental comparison to some other
triangular decomposition algorithms.Comment: arXiv admin note: substantial text overlap with arXiv:1008.376
N,N'-dimethylperylene-3,4,9,10-bis(dicarboximide) on alkali halide(001) surfaces
The growth of N,N'-dimethylperylene-3,4,9,10-bis(dicarboximide) (DiMe-PTCDI)
on KBr(001) and NaCl(001) surfaces has been studied. Experimental results have
been achieved using frequency modulation atomic force microscopy at room
temperature under ultra-high vacuum conditions. On both substrates, DiMe-PTCDI
forms molecular wires with a width of 10 nm, typically, and a length of up to
600 nm at low coverages. All wires grow along the [110] direction (or
[10] direction, respectively) of the alkali halide (001) substrates.
There is no wetting layer of molecules: Atomic resolution of the substrates can
be achieved between the wires. The wires are mobile on KBr surface but
substantially more stable on NaCl. A p(2 x 2) superstructure in brickwall
arrangement on the ionic crystal surfaces is proposed based on electrostatic
considerations. Calculations and Monte-Carlo simulations using empirical
potentials reveal possible growth mechanisms for molecules within the first
layer for both substrates, also showing a significantly higher binding energy
for NaCl(001). For KBr, the p(2 x 2) superstructure is confirmed by the
simulations, for NaCl, a less dense, incommensurate superstructure is
predicted.Comment: 5 pages, 5 figure
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