585 research outputs found
Computing elements of certain form in ideals to prove properties of operators
Proving statements about linear operators expressed in terms of identities
often leads to finding elements of certain form in noncommutative polynomial
ideals. We illustrate this by examples coming from actual operator statements
and discuss relevant algorithmic methods for finding such polynomials based on
noncommutative Gr\"obner bases. In particular, we present algorithms for
computing the intersection of a two-sided ideal with a one-sided ideal as well
as for computing homogeneous polynomials in two-sided ideals and monomials in
one-sided ideals. All methods presented in this work are implemented in the
Mathematica package OperatorGB.Comment: 26 page
A mathematical framework for yield (versus rate) optimization in constraint-based modeling and applications in metabolic engineering
Formal proofs of operator identities by a single formal computation
A formal computation proving a new operator identity from known ones is, in
principle, restricted by domains and codomains of linear operators involved,
since not any two operators can be added or composed. Algebraically, identities
can be modelled by noncommutative polynomials and such a formal computation
proves that the polynomial corresponding to the new identity lies in the ideal
generated by the polynomials corresponding to the known identities. In order to
prove an operator identity, however, just proving membership of the polynomial
in the ideal is not enough, since the ring of noncommutative polynomials
ignores domains and codomains. We show that it suffices to additionally verify
compatibility of this polynomial and of the generators of the ideal with the
labelled quiver that encodes which polynomials can be realized as linear
operators. Then, for every consistent representation of such a quiver in a
linear category, there exists a computation in the category that proves the
corresponding instance of the identity. Moreover, by assigning the same label
to several edges of the quiver, the algebraic framework developed allows to
model different versions of an operator by the same indeterminate in the
noncommutative polynomials.Comment: 22 page
Compatible rewriting of noncommutative polynomials for proving operator identities
The goal of this paper is to prove operator identities using equalities
between noncommutative polynomials. In general, a polynomial expression is not
valid in terms of operators, since it may not be compatible with domains and
codomains of the corresponding operators. Recently, some of the authors
introduced a framework based on labelled quivers to rigorously translate
polynomial identities to operator identities. In the present paper, we extend
and adapt the framework to the context of rewriting and polynomial reduction.
We give a sufficient condition on the polynomials used for rewriting to ensure
that standard polynomial reduction automatically respects domains and codomains
of operators. Finally, we adapt the noncommutative Buchberger procedure to
compute additional compatible polynomials for rewriting. In the package
OperatorGB, we also provide an implementation of the concepts developed.Comment: 17 page
Algebraic proof methods for identities of matrices and operators: improvements of Hartwig's triple reverse order law
When improving results about generalized inverses, the aim often is to do
this in the most general setting possible by eliminating superfluous
assumptions and by simplifying some of the conditions in statements. In this
paper, we use Hartwig's well-known triple reverse order law as an example for
showing how this can be done using a recent framework for algebraic proofs and
the software package OperatorGB. Our improvements of Hartwig's result are
proven in rings with involution and we discuss computer-assisted proofs that
show these results in other settings based on the framework and a single
computation with noncommutative polynomials
Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems
We consider solution operators of linear ordinary boundary problems with "too
many" boundary conditions, which are not always solvable. These generalized
Green's operators are a certain kind of generalized inverses of differential
operators. We answer the question when the product of two generalized Green's
operators is again a generalized Green's operator for the product of the
corresponding differential operators and which boundary problem it solves.
Moreover, we show that---provided a factorization of the underlying
differential operator---a generalized boundary problem can be factored into
lower order problems corresponding to a factorization of the respective Green's
operators. We illustrate our results by examples using the Maple package
IntDiffOp, where the presented algorithms are implemented.Comment: 19 page
Experimental Measurement of the Berry Curvature from Anomalous Transport
Geometrical properties of energy bands underlie fascinating phenomena in a
wide-range of systems, including solid-state materials, ultracold gases and
photonics. Most famously, local geometrical characteristics like the Berry
curvature can be related to global topological invariants such as those
classifying quantum Hall states or topological insulators. Regardless of the
band topology, however, any non-zero Berry curvature can have important
consequences, such as in the semi-classical evolution of a wave packet. Here,
we experimentally demonstrate for the first time that wave packet dynamics can
be used to directly map out the Berry curvature. To this end, we use optical
pulses in two coupled fibre loops to study the discrete time-evolution of a
wave packet in a 1D geometrical "charge" pump, where the Berry curvature leads
to an anomalous displacement of the wave packet under pumping. This is both the
first direct observation of Berry curvature effects in an optical system, and,
more generally, the proof-of-principle demonstration that semi-classical
dynamics can serve as a high-resolution tool for mapping out geometrical
properties
Evidence for the η_b(1S) Meson in Radiative Υ(2S) Decay
We have performed a search for the η_b(1S) meson in the radiative decay of the Υ(2S) resonance using a sample of 91.6 × 10^6 Υ(2S) events recorded with the BABAR detector at the PEP-II B factory at the SLAC National Accelerator Laboratory. We observe a peak in the photon energy spectrum at E_γ = 609.3^(+4.6)_(-4.5)(stat)±1.9(syst) MeV, corresponding to an η_b(1S) mass of 9394.2^(+4.8)_(-4.9)(stat) ± 2.0(syst) MeV/c^2. The branching fraction for the decay Υ(2S) → γη_b(1S) is determined to be [3.9 ± 1.1(stat)^(+1.1)_(-0.9)(syst)] × 10^(-4). We find the ratio of branching fractions B[Υ(2S) → γη_b(1S)]/B[Υ(3S) → γη_b(1S)]= 0.82 ± 0.24(stat)^(+0.20)_(-0.19)(syst)
Solitary waves in the Nonlinear Dirac Equation
In the present work, we consider the existence, stability, and dynamics of
solitary waves in the nonlinear Dirac equation. We start by introducing the
Soler model of self-interacting spinors, and discuss its localized waveforms in
one, two, and three spatial dimensions and the equations they satisfy. We
present the associated explicit solutions in one dimension and numerically
obtain their analogues in higher dimensions. The stability is subsequently
discussed from a theoretical perspective and then complemented with numerical
computations. Finally, the dynamics of the solutions is explored and compared
to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger
equation. A few special topics are also explored, including the discrete
variant of the nonlinear Dirac equation and its solitary wave properties, as
well as the PT-symmetric variant of the model
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