648 research outputs found

### New non-unitary representations in a Dirac hydrogen atom

New non-unitary representations of the SU(2) algebra are introduced for the
case of the Dirac equation with a Coulomb potential; an extra phase, needed to
close the algebra, is also introduced. The new representations does not require
integer or half integer labels. The set of operators defined are used to span
the complete space of bound state eigenstates of the problem thus solving it in
an essentially algebraic way

### Efficient Simulation of Quantum State Reduction

The energy-based stochastic extension of the Schrodinger equation is a rather
special nonlinear stochastic differential equation on Hilbert space, involving
a single free parameter, that has been shown to be very useful for modelling
the phenomenon of quantum state reduction. Here we construct a general closed
form solution to this equation, for any given initial condition, in terms of a
random variable representing the terminal value of the energy and an
independent Brownian motion. The solution is essentially algebraic in
character, involving no integration, and is thus suitable as a basis for
efficient simulation studies of state reduction in complex systems.Comment: 4 pages, No Figur

### Essentially Algebraic Descriptions of Locally Presentable Categories

Local presentability has turned out to be one of the most fruitful concepts in category theory. The fact, that a category is locally finitely presentable iff it is equivalent to the category of models of an essentially algebraic, finitary theory, is widely known. Unfortunately, the existing approaches in literature are either unsatisfactory - with respect to existing examples and to the number of sorts - or even wrong. Now the aim of the main part of this thesis is to give an intuitive proof of the mentioned fact, which covers existing examples, and can be generalized to the non-finitary case under mild assumptions. Here the set of sorts of such a description of a locally finitely presentable category is given by a strong generator of finitely presentables in this category. Also, this construction provides a new approach to the known characterization of quasivarieties. Further, the theory constructed for a locally finitely presentable category is some kind of clone. A non-trivial example of a locally finitely presentable category with a managable strong generator of finitely presentables is given by the category of coalgebras for a polynomial set-endofunctor. In the second part of this dissertation we show that this category is even equivalent to some variety of unary algebras without equations. Moreover, we characterize polynomial set-endofunctors by the property that the corresponding category of coalgebras is concretely equivalent to some presheaf category. Finally, we introduce for an endofunctor the concept of polynomiality w.r.t. a family of functors, and show that - under certain assumptions - several constructions and properties can be lifted to the corresponding category of coalgebras

### Oscillatory integral operators with homogeneous polynomial phases in several variables

We obtain $L^2$ decay estimates in $\lambda$ for oscillatory integral
operators whose phase functions are homogeneous polynomials of degree m and
satisfy various genericity assumptions. The decay rates obtained are optimal in
the case of (2+2)--dimensions for any m while, in higher dimensions, the result
is sharp for m sufficiently large. The proof for large $m$ follows from
essentially algebraic considerations. For cubics in (2+2)--dimensions, the
proof involves decomposing the operator near the conic zero variety of the
determinant of the Hessian of the phase function, using an elaboration of the
general approach of Phong and Stein [1994].Comment: 39 pages, 2 figures; minor corrections; to appear in Journal of
Functional Analysi

### Taub-NUT Dynamics with a Magnetic Field

We study classical and quantum dynamics on the Euclidean Taub-NUT geometry
coupled to an abelian gauge field with self-dual curvature and show that, even
though Taub-NUT has neither bounded orbits nor quantum bound states, the
magnetic binding via the gauge field produces both. The conserved Runge-Lenz
vector of Taub-NUT dynamics survives, in a modified form, in the gauged model
and allows for an essentially algebraic computation of classical trajectories
and energies of quantum bound states. We also compute scattering cross sections
and find a surprising electric-magnetic duality. Finally, we exhibit the
dynamical symmetry behind the conserved Runge-Lenz and angular momentum vectors
in terms of a twistorial formulation of phase space.Comment: 36 pages, three figure

### On the Model of Computation of Place/Transition Petri Nets

In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical "token game", one can model the behaviour of Petri nets via non-sequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. More precisely, we introduce the new notion of decorated processes of Petri nets and we show that they induce on nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net N can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification

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