55 research outputs found
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Urn models & operator ordering procedures
Ordering of operators is purely combinatorial task involving a number
of commutators shuffling components of operator expression to desired
form. Here we show how it can be illustrated by simple urn models in
which normal ordering procedure is equivalent to enumeration of urn
histories
On Urn Models, Non-commutativity and Operator Normal Forms
Non-commutativity is ubiquitous in mathematical modeling of reality and in
many cases same algebraic structures are implemented in different situations.
Here we consider the canonical commutation relation of quantum theory and
discuss a simple urn model of the latter. It is shown that enumeration of urn
histories provides a faithful realization of the Heisenberg-Weyl algebra.
Drawing on this analogy we demonstrate how the operator normal forms facilitate
counting of histories via generating functions, which in turn yields an
intuitive combinatorial picture of the ordering procedure itself.Comment: 7 pages, 2 figure
Combinatorial Route to Algebra: The Art of Composition & Decomposition
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co-multiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions
On safe post-selection for Bell tests with ideal detectors: Causal diagram approach
Reasoning about Bell nonlocality from the correlations observed in
post-selected data is always a matter of concern. This is because conditioning
on the outcomes is a source of non-causal correlations, known as a selection
bias, rising doubts whether the conclusion concerns the actual causal process
or maybe it is just an effect of processing the data. Yet, even in the
idealised case without detection inefficiencies, post-selection is an integral
part of experimental designs, not least because it is a part of the
entanglement generation process itself. In this paper we discuss a broad class
of scenarios with post-selection on multiple spatially distributed outcomes. A
simple criterion is worked out, called the all-but-one principle, showing when
the conclusions about nonlocality from breaking Bell inequalities with
post-selected data remain in force. Generality of this result, attained by
adopting the high-level diagrammatic tools of causal inference, provides safe
grounds for systematic reasoning based on the standard form of multipartite
Bell inequalities in a wide array of entanglement generation schemes, without
worrying about the dangers of selection bias. In particular, it can be applied
to post-selection defined by single-particle events in each detection chanel
when the number of particles in the system is conserved.Comment: 16 pages, 9 figure
One-parameter groups and combinatorial physics
In this communication, we consider the normal ordering of sums of elements of
the form (a*^r a a*^s), where a* and a are boson creation and annihilation
operators. We discuss the integration of the associated one-parameter groups
and their combinatorial by-products. In particular, we show how these groups
can be realized as groups of substitutions with prefunctions.Comment: 15 pages, 23 references. Presented at the Third International
Workshop on Contemporary Problems in Mathematical Physics (COPROMAPH3),
Porto-Novo (Benin), November 200
Proportional valve with a piezoelectric actuator
The article concerns a slotted proportional valve for use in pneumatic and hydraulic systems. There is a growing demand for both hydraulic and pneumatic ultrafast proportional valves. The conducted analysis of literature confirms the lack of such solutions for proportional valves. The currently used pneumatic systems for selection and segregation of parts and objects require ultrafast valves. The presented solution for the proportional valve can significantly improve and accelerate this type of technological processes. Furthermore, fast proportional valves can be successfully used for positional control of pneumatic and hydraulic drives. The article presents the design of a slotted divide valve and sets the maximum mass flow rate for service roads
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