32,073 research outputs found
Non-degenerate umbilics, the path formulation and gradient bifurcation problems
Parametrised contact-equivalence is successful for the understanding and classification of the qualitative local behaviour of bifurcation
diagrams and their perturbations. Path formulation is an alternative point of view. It makes explicit the singular behaviour due to the core of the bifurcation germ (when the parameters vanish) from the effects of the way parameters enter.
Here we show how path formulation can be used to classify and structure efficiently multiparameter bifurcation problems in corank 2 problems. In particular, the non degenerate umbilics singularities are the generic cores in four situations: the general or gradient problems and the Z_2-equivariant (general or gradient) problems
where Z_2 acts on the second component of R^2 via
the reflection kappa(x,y)=(x,-y). The universal unfolding of the umbilic singularities have an interesting 'Russian doll' type of structure
of universal unfoldings in all those categories.
In our approach we can handle one, or more, parameter situations using the same framework. We can even consider some special parameter structure (for instance some internal hierarchy). We classify the generic bifurcations with 1, 2 or 3 parameters that occur in those cases. Some results are known with one bifurcation parameter, but the others are new.
We discuss some application to the bifurcation of a cylindrical panel under different loads structure. This problem has many natural parameters that provide concrete examples of our generic diagrams around the first interaction of the buckling modes
Many-Body Expanded Full Configuration Interaction. II. Strongly Correlated Regime
In this second part of our series on the recently proposed many-body expanded
full configuration interaction (MBE-FCI) method, we introduce the concept of
multideterminantal expansion references. Through theoretical arguments and
numerical validations, the use of this class of starting points is shown to
result in a focussed compression of the MBE decomposition of the FCI energy,
thus allowing chemical problems dominated by strong correlation to be addressed
by the method. The general applicability and performance enhancements of
MBE-FCI are verified for standard stress tests such as the bond dissociations
in HO, N, C, and a linear H chain. Furthermore, the benefits
of employing a multideterminantal expansion reference in accelerating
calculations of high accuracy are discussed, with an emphasis on calculations
in extended basis sets. As an illustration of this latter quality of the
MBE-FCI method, results for HO and C in basis sets ranging from double-
to pentuple- quality are presented, demonstrating near-ideal parallel
scaling on up to almost processing units.Comment: 41 pages, 4 tables, 10 figures, 1 SI attached as an ancillary fil
Social choice theory, game theory, and positive political theory
We consider the relationships between the collective preference and non-cooperative game theory approaches to positive political theory. In particular, we show that an apparently decisive difference between the two approachesthat in sufficiently complex environments (e.g. high-dimensional choice spaces) direct preference aggregation models are incapable of generating any prediction at all, whereas non-cooperative game-theoretic models almost always generate predictionis indeed only an apparent difference. More generally, we argue that when modeling collective decisions there is a fundamental tension between insuring existence of well-defined predictions, a criterion of minimal democracy, and general applicability to complex environments; while any two of the three are compatible under either approach, neither collective preference nor non-cooperative game theory can support models that simultaneously satisfy all three desiderata
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