1,734,666 research outputs found
From IF to BI: a tale of dependence and separation
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their compositional
semantics due to Hodges. We show how Hodges' semantics can be seen as a special
case of a general construction, which provides a context for a useful
completeness theorem with respect to a wider class of models. We shed some new
light on each aspect of the logic. We show that the natural propositional logic
carried by the semantics is the logic of Bunched Implications due to Pym and
O'Hearn, which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics of
informational dependence, but which we show play a very natural role, most
notably intuitionistic implication. As regards the quantifiers, we show that
their interpretation in the Hodges semantics is forced, in that they are the
image under the general construction of the usual Tarski semantics; this
implies that they are adjoints to substitution, and hence uniquely determined.
As for the dependence predicate, we show that this is definable from a simpler
predicate, of constancy or dependence on nothing. This makes essential use of
the intuitionistic implication. The Armstrong axioms for functional dependence
are then recovered as a standard set of axioms for intuitionistic implication.
We also prove a full abstraction result in the style of Hodges, in which the
intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio
Characterization of Birkhoff’s Conditions by Means of Cover-Preserving and Partially Cover-Preserving Sublattices
In the paper we investigate Birkhoff’s conditions (Bi) and (Bi*). We prove that a discrete lattice L satisfies the condition (Bi) (the condition (Bi*)) if and only if L is a 4-cell lattice not containing a cover-preserving sublattice isomorphic to the lattice S*7 (the lattice S7). As a corollary we obtain a well known result of J. Jakubík from [6]. Furthermore, lattices S7 and S*7 are considered as so-called partially cover-preserving sublattices of a given lattice L, S7 ≪ L and S*7 ≪ L, in symbols. It is shown that an upper continuous lattice L satisfies (Bi*) if and only if L is a 4-cell lattice such that S7 ≪ L. The final corollary is a generalization of Jakubík’s theorem for upper continuous and strongly atomic lattices
On Quasi-inversions
Given a bounded domain strictly starlike with
respect to we define a quasi-inversion w.r.t. the boundary
We show that the quasi-inversion is bi-Lipschitz w.r.t. the
chordal metric if and only if every "tangent line" of is far away
from the origin. Moreover, the bi-Lipschitz constant tends to when
approaches the unit sphere in a suitable way. For the formulation
of our results we use the concept of the -tangent condition due to F.
W. Gehring and J. V\"ais\"al\"a (Acta Math. 1965). This condition is shown to
be equivalent to the bi-Lipschitz and quasiconformal extension property of what
we call the polar parametrization of . In addition, we show that
the polar parametrization, which is a mapping of the unit sphere onto is bi-Lipschitz if and only if satisfies the -tangent
condition.Comment: 22 pages; 5 figure
Plastic Flow in Two-Dimensional Solids
A time-dependent Ginzburg-Landau model of plastic deformation in
two-dimensional solids is presented. The fundamental dynamic variables are the
displacement field \bi u and the lattice velocity {\bi v}=\p {\bi u}/\p t.
Damping is assumed to arise from the shear viscosity in the momentum equation.
The elastic energy density is a periodic function of the shear and tetragonal
strains, which enables formation of slips at large strains. In this work we
neglect defects such as vacancies, interstitials, or grain boundaries. The
simplest slip consists of two edge dislocations with opposite Burgers vectors.
The formation energy of a slip is minimized if its orientation is parallel or
perpendicular to the flow in simple shear deformation and if it makes angles of
with respect to the stretched direction in uniaxial stretching.
High-density dislocations produced in plastic flow do not disappear even if
the flow is stopped. Thus large applied strains give rise to metastable,
structurally disordered states. We divide the elastic energy into an elastic
part due to affine deformation and a defect part. The latter represents degree
of disorder and is nearly constant in plastic flow under cyclic straining.Comment: 16pages, Figures can be obtained at
http://stat.scphys.kyoto-u.ac.jp/index-e.htm
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