Sentence examples for minimal properties from inspiring English sources

Fraser argues, first, that the representations for free and interacting systems are unavoidably unitarily inequivalent, and second, that the representation for an interacting system does not have the minimal properties that are needed for any particle interpretation e.g.g

For the conclusions of this article to hold for networks constructed with real cells, the minimal properties of Table 1 only need to be approximations of actual complex neuron responses.

The "goal matrix", G, contains all desirable EM, which define the minimal properties of the NMF and must therefore be kept.

When Russian, Kazakh or American billionaires buy mansions, the government ensures they pay minimal property and foreign earnings taxes.

It is called that a subset (Asubset Y) has the D-weak minimal property, if for all (yin A) there exists a weakly minimal a of A such that (a-yin -operatorname{int} C)cup{0}).

If (F x nless^{mathrm{lw}} F bar{x})), (operatorname{WMin}[F bar{x}),C]={ bar{y}}) and (F bar{x})) has the C-weak minimal property, one has bigl(F x -bar{y} bigr)cap (-operatorname{int}C )=emptyset.

If (operatorname{WMin}[F bar{x}),C]={bar{y}}) and (F bar {x})) has the C-weak minimal property, then bigl overline{D}F bar{x},bar{y}) (x) bigr cap (-operatorname{int}C )=emptyset,quad textit{for all } xin S. (4.2).

If (operatorname{WMin}[F bar{x}),C]={bar{y}}) and (F bar {x})) has the C-weak minimal property then underline{D}F bar{x},bar{y}) (x cap -operatorname{int} C)=emptyset,quad tex cap -operatorname{intbar{x}).

On the other hand, it follows from (operatorname{WMin}[F bar{x}),C]={bar {y}}) and the C-weak minimal property of (F bar{x})) that, for all (bar{y}'in F bar{x})), there is (d'inoperatorname{int}Ccup{0}) such that bar{y}'=bar{y}+d'in F x)+operatorname{int}C. Therefore, we derive that F bar{x})subset F x)+operatorname{int}C, which means (F x)<^{mathrm{lw}} F bar{x})).

The subsequent operations were no more successful as the five large Japanese submarines sank only three merchant ships and caused minimal property damage during the two bombardments.

Thus the problem of finding G nt+r,n) for large t is equivalent to finding a regular graph, whose degree of regularity and number of vertices depend only on n and r, with a certain "trace-minimal" property.