Time domain study of the Drude–Born–Fedorov model for a class of heterogeneous chiral materials

We deal with the well‐posedness of the transient Maxwell equations in a particular class of heterogeneous isotropic chiral material modeled by the Drude–Born–Fedorov constitutive relations with different boundary conditions. A new formulation of the underlying evolution problems allows us to establish existence and uniqueness results. The selfadjointness of the curl operator multiplied by a piecewise smooth function in an appropriated Hilbert space is also proved. Copyright © 2012 John Wiley & Sons, Ltd.     

Selection of variables in two-group discriminant analysis by error rate and Akaike's information criteria

This paper deals with two criteria for selection of variables for the discriminant analysis in the case of two multivariate normal populations with different means and a common covariance matrix. One is based on the estimated error rate of misclassification. The other uses Akaike's information criterion. The asymptotic distributions and error rate risks of the criteria are obtained. The result will prove that the two criteria are asymptotically equivalent in the sense of their asymptotic distributions and error rate risks being identical.     

Random utility models with all choice probabilities expressible as ‘functions’ of the binary choice probabilities

Given the absence of non-trivial decision rules which are strategy-proof in the Gibbard-Satterthwaite sense, Peleg and Sen have suggested that one might search for decision rules under which there will be always some Nash equilibrium yielding the same outcome as would be yielded by honest voting. This paper shows that most decision rules based on pairwise comparison will fail to satisfy the requirement of Peleg and Sen if one imposes the additional condition that the relevant Nash equilibrium should consist of undominated voting strategies only (a most reasonable condition in the absence of cooperation). It is also shown that under most decision rules based on pairwise comparisons, it will not be possible either to ensure that the outcome of honest voting will be always an outcome of sophisticated voting (in Farquharson's sense) or to ensure that the outcome of honest voting will be always a core outcome.     

On Birman's sequence of Hardy–Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_n\}$, for $n\in \mathbb{N}$, valid for functions in $C_0^n((0,\infty ))?L^2((0,\infty ))$. In particular, $I_1$ is the classical (integral) Hardy inequality and $I_2$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_n([0,\infty ))$ of functions defined on $[0,\infty )$. Moreover, $f\in H_n([0,\infty ))$ implies $f^{\prime }\in H_{n?1}([0,\infty ))$; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0$, these inequalities hold on the standard Sobolev space $H_0^n((0,b))$. Furthermore, in all cases, the Birman constants ${[(2n?1)!!]}^2/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^2((0,\infty ))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.