Substitute Arguments in Constitutional Law

In this article, I argue that that substitution is crucial to our practice of constitutional law. Of course, if one wished, one could easily extend the domain of substitution beyond these boundaries. Substitute arguments are an important aspect of law more generally and, indeed, of life. I have nonetheless chosen to limit my discussion to constitutional substitution because, I believe, overt discussion of substitution in this particular area illuminates important aspects of our constitutional regime-–aspects that substitution itself regularly obscures. To put my central point directly, I hope to show that constitutional law amounts to one, giant substitute argument

Ligand Substitution Dynamics

Substitution of a ligand in an inner sphere complex by an outside group is the most fundamental reaction in metal ion chemistr

Substitution Delone Sets

This paper addresses the problem of describing aperiodic discrete structures that have a self-similar or self-affine structure. Substitution Delone set families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an inflation functional equation under the action of an expanding integer matrix in R^d. This paper studies such functional equation in which each X_i is a discrete multiset (a set whose elements are counted with a finite multiplicity). It gives necessary conditions on the coefficients of the functional equation for discrete solutions to exist. It treats the case where the equation has Delone set solutions. Finally, it gives a large set of examples showing limits to the results obtained.Comment: 34 pages, latex file; some results in Sect 5 rearranged and theorems reformulate

Tridiagonal substitution Hamiltonians

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schroedinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schoedinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schroedinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.Comment: 41 pages, 5 figures, 81 reference

Theoretical Analysis of the Reduction of Neel Temperature in La2_{2}(Cu1−x_{1-x}Zn(or Mg)x)_x)O4_{4}

Using Tyablikov's decoupling approximation, we calculate the initial suppression rate of the Neel temperature, $R_{IS}=-lim_{x-> 0} T^{-1}_{N} dT_{N}/dx$, in a quasi two-dimensional diluted Heisenberg antiferromagnet with nonmagnetic impurities of concentration $x$. In order to explain an experimental fact that $R^{(Zn)}_{IS}=3.4$ of the Zn-substitution is different from $R^{(Mg)}_{IS}=3.0$ of the Mg-substitution, we propose a model in which impurity substitution reduces the intra-plane exchange couplings surrounding impurities, as well as dilution of spin systems. The decrease of 12% in exchange coupling constants by Zn substitution and decrease of 6% by Mg substitution explain those two experimental results, when an appropriate value of the interplane coupling is used.Comment: 2 pages, 3 figure