We prove that any bounded degree regular graph with sufficiently strong spectral expansion contains an induced path of linear length. This is the first such result for expanders, strengthening an analogous result in the random setting by Draganić, Glock, and Krivelevich. More generally, we find long induced paths in sparse graphs that satisfy a mild upper-uniformity edge-distribution condition.

Smectite (from South Dakota, Wyoming, and Mississippi) and Vermiculite (Transvaal) were treated with solutions of Al(OH)B(3-B)+, with B varying from 0 to 2.5. The average basicity (OH/Al = B) of the Al adsorbed differed very much from the basicity of the Al added. The average basicity of the Al adsorbed by smectite was always above the average basicity of the Al added. In contrast to smectite, Vermiculite adsorbed smaller hydroxy-Al complexes. One reason for the different selective behavior was the difference in expansion between smectite (about 18 Å) and vermiculite (about 14 Å). Because of the adsorption of the relatively more basic OH-Al by smectite, smectite adsorbed considerably more Al than vermiculite. The total amount of aluminum in the interlayer generally could not be calculated by the difference between Al added and that remaining in solution after the reaction because of possible protonation of the clay mineral and adsorption of structural Al and other cations, which is more pronounced for vermiculite. The results in the present study demonstrated that neither the quantitative nor the qualitative composition of an Al(OH)B-treated exchanger can be deduced from B of the Al salts added. These points are frequently overlooked when cation exchangers are pretreated with Al of variable basicity and are used for further investigations, such as studies of CEC, surface area, interlayer spacing, anion reactions, the formation of gibbsite, etc. Before these kinds of investigations are conducted employing the pretreated OH-Al-exchangers, their composition should be known precisely.

Kaolinites of all kinds (fine, ‘fireclay,’ ‘type IV,’ etc.), some of which do not expand or expand incompletely with the usual intercalation methods used for comparison, are expanded completely by treatment of dry (110°C) clay with dry CsCl salt, followed by contact with hydrazine for 1 day at 65°C and then with DMSO overnight at 90°C. Comparison treatments were grinding in KOAc, soaking in hydrazine, and Li-DMSO, as well as combination of these. Following the Cs-hydrazine-DMSO treatment, the 7.2 Å spacing of 1:1 dioctahedral layer silicates shifts to 11.2 Å and the 11.2 Å/(7.2 + 11.2 Å) ratio ≃1.0. The trioctahedral 1:1 layer silicates and chlorite are not expanded by the Cs-hydrazine-DMSO procedure.

Using a simple ionic model, the energy necessary to expand a layer structure by a certain distance can be calculated. This has been done for a series of 15 structures including hydroxides, 2:1 and 1:1 structures of various types. Plots of energy versus separation distance show three major groups which have common bonding properties. For large separations, the group with the strongest interlayer bonds contains the brittle micas, the hydroxides, and the 1:1 structures. Intermediate bonding structures are the normal micas and the weakest bonds occur in the zero layer charge 2:1 structures. The relative energies needed for a given separation are not constant so that for small separations the zero layer charge structures such as talc and pyrophyllite are more strongly bonded than the normal micas. These groupings correlate very well with the expandability of the structures by water and other substances. It is proposed that this approach to the study of the layer structures will provide a simple theory explaining the expansion properties of layer silicates.

From swelling and surface area measurements, it was found that the swelling of a montmorillonite depends linearly on the fraction of its layers that fully expand in water and that this fraction, in turn, depends linearly on the b dimension of the unit cell. Therefore, swelling is a linear function of the b dimension. However, the specific surface area of a montmorillonite is a linear function of its b dimension only if no partially expanded layers exist. It was also found that the distance between fully expanded layers at a given applied pressure is the same for all montmorillonites.

To determine the reason why the adsorption of ethylene glycol on organo-smectites does not result in an expansion along the c-axis of the clays, smectites containing relatively small organo-ammonium ions (lauryl-, benzyl-, dibenzyl-, and dicyclohexylammonium), larger organic cations (dimethylbenzyloctadeyl- and methylbenzyldioctadecylammonium), and the heterocyclic organo-ammonium ion 1,4a-dimethyl-7-isopropyl-1,2,3,4,4a,9,10,10a-octahydro-1-phenanthrenemethylammonium and the corresponding ethoxylated compound were exposed to ethylene glycol vapor for up to several months and examined by X-ray powder diffraction (XRD), surface area, and thermogravimetric methods. Weight loss data showed that all samples adsorbed ethylene glycol. XRD data for oriented samples indicated that lauryl-, benzyl-, dicyclohexyl-, and ethoxylated heterocyclic ammonium clays expanded by one layer of ethylene glycol and that methylbenzyldioctadecylammonium smectite expanded by two layers. Dibenzyl-, dimethylbenzyloctadecyl-, and heterocyclic smectites did not expand because the clay oriented in such a manner as to leave free clay surface between the organo-ammonium cations.

The coarse (2-0.2 μm) and fine (<0.2 μm) size fractions of several soil and reference kaolinite samples were completely intercalated and expanded to 11.2 Å using a simplified CsCl-hydrazine-di-methylsulfoxide (DMSO) treatment. Rapid equilibration of the clay in hot (80°C) hydrazine monohydrate and hot (100°C) DMSO, and the use of ceramic tile mounts, limited the sample pretreatment time to only 15 min. The fine size fraction of kaolinite may be X-rayed immediately after pretreatment, though analysis of the sample 2 or 12 hr after pretreatment produced more intense, sharp basal reflections. This difference may be due to a better ordering of the DMSO-kaolinite complex with time and to a drying of the excess DMSO. The coarse size fraction of kaolinite did not entirely expand to 11.2 Å when analyzed immediately after pretreatment. A 9.6-Å peak was also present and possibly represents a mixed layering of expanded and nonexpanded kaolinite layers. The larger crystals seem to require additional time for the reaction to become complete as evidenced by the presence of an intense, sharp 11.2-Å peak and absence of the 9.6-Å peak when the coarse clay was analyzed 2 or 12 hr after sample pretreatment. Kaolinite particles <50 μm did not react completely even when they were analyzed 24 hr after pretreatment. Therefore, this technique should be limited to <2-μm particles.

The structure of sepiolite from a piston core obtained on Ninetyeast Ridge in the Indian Ocean was modified by exposure to ethylene glycol vapor. With ethylene glycol, the sepiolite Oil X-ray powder diffraction peak expanded from 12.4 to 12.8 Å, and the 130 peak contracted from 4.53 to about 4.45 Å. The cell modification is consistent with concomitant expansion along the c-axis and contraction along the b-axis. This structural distortion is not permanent, however, inasmuch as the sepiolite returned to its original state 6 to 12 hr after it was removed from the ethylene glycol-saturated atmosphere.

Treatment of smectite with laurylamine hydrochloride was verified to cause expansion of d(001) which may be retained and observed in ion-milled samples by transmission electron microscopy. The spacings between layers as observed in lattice fringe images, however, are variable and may be as small as 10 Å. The method therefore produces ambiguities in differentiating between some smectites and illites, similar to those that have been found for untreated samples; e.g., on this basis, expanded layers may be inferred to be smectite, but layers with d-values approaching 10 Å may be either illite or smectite. Expansion also destroys the original rock texture, which, therefore, must be observed using only untreated samples.

According to the real business cycle theory, business cycles mainly result from random exogenous shocks. In this paper, this argument is tested. I extend the Wald–Wolfowitz runs test under the assumption that a recession lasts for two periods at least and an expansion lasts for $k$ periods at least with k ≥ 2. I apply the extended runs test to the three two-valued data recession-expansion series generated by the National Bureau of Economic Research and the Center for Economic and Policy Research. The test results reject the null hypothesis that they are generated in a random way for any $k$ even at the 1% significance level.