For $\lambda \in (0,\,1/2]$ let $K_\lambda \subset \mathbb {R}$ be a self-similar set generated by the iterated function system $\{\lambda x,\, \lambda x+1-\lambda \}$. Given $x\in (0,\,1/2)$, let $\Lambda (x)$ be the set of $\lambda \in (0,\,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\,\ldots,\, y_p\in (0,\,1/2)$ there exists a full Hausdorff dimensional set of $\lambda \in (0,\,1/2]$ such that $y_1,\,\ldots,\, y_p \in K_\lambda$.

An indirect Fourier transformation applied to small-angle X-ray scattering data has been used to determine the thickness and surface properties of two common clay minerals. For an illite system, the particle density distribution function (PDDF) generated by the analysis gave a correct description of particle geometry, and the calculated electron density profile was in accordance with the theoretical electron density distribution for this mineral. This approach provides the opportunity to determine the thickness of fundamental particles of illite while avoiding the difficulties encountered in other methods. Both the PDDF and the electron density profile accurately predict the thickness of Na-montmorillonite layers, and the results suggest that an electron inhomogeneity exists at the interface of this mineral.

We show that products of sufficiently thick Cantor sets generate trees in the plane with constant distance between adjacent vertices. Moreover, we prove that the set of choices for this distance has non-empty interior. We allow our trees to be countably infinite, which further distinguishes this work from previous results on patterns in fractal sets. This builds on the authors’ previous work on graphs and distance sets over products of Cantor sets of sufficient Newhouse thickness.

The ability to accurately and precisely measure the thickness of biomaterial constructs is critical for characterizing both specific dimensional features and related mechanical properties. However, in the absence of a standardized approach for thickness measurements, a variety of imaging modalities have been employed, which have been associated with varying limits of accuracy, particularly for ultrathin hydrated structures. Electron microscopy (EM), a commonly used modality, yields thickness values for extensively processed and nonhydrated constructs, potentially resulting in overestimated mechanical properties, including elastic modulus and ultimate tensile strength. Confocal laser scanning microscopy (CLSM) has often been used as a nondestructive imaging alternative. However, published CLSM-derived image analysis protocols use arbitrary signal intensity cutoffs and provide minimal information regarding thickness variability across imaged surfaces. To address the aforementioned limitations, we present a standardized, user-independent CLSM image acquisition and analysis approach developed as a custom ImageJ macro and validated with collagen-based scaffolds. In the process, we also quantify thickness discrepancies in collagen-based scaffolds between CLSM and EM techniques, further illustrating the need for improved strategies. Employing the same image acquisition protocol, we also demonstrate that this approach can be used to estimate the surface roughness of the same scaffolds without the use of specialized instrumentation.

The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G$. A $t$-minimal graph is a graph of thickness $t$ that contains no proper subgraph of thickness $t$. In this paper, upper and lower bounds are obtained for the thickness, $t\left( G\,\square \,H \right)$, of the Cartesian product of two graphs $G$ and $H$, in terms of the thickness $t\left( G \right)$ and $t\left( H \right)$. Furthermore, the thickness of the Cartesian product of two planar graphs and of a $t$-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph ${{K}_{4k,\,4k}}$, the thickness of the Cartesian product of two complete bipartite graphs ${{K}_{n,n}}$ and ${{K}_{n,n}}$ is also given for $n\,\ne \,4k\,+\,1$.

Characterization of the topmost surface of biomaterials is crucial to understanding their properties and interactions with the local environment. In this study, the oxide layer microstructure of plasma-modified 316L stainless steel (SS316L) samples was analyzed by a combination of electron backscatter diffraction and electron channeling contrast imaging using low-energy incident electrons. Both techniques allowed clear identification of a nano-thick amorphous oxide layer, on top of the polycrystalline substrate, for the plasma-modified samples. A methodology was developed using Monte Carlo simulations combined with the experimental results to estimate thickness of the amorphous layer for different surface conditions. X-ray photoelectron spectroscopy depth profiles were used to validate these estimations.

The measurement of the thickness of a coating layer is a traditional industrial problem. Many methods of measurement are already classically used for that purpose. However, the majority of these methods require either contact with the sample to be studied, or its destruction. These constraints limit their applicability. Photothermal radiometry is already largely used in the field of thermophysical properties measurement and for the detection and characterization of defects. This method does not impose the constraints mentioned above. It could thus be a complementary technique of the classical ones. The purpose of this work is then to approach the possibilities of this photothermal method, implemented under pulse excitation and front face analysis, for the measurement of the coating thickness. Using theoretical and experimental studies, we show that the method allows a good estimation of the thickness of chromium deposits on a steel substrate, at distance, without contact and in a non-destructive way.

Previous research suggests that increasing beverage protein content enhances subsequent satiety, but whether this effect is entirely attributable to post-ingestive effects of protein or is partly caused by the distinct sensory characteristics imparted by the presence of protein remains unclear. To try and discriminate nutritive from sensory effects of added protein, we contrasted effects of three higher-energy (about 1·2 MJ) and one lower-energy (LE: 0·35 MJ) drink preloads on subsequent appetite and lunch intake. Two higher-energy drinks had 44 % of energy from protein, one with the sensory characteristics of a juice drink (HP − , low-sensory protein) and the second a thicker and creamier (HP+, high-sensory protein) drink. The high-carbohydrate preload (HC+, high-sensory carbohydrate) was matched for thickness and creaminess to the HP+ drink. Participants (healthy male volunteers, n 26) consumed significantly less at lunch after the HP+(566 g) and HC+(572 g) than after HP −  (623 g) and LE (668 g) drinks, although the compensation for drink energy accounted for only 50 % of extra energy at best. Appetite ratings indicated that participants felt significantly less hungry and more full immediately before lunch in HP+ and HC+ groups compared with LE, with HP −  being intermediate. The finding that protein generated stronger satiety in the context of a thicker creamier drink (HP+ but not HP − ) and that an isoenergetic carbohydrate drink (HC+), matched in thickness and creaminess to the HP+ drink, generated the same pattern of satiety as HP+, both suggest an important role for these sensory cues in the development of protein-based satiety.

The relation between image resolution and information transfer is explored. It is shown that the existence of higher frequency transfer in the image is just a necessary but not sufficient condition for the achievement of higher resolution. Adopting a two-point resolution criterion, we suggest that a 10% contrast level between two features in an image should be used as a practical definition of resolution. In the context of scanning transmission electron microscopy, it is shown that the channeling effect does not have a direct connection with image resolution because sharp channeling peaks do not move with the scanning probe. Through a quantitative comparison between experimental image and simulation, a Fourier-space approach is proposed to estimate defocus and sample thickness. The effective atom size in Z-contrast imaging depends on the annular detector's inner angle. Therefore, an optimum angle exists for the highest resolution as a trade-off between reduced atom size and reduced signal with limited information transfer due to noise.

Given a centrally symmetric convex body $B$ in ${{\mathbb{E}}^{d}}$, we denote by ${{\mathcal{M}}^{d}}\left( B \right)$ the Minkowski space (i.e., finite dimensional Banach space) with unit ball $B$. Let $K$ be an arbitrary convex body in ${{\mathcal{M}}^{d}}\left( B \right)$. The relationship between volume $V\left( K \right)$ and the Minkowskian thickness (= minimal width) ${{\Delta }_{B}}\left( K \right)$ of $K$ can naturally be given by the sharp geometric inequality $V\left( K \right)\ge \alpha \left( B \right)\cdot {{\Delta }_{B}}{{\left( K \right)}^{d}}$, where $\alpha \left( B \right)>0$. As a simple corollary of the Rogers-Shephard inequality we obtain that ${{\left( _{d}^{2d} \right)}^{-1}}\le \alpha \left( B \right)/V\left( B \right)\le {{2}^{-d}}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.

Mechanisms underlying the development of the primate area of high acuity (AHA) remain poorly understood. Finite-element models have identified retinal stretch and intraocular pressure (IOP) as possible mechanical forces that can form a pit (Springer & Hendrickson, 2004). A series of Macaca nemestrina monkey retinas between 68 days postconception (dpc) and adult were used to quantify growth and morphological changes. Retinal and pars plana length, optic disc diameter, disc-pit distance, and inner and outer retinal laminar thickness were measured over development to identify when and where IOP or stretch might operate. Horizontal optic disc diameter increased 500 μm between 115 dpc and 2 months after birth when it reached adult diameter. Disc growth mainly influences the immediate surrounding retina, presumably displacing retinal tissue centrifugally. Pars plana elongation also began at 115 dpc and continued steadily to 3–4 years postnatal, so its influence would be relatively constant over retinal development. Unexpectedly, horizontal retinal length showed nonlinear growth, divided into distinct phases. Retinal length increased rapidly until 115 dpc and then remained unchanged (quiescent phase) between 115–180 dpc. After birth, the retina grew rapidly for 3 months and then very slowly into adulthood. The onset of pit development overlapped the late fetal quiescent phase, suggesting that the major mechanical factor initiating pit formation is IOP, not retinal growth-induced stretch. Developmental changes in the thickness of retinal layers were different for inner and outer retina at many, but not all, of the ten eccentricities examined. Peripheral inner and outer retinal layers thinned appreciably with age, consistent with retinal stretch having a larger effect on the retinal periphery. Central inner retina around the area of high acuity (AHA) changed tri-phasically. It increased in thickness prenatally, thinned transiently after birth, and then resumed thickening. Transient postnatal inner retinal thinning around the pit coincided with the resumption of retinal growth and with cone packing providing evidence that a small amount of growth-induced central retinal stretch may account for cone packing as previously hypothesized (Springer, 1999). Central outer retina around the AHA progressively thickened over the fetal period. It reached asymptotic thickness at birth and continued to thicken into adulthood at some temporal, but not nasal, central eccentricities. These data indicate that peripheral outer and inner retina progressively thin with age because of eye growth-induced stretch, while central retina is minimally affected by stretch. Outer and inner retinal laminar thickness at the same locus can change in different directions, suggesting that they shear with respect to one another. This shearing induces the elongation of Henle axons, while their angle reflects the direction of shear.

Leaf anatomical parameters such as leaf mass per area (LMA) and biochemical composition can be used as indicators of leaf photosynthetic capacity. The aims of this study are to evaluate the potential of reflectance spectroscopy of fresh leaves for assessing and predicting various parameters, anatomical (LMA and tissue thickness) and biochemical (nitrogen concentration). This paper describes results obtained with fresh leaves of holm oak (Quercus ilex), an evergreen oak that is widely distributed from mesic to xeric habitats in the Mediterranean. Fresh leaves (560) were collected over 3 yr at six different sites, from the top to the bottom of the canopy. The reflectance of each leaf was obtained within 1 h of sampling with an NIRSystems 6500 spectrophotometer over the range 400–2500 nm. LMA was determined for all samples; biochemical and anatomical measurements were conducted over representative subsample populations of 92 and 87 leaves, respectively. Stepwise regression calibrations and partial least squares (PLS) calibrations were developed and compared with different spectral regions and mathematical treatments. Calibration equations had high coefficients of determination (r2 ranging from 0.94 for nitrogen to 0.98 for LMA and tissue thickness). The PLS regressions gave better results than stepwise regressions for all parameters studied. Compared with regressions calculated on raw spectral data, calculations on second derivatives of spectra improved results in all cases. The use of scatter corrections also improved results. These results show that visible and near-infra red reflectance can be used for accurately predicting anatomical parameters and the nitrogen concentration of fresh holm oak leaves. The results support the suggestion that high spectral resolution imaging spectrometry can be a useful tool for assessing functional processes in forest ecosystems.

For a two parameter family of C3 diffeomorphisms having a homoclinic orbit of tangency derived from a horseshoe, the relationship between the measure of the parameter values at which the diffeomorphism (restricted to a certain compact invariant set containing the horseshoe) is not expansive and the Hausdorff dimension of the horseshoe associated to the homoclinic orbit of tangency is investigated. This is a simple application of the Newhouse-Palis-Takens-Yoccoz theory.

We analyse by means of transmission electron microscopy (TEM) and atomic force microscopy (AFM) the strain relaxation mechanisms in InGaN layers on GaN as dependent on the In content. At the experimentally given thickness of 100 nm, the layers remain coherently strained, up to an In concentration of 14 %. We show that part of the strain is reduced elastically by formation of hexagonally facetted pinholes. First misfit dislocations are observed to form at pinholes that reach the InGaN/GaN interface. We discuss these results in the framework of the Matthews-Blakeslee model for the critical thickness considering the Peierls force for glide of threading dislocations in the different slip systems of the wurtzite lattice.

Time-resolved photoluminescence spectra have been recorded on three GaN epitaxial layers of thickness 2.5 μm, 7 μm and 16 μm, at various temperatures ranging from 8K to 300K. The layers were deposited by MOVPE on (0001) sapphire substrates with standard AlN buffer layers. To achieve good homogeneities, the growth was in-situ monitored by laser reflectometry. All GaN layers showed sharp excitonic peaks in cw PL and three excitonic contributions were seen by reflectivity. The recombination dynamics of excitons depends strongly upon the layer thickness. For the thinnest layer, exponential decays with τ ~ 35 ps have been measured for both XA and XB free excitons. For the thickest layer, the decay becomes biexponential with τ1 ~ 80 ps and τ2 ~ 250 ps. These values are preserved up to room temperature. By solving coupled rate equations in a four-level model, this evolution is interpreted in terms of the reduction of density of both shallow impurities and deep traps, versus layer thickness, roughly following a L−1 law.