Packing Dimension Research Articles

Gas emission strength and evolution of the molar ratio of BrO/SO2 in the plume of Nyiragongo in comparison to Etna

AbstractAirborne and ground‐based differential optical absorption spectroscopy observations have been carried out at the volcano Nyiragongo (Democratic Republic of Congo) to measure SO2 and bromine monoxide (BrO) in the plume in March 2004 and June 2007, respectively. Additionally filter pack and multicomponent gas analyzer system (Multi‐GAS) measurements were carried out in June 2007. Our measurements provide valuable information on the chemical composition of the volcanic plume emitted from the lava lake of Nyiragongo. The main interest of this study has been to investigate for the first time the bromine emission flux of Nyiragongo (a rift volcano) and the BrO formation in its volcanic plume. Measurement data and results from a numerical model of the evolution of BrO in Nyiragongo volcanic plume are compared with earlier studies of the volcanic plume of Etna (Italy). Even though the bromine flux from Nyiragongo (2.6 t/d) is slightly greater than that from Etna (1.9 t/d), the BrO/SO2 ratio (maximum 7 × 10−5) is smaller than in the plume of Etna (maximum 2.1 × 10−4). A one‐dimensional photochemical model to investigate halogen chemistry in the volcanic plumes of Etna and Nyiragongo was initialized using data from Multi‐GAS and filter pack measurements. Model runs showed that the differences in the composition of volcanic volatiles led to a smaller fraction of total bromine being present as BrO in the Nyiragongo plume and to a smaller BrO/SO2 ratio.

Packing and Hausdorff Measures of Cantor Sets Associated with Series

We study a generalization of Moran’s sum sets, obtaining information about the h-Hausdorff and h-packing measures of these sets and certain of their subsets.

Packing spectra for Bernoulli measures supported on Bedford–McMullen carpets

We consider the packing spectra for the local dimension of Bernoulli measures supported on Bedford–McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We also cons

Hausdorff and Packing Measures of Balanced Cantor Sets

We estimate the \(h\)-Hausdorff and \(h\)-packing measures of balanced Cantor sets, and characterize the corresponding dimension partitions. This generalizes results known for Cantor sets associated with positive decreasing summable sequences and central Cantor sets.

Unified characterizations of lowness properties via Kolmogorov complexity

Consider a randomness notion $${\mathcal{C}}$$ C . A uniform test in the sense of $${\mathcal{C}}$$ C is a total computable procedure that each oracle X produces a test relative to X in the sense of $${\mathcal{C}}$$ C . We say that a binary sequence Y is $${\mathcal{C}}$$ C -random uniformly relative to X if Y passes all uniform $${\mathcal{C}}$$ C tests relative to X. Suppose now we have a pair of randomness notions $${\mathcal{C}}$$ C and $${\mathcal{D}}$$ D where $${\mathcal{C} \subseteq \mathcal{D}}$$ C ⊆ D , for instance Martin-Lof randomness and Schnorr randomness. Several authors have characterized classes of the form Low( $${\mathcal{C}, \mathcal{D}}$$ C , D ) which consist of the oracles X that are so feeble that $${\mathcal{C} \subseteq \mathcal{D}^X}$$ C ⊆ D X . Our goal is to do the same when the randomness notion $${\mathcal{D}}$$ D is relativized uniformly: denote by Low $${\star(\mathcal{C},\mathcal{D})}$$ ( C , D ) the class of oracles X such that every $${\mathcal{C}}$$ C -random is uniformly $${\mathcal{D}}$$ D -random relative to X. (1) We show that $${X\in Low ^\star}$$ X ? L o w ? (MLR, SR) if and only if X is c.e. tt-traceable if and only if X is anticomplex if and only if X is Martin-Lof packing measure zero with respect to all computable dimension functions. (2) We also show that $${X\in Low^\star}$$ X ? L o w ? (SR, WR) if and only if X is computably i.o. tt-traceable if and only if X is not totally complex if and only if X is Schnorr Hausdorff measure zero with respect to all computable dimension functions.

Dimension of slices of Sierpiński-like carpets

We investigate the dimension of intersections of the Sierpiński-like carpets with lines. We show a sufficient condition that for a fixed rational slope the dimension of almost every intersection with respect to the natural measure is strictly greater than s–1 , and almost every intersection with respect to the Lebesgue measure is strictly less than s–1 , where s is the Hausdorff dimension of the carpet. Moreover, we give partial multifractal spectra for the Hausdorff and packing dimension of slices.

On quasisymmetric minimality of Cantor sets

We prove that uniform Cantor sets of Hausdorff dimension 1 are all quasisymmetrically minimal for Hausdorff dimension. An analog of this result for packing dimension is also obtained. From the proof a general sufficient condition for minimal Cantor sets can be formulated.

Structural and magnetic properties of cobalt nanostructures on SiO2/Si(1 1 1) substrates

2D architectures of cobalt onto silicon (111) surfaces were elaborated by patterning of magnetic cobalt in the nanometer scale. A continuous cobalt layer of 1, 3 and 10nm thickness, respectively, was first deposited by means of thermoionic vacuum arc technique and then, thermally annealed in vacuum at temperatures ranging from 450 to 800°C. Surface structure was analyzed by atomic force and field emission-scanning electron microscopies. Above 750°C, regular triangular shape cobalt nanostructures are formed with pattern dimensions varying between 10 and 200nm. Good control of shape and packing density could be achieved by adjusting the initial thickness and the thermal and hydrogen plasma treatments. Magnetic properties were investigated using vibrating sample magnetometer technique. The evolution of the coercive field versus packing density and dimensions of the nanostructures was studied and compared to micromagnetic calculations. The observed nanostructures have been modeled by a series of shapes tending to a fractal curve.

PACKING DIMENSIONS OF GENERALIZED RANDOM MORAN SETS

We consider random fractal sets with random recursive constructions in which the contracting vectors have different distributions at different stages. We prove that the random fractal associated with such construction has a constant packing dimension almost surely and give an explicit formula to determine it.

Homogeneous sets and their quasi-Lipschitz equivalence

We define and study a class of homogeneous fractals, including all (quasi) Ahlfors-David regular sets and many irregular Moran sets. We say a fractal is irregular if its Hausdorff dimension is not equal to packing dimension. For homogeneous fractals, we obtain their fractal dimensions and give a necessary and sufficient condition for two homogeneous fractals to be quasi-Lipschitz equivalent, and then we apply these results to irregular Moran sets.

Exact Hausdorff and packing measures of Cantor sets with overlaps

Let $K$ be the attractor of a linear iterated function system (IFS) $S_{j}(x)={\it\rho}_{j}x+b_{j},j=1,\ldots ,m$, on the real line $\mathbb{R}$ satisfying the generalized finite type condition (whose invariant open set ${\mathcal{O}}$ is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math.208 (2007), 647–671] as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let ${\it\alpha}$ be the dimension of $K$. In this paper, we state that $$\begin{eqnarray}{\mathcal{H}}^{{\it\alpha}}(K\cap J)\leq |J|^{{\it\alpha}}\end{eqnarray}$$ for all intervals $J\subset \overline{{\mathcal{O}}}$, and $$\begin{eqnarray}{\mathcal{P}}^{{\it\alpha}}(K\cap J)\geq |J|^{{\it\alpha}}\end{eqnarray}$$ for all intervals $J\subset \overline{{\mathcal{O}}}$ centered in $K$, where ${\mathcal{H}}^{{\it\alpha}}$ denotes the ${\it\alpha}$-dimensional Hausdorff measure and ${\mathcal{P}}^{{\it\alpha}}$ denotes the ${\it\alpha}$-dimensional packing measure. This result extends a recent work of Olsen [Density theorems for Hausdorff and packing measures of self-similar sets. Aequationes Math.75 (2008), 208–225] where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of $K$. Moreover, using these density theorems, we describe a scheme for computing ${\mathcal{H}}^{{\it\alpha}}(K)$ exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing ${\mathcal{P}}^{{\it\alpha}}(K)$ as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer and Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets. Trans. Amer. Math. Soc.351 (1999), 3725–3741] and by Feng [Exact packing measure of Cantor sets. Math. Natchr.248–249 (2003), 102–109], respectively, and apply to some new classes allowing us to include Cantor sets in $\mathbb{R}$ with overlaps.

Aneurysm permeability following coil embolization: packing density and coil distribution

BackgroundRates of durable aneurysm occlusion following coil embolization vary widely, and a better understanding of coil mass mechanics is desired. The goal of this study is to evaluate the impact...

E-060 Effect of Packing Density and Coil Uniformity on Coil Permeability

IntroductionDurability of coiling procedures varies widely, and a better understanding of coil dynamics may help improve the treatment strategy. The goal of this study is to evaluate how packing density...

A note on the hitting probabilities of random covering sets

Let E = limsup n!1 (gn + �n) be the random covering set on the torus T d , where {gn} is a sequence of ball-like sets andn is a sequence of independent random variables uniformly distributed on T d . We prove that E ∩ F 6 ∅ almost surely whenever F ⊂ T d is an analytic set with Hausdorff dimension, dimH(F) > d−�, whereis the almost sure Hausdorff dimension of E. Moreover, examples are given to show that the condition on dimH(F) cannot be replaced by the packing dimension of F.

Hausdorff and Packing Spectra, Large Deviations, and Free Energy for Branching Random Walks in $${\mathbb{R}^d}$$ R d

Consider an $\R^d$-valued branching random walk (BRW) on a supercritical Galton Watson tree. Without any assumption on the distribution of this BRW we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets $E(K)$ of infinite branches in the boundary of the tree (endowed with its standard metric) along which the averages of the BRW have a given closed connected set of limit points $K$. This goes beyond multifractal analysis, which only considers those level sets when $K$ ranges in the set of singletons $\{\alpha\}$, $\alpha\in\R^d$. We also give a $0$-$\infty$ law for the Hausdorff and packing measures of the level sets $E(\{\alpha\})$, and compute the free energy of the associated logarithmically correlated random energy model in full generality. Moreover, our results complete the previous works on multifractal analysis by including the levels $\alpha$ which do not belong to the range of the gradient of the free energy. This covers in particular a situation until now badly understood, namely the case where a first order phase transition occurs. As a consequence of our study, we can also describe the whole singularity spectrum of Mandelbrot measures, as well as the associated free energy function (or $L^q$-spectrum), when a first order phase transition occurs.

On packing measures and a theorem of Besicovitch

Let H h \mathcal {H}^h be the h h -dimensional Hausdorff measure on R d \mathbb {R}^d . Besicovitch showed that if a set E E is null for H h \mathcal {H}^h , then it is null for H g \mathcal {H}^g , for some dimension g g smaller than h h . We prove that this is not true for packing measures. Moreover, we consider the corresponding questions for sets of non- σ \sigma -finite packing measure and for pre-packing measure instead of packing measure.

On restricted families of projections in ℝ3

We study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 1 2 -dimensional sets B ⊂ R 3 is typically preserved under one-dimensional families of projections onto lines. We improve the result by an ε, proving that if dim H B = s > 1 2 , then the packing dimension of the projections is almost surely at least σ ( s ) > 1 2 . For projections onto planes, we obtain a similar bound, with the threshold 1 2 replaced by 1. In the special case of self-similar sets K ⊂ R 3 without rotations, we obtain a full Marstrand-type projection theorem for 1-parameter families of projections onto lines. The dim H K ⩽ 1 case of the result follows from recent work of M. Hochman, but the dim H K > 1 part is new: with this assumption, we prove that the projections have positive length almost surely.

On the packing dimension and category of exceptional sets of orthogonal projections

We consider several classical results related to the Hausdorff dimension of exceptional sets of orthogonal projections and find out whether they have reasonable formulations in terms of packing dimension. We also investigate the existence of category versions for Marstrand and Falconer–Howroyd-type projection results.

Hausdorff and upper box dimension estimate of hyperbolic recurrent sets

In this paper, under the open set condition we have determined the lower and upper bounds of the Hausdorff dimensions and the upper box-counting dimensions of the limit sets of hyperbolic recurrent iterated function systems in terms of the unique zeros h and H of the two pressure functions. In addition, we have estimated the bounds of h, H-dimensional Hausdorff and packing measures. The result in this paper generalizes existing results about fractal dimensions of many other iterated function systems.

Lithography-free synthesis of nanostructured cobalt on Si (111) surfaces: structural and magnetic properties

We illustrate the concept of lithography-free synthesis and patterning of magnetic cobalt in the nanometric scale. Our elaboration method allows fabricating 2D architectures of cobalt and cobalt silicide onto silicon (111) surfaces. A continuous cobalt layer of 1, 3 and 10 nm thickness was first deposited by using thermoionic vacuum arc (TVA) technology and then, thermally annealed on vacuum at temperatures from 450° C to 800° C. Surface structure was analyzed by atomic force and field emission-scanning electron microscopies. Above 750° C, regular triangular shape cobalt nanostructures are formed with pattern dimensions varying between 10 and 200 nm. Good control of shape and packing density could be achieved by adjusting the initial thickness and the substrate temperature. Magnetic properties were investigated by means of vibrating sample magnetometer (VSM) technique. The evolution of the coercive field versus packing density and dimensions of the nanostructures was studied and compared to micromagnetic calculations. The observed nanostructures have been modelled by a series of shapes tending to a fractal curve.