Packing Dimension Research Articles

Geant4.10 simulation of geometric model for metaphase chromosome

In this paper, a geometric model of metaphase chromosome is explained. The model is constructed according to the packing ratio and dimension of the structure from nucleosome up to chromosome. A B-DNA base pair is used to construct 200 base pairs of nucleosomes. Each chromatin fiber loop, which is the unit of repeat, has 49,200bp. This geometry is entered in Geant4.10 Monte Carlo simulation toolkit and can be extended to the whole metaphase chromosomes and any application in which a DNA geometrical model is needed. The chromosome base pairs, chromosome length, and relative length of chromosomes are calculated. The calculated relative length is compared to the relative length of human chromosomes.

柱形Pt-SDB催化剂的水–氢交换高处理量工艺

摘要：研究了基于柱形Pt-SDB催化剂的水–氢催化交换工艺, 进行了柱形Pt-SDB催化剂的流体力学性能实验并讨论了填料规格、填料催化剂装填比、反应温度、气液比、处理量等对水–氢交换传质单元高度和传质系数的影响。结果表明，催化活性较低的柱形Pt-SDB催化剂，具有液泛气速高、持液量稳定、单位床层高度压降低等优点；柱形Pt-SDB催化剂水–氢催化交换的高处理量工艺条件为θ环填料与催化剂填装比例1:1、反应温度70℃~75℃、气液比1:1、空塔气速0.3 m/s。 The process conditions are experimentally studied for H2O-H2 liquid catalytic isotopic exchange with Pt-SDB as hydrophobic cylindrical catalyst, and hydrodynamic experiments are conducted for obtaining the correlating expressions of pressure drop △P/Z, flooding gas velocity uF. The results indicate that the Pt-SDB cylindrical catalyst with little catalytic activity has a number of advantages, including low pressure drop, higher flooding gas velocity and stable liquid holdup. The height of transfer unit (HTU) is also changed to be higher with the increasing of the air velocity in empty tower and the packing dimension. The efficiency of catalytic exchange reaction is high with a packing ratio of 1:1 of hydrophilic packing and hydrophobic catalyst in separated layers. The HTU decreases with increasing operating temperature, but the trend is slowed down when the temperature is above 70˚C. The HTU increases with increasing hydrogen flow rate and decreases with increasing diluted heavy water flow rate. When the air velocity of empty tower is 0.3 m/s, the process capacity is higher.

Controlling Effective Packing Dimension of Δ20 Degrees

This paper presents a refinement of a result by Conidis, who proved that there is a real X of effective packing dimension 0<α<1 which cannot compute any real of effective packing dimension 1. The original construction was carried out below ∅'', and this paper’s result is an improvement in the effectiveness of the argument, constructing such an X by a limit-computable approximation to get X≤T∅'.

On packing dimension preservation by distribution functions of random variables with independent $\tilde{Q}$-digits

The article is devoted to finding conditions for the packing dimension preservation by distribution functions of random variables with independent $\tilde{Q}$-digits. The notion of “faithfulness of fine packing systems for packing dimension calculation” is introduced, and connections between this notion and packing dimension preservation are found.

Exact packing measure of the range of $$\psi $$ ψ -Super Brownian motions

We consider super processes whose spatial motion is the d-dimensional Brownian motion and whose branching mechanism $$\psi $$ is critical or subcritical; such processes are called $$\psi $$ -super Brownian motions. If $$d>2\varvec{\gamma }/(\varvec{\gamma }-1)$$ , where $$\varvec{\gamma }\in (1,2]$$ is the lower index of $$\psi $$ at $$\infty $$ , then the total range of the $$\psi $$ -super Brownian motion has an exact packing measure whose gauge function is $$g(r) = (\log \log 1/r) / \varphi ^{-1} ( (1/r\log \log 1/r)^{2})$$ , where $$\varphi = \psi ^\prime \circ \psi ^{-1}$$ . More precisely, we show that the occupation measure of the $$\psi $$ -super Brownian motion is the g-packing measure restricted to its total range, up to a deterministic multiplicative constant only depending on d and $$\psi $$ . This generalizes the main result of Duquesne (Ann Probab 37(6):2431–2458, 2009) that treats the quadratic branching case. For a wide class of $$\psi $$ , the constant $$2\varvec{\gamma }/(\varvec{\gamma }-1)$$ is shown to be equal to the packing dimension of the total range.

Hitting probabilities and the Hausdorff dimension of the inverse images of a class of anisotropic random fields

Let X = {X(t): t ∈ RN} be an anisotropic random field with values in Rd. Under certain conditions on X, we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel–Riesz capacity. We also obtain the Hausdorff dimension of its inverse image, and the Hausdorff and packing dimensions of its level sets. These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.

Design and testing of foam-inflated wings for small unmanned aerial vehicles

Inflatable wings can greatly improve the portability of small UAVs. The foam-inflated wing concept consists of a hybrid structure of rigid and flexible elements. This approach potentially reduces the packaging and logistical needs before deployment of a small UAV. The small packaging volume of the wing is achieved using spars and wing skin that are made out of flexible polyester-film material. For deployment, the cylindrical spars are rigidized using expanding foam that increases the buckling stiffness of the spars. Solid ribs that were previously attached to those spars provide high shape compliance for the external skin. Due to the solid rib structure, the chord length of the wing defines the maximum packing dimension before inflation. As demonstrated in tests, the inflation process of the hybrid wing is uncomplicated and swift. The subsequent foam-inflated structure is lightweight and can be tailored to the expected loads. In addition, the underlying ribs ensure a high aerodynamic quality, despite the inflatable nature of the hybrid wing structure. The feasibility of the concept of the hybrid wing structure has been demonstrated in flight tests.

Graphs of continuous functions and packing dimension

We prove that for a typical continuous function f∈C(X) over an uncountable compact metric space X, the packing dimension of its graphGf={(x,f(x))|x∈X} is dimP⁡(X)+1; we also consider decompositions of functions in C([0,1]) in terms of upper box dimension as well as packing dimension, which are quite different from the case of Hausdorff dimension.

ON SIMULTANEOUS LOCAL DIMENSION FUNCTIONS OF SUBSETS OF ℝd

For a subset <TEX>$E{\subseteq}\mathbb{R}^d$</TEX> and <TEX>$x{\in}\mathbb{R}^d$</TEX>, the local Hausdorff dimension function of E at x and the local packing dimension function of E at x are defined by <TEX>$$dim_{H,loc}(x,E)=\lim_{r{\searrow}0}dim_H(E{\cap}B(x,r))$$</TEX>, <TEX>$$dim_{P,loc}(x,E)=\lim_{r{\searrow}0}dim_P(E{\cap}B(x,r))$$</TEX>, where <TEX>$dim_H$</TEX> and <TEX>$dim_P$</TEX> denote the Hausdorff dimension and the packing dimension, respectively. In this note we give a short and simple proof showing that for any pair of continuous functions <TEX>$f,g:\mathbb{R}^d{\rightarrow}[0,d]$</TEX> with <TEX>$f{\leq}g$</TEX>, it is possible to choose a set E that simultaneously has f as its local Hausdorff dimension function and g as its local packing dimension function.

On the packing measure of slices of self-similar sets

Let K \subset \mathbb R^{2} be a rotation and reflection free self-similar set satisfying the strong separation condition, with dimension \mathrm {dim} \: K = s &gt; 1 . Intersecting K with translates of a fixed line, one can study the (s - 1) -dimensional Hausdorff and packing measures of the generic non-empty line sections. In a recent article, T. Kempton gave a necessary and sufficient condition for the Hausdorff measures of the sections to be positive. In this paper, I consider the packing measures: it turns out that the generic section has infinite (s - 1) -dimensional packing measure under relatively mild assumptions.

On the equality of Hausdorff measure and Hausdorff content

We are interested in situations where the Hausdor measure and Hausdorff content of a set are equal in the critical dimension. Our main result shows that this equality holds for any subset of a self-similar set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph-directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from ‘self-similar’. For example, it fails in general for self-conformal sets, self-affine sets and Julia sets. We also give applications of our results concerning Ahlfors regularity. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and \delta -approximate packing pre-measure coincide for sufficiently small \delta&gt;0 .

Packing dimensions of the images of Gaussian random fields

Let X={X(t):t∈RN} be a Gaussian random field with values in Rd and let E⊆RN be a Borel set. We determine the packing dimension of the image set X(E) in terms of the packing dimension profiles in the canonical metric ρ of X, which are extensions of the packing dimension profiles of Falconer and Howroyd (1997) and the box-counting dimension profiles of Howroyd (2001).

On interpreting Patterson–Sullivan measures of geometrically finite groups as Hausdorff and packing measures

We provide a new proof of a theorem whose proof was sketched by Sullivan [Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics.Acta Math.149(3–4) (1982), 215–237], namely that if the Poincaré exponent of a geometrically finite Kleinian group$G$is strictly between its minimal and maximal cusp ranks, then the Patterson–Sullivan measure of$G$is not proportional to the Hausdorff or packing measure of any gauge function. This disproves a conjecture of Stratmann [Multiple fractal aspects of conformal measures; a survey.Workshop on Fractals and Dynamics (Mathematica Gottingensis, 5). Eds. M. Denker, S.-M. Heinemann and B. Stratmann. Springer, Berlin, 1997, pp. 65–71; Fractal geometry on hyperbolic manifolds.Non-Euclidean Geometries (Mathematical Applications (N.Y.), 581). Springer, New York, 2006, pp. 227–247].

On the dimensions of a family of overlapping self-affine carpets

We consider the dimensions of a family of self-affine sets related to the Bedford–McMullen carpets. In particular, we fix a Bedford–McMullen system and then randomize the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford–McMullen set-up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, Hochman’s recent work on the dimensions of self-similar sets and measures.

Bilipschitz embedding of homogeneous fractals

In this paper, we introduce a class of fractals named homogeneous sets based on some measure versions of homogeneity, uniform perfectness and doubling. This fractal class includes all Ahlfors–David regular sets, but most of them are irregular in the sense that they may have different Hausdorff dimensions and packing dimensions. Using Moran sets as the main tool, we study the dimensions, bilipschitz embedding and quasi-Lipschitz equivalence of homogeneous fractals.

On the 𝐿^{𝑞}-spectrum of planar self-affine measures

We study the dimension theory of a class of planar self-affine multifractal measures. These measures are the Bernoulli measures supported on box-like self-affine sets, introduced by the author, which are the attractors of iterated function systems consisting of contracting affine maps which take the unit square to rectangles with sides parallel to the axes. This class contains the self-affine measures recently considered by Feng and Wang as well as many other measures. In particular, we allow the defining maps to have non-trivial rotational and reflectional components. Assuming the rectangular open set condition, we compute the L q L^q -spectrum by means of a q q -modified singular value function. A key application of our results is a closed form expression for the L q L^q -spectrum in the case where there are no mappings that switch the coordinate axes. This is useful for computational purposes and also allows us to prove differentiability of the L q L^q -spectrum at q = 1 q=1 in the more difficult ‘non-multiplicative’ situation. This has applications concerning the Hausdorff, packing and entropy dimension of the measure as well as the Hausdorff and packing dimension of the support. Due to the possible inclusion of axis reversing maps, we are led to extend some results of Peres and Solomyak on the existence of the L q L^q -spectrum of self-similar measures to the graph-directed case.

Too packed to change: side-chain packing and site-specific substitution rates in protein evolution.

In protein evolution, due to functional and biophysical constraints, the rates of amino acid substitution differ from site to site. Among the best predictors of site-specific rates are solvent accessibility and packing density. The packing density measure that best correlates with rates is the weighted contact number (WCN), the sum of inverse square distances between a site’s Cα and the Cα of the other sites. According to a mechanistic stress model proposed recently, rates are determined by packing because mutating packed sites stresses and destabilizes the protein’s active conformation. While WCN is a measure of Cα packing, mutations replace side chains. Here, we consider whether a site’s evolutionary divergence is constrained by main-chain packing or side-chain packing. To address this issue, we extended the stress theory to model side chains explicitly. The theory predicts that rates should depend solely on side-chain contact density. We tested this prediction on a data set of structurally and functionally diverse monomeric enzymes. We compared side-chain contact density with main-chain contact density measures and with relative solvent accessibility (RSA). We found that side-chain contact density is the best predictor of rate variation among sites (it explains 39.2% of the variation). Moreover, the independent contribution of main-chain contact density measures and RSA are negligible. Thus, as predicted by the stress theory, site-specific evolutionary rates are determined by side-chain packing.

Dynamics of the scenery flow and geometry of measures

We employ the ergodic theoretic machinery of scenery flows to address classical geometric measure theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases.

Multifractal Analysis of Some Inhomogeneous Multinomial Measures with Distinct Analytic Olsen’s $$b$$ b and $$B$$ B Functions

We construct exact dimensional measures whose support is the whole interval \([0,1]\) and whose Olsen’s multifractal functions \(b\) and \(B\) are real analytic and agree at two points only. These measures \(\nu \) satisfy an extended multifractal formalism in the sense that, for \(\alpha \) in some interval, the Hausdorff dimension of the level sets \(X(\alpha )\) of the local Holder exponent of \(\nu \) is the Legendre transform of \(b\) whereas their packing dimension is the Legendre transform of \(B\). We first construct such measures \(\mu \) on a symbolic space. Then we obtain the measures \(\nu \) by projecting \(\mu \) on \([0,1]\) after composition with a Gray code.

Computability of the packing measure of totally disconnected self-similar sets

We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called Strong Separation Condition (SSC) and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show both the accuracy of the algorithm for the most regular cases and the possibility of using the additional information provided by it to obtain formulas for the packing measure of certain self-similar sets. For example, we are able to obtain a formula for the packing measure of any Sierpinski gasket with contraction factor in the interval $(0,\frac{1}{3}]$.