NODE-WEIGHTED AVERAGE FERMAT DISTANCES OF FRACTAL TREE NETWORKS

In this paper, we investigate a class of self-similar networks modeled on a self-similar fractal tree, and use the self-similar measure and the method of finite pattern to obtain the asymptotic formula of node-weighted average Fermat distances on fractal tree networks.

Equidistribution Results for Self-Similar Measures

A well-known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one. In this paper, we give sufficient conditions for an analogue of this theorem to hold for a self-similar measure. Our approach applies more generally to sequences of the form $(f_{n}(x))_{n=1}^{\infty }$ where $(f_n)_{n=1}^{\infty }$ is a sequence of sufficiently smooth real-valued functions satisfying some nonlinearity conditions. As a corollary of our main result, we show that if $C$ is equal to the middle 3rd Cantor set and $t\geq 1$, then with respect to the natural measure on $C+t,$ for almost every $x$, the sequence $(x^n)_{n=1}^{\infty }$ is uniformly distributed modulo one.

Weighted average geodesic distance of Vicsek network in three-dimensional space

Fractal generally has self-similarity. Using the self-similarity of fractal, we can obtain some important theories about complex networks. In this paper, we concern the Vicsek fractal in three-dimensional space, which provides a natural generalization of Vicsek fractal. Concretely, the Vicsek fractal in three-dimensional space is obtained by repeatedly removing equilateral cubes from an initial equilateral cube of unit side length, at each stage each remaining cube is divided into [Formula: see text] smaller cubes of which [Formula: see text] are kept and the rest discarded, where [Formula: see text] is odd. In addition, we obtain the skeleton network of the Vicsek fractal in three-dimensional space. Then we focus on weighted average geodesic distance of the Vicsek fractal in three-dimensional space. Take [Formula: see text] as an example, we define a similar measure on the Vicsek fractal in three-dimensional space by weight vector and calculate the weighted average geodesic distance. At the same time, asymptotic formula of weighted average geodesic distance on the skeleton network is also obtained. Finally, the general formula of weighted average geodesic distance should be applicable to the models when [Formula: see text], the base of a power, is odd.

WSN Based on Bio-Inspired Algorithm in Cluster Head Formation

A sort of inclusion improvement technique dependent on bio-inspired algorithm was implemented so as to fathom the irrationality and low system inclusion of sensor hub in WSN at the irregular dispersion. Right off the bat, the ebb and flow investigate status of WSN inclusion was broke down, the hub inclusion and territorial inclusion in WSN on the premise were examined, the comparing scientific model was set up, the bio-enlivened calculation was taken to tackle the built up numerical model, and the WSN inclusion advancement program dependent on the bio-roused was gotten. At last, MATLAB was utilized for the recreation analyses, and the reproduction results demonstrated that the presentation of bio-motivated calculation improved the hub inclusion in WSN viably; the inclusion territory was huger at a similar measure of hubs. In addition, the calculation can get the ideal arrangement in the worldwide extension, and arrive at the better system inclusion advancement impact with less sensor hubs, and the quantity of cycles was diminished altogether.

SPECTRALITY AND NON-SPECTRALITY OF PLANAR SELF-SIMILAR MEASURES WITH FOUR-ELEMENT DIGIT SETS

Let [Formula: see text] be the unit matrix and [Formula: see text]. In this paper, we consider the self-similar measure [Formula: see text] on [Formula: see text] generated by the iterated function system [Formula: see text] where [Formula: see text]. We prove that there exists [Formula: see text] such that [Formula: see text] is an orthonormal basis for [Formula: see text] if and only if [Formula: see text] for some integer [Formula: see text].

A strongly irreducible affine iterated function system with two invariant measures of maximal dimension

A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.

WEIGHTED AVERAGE GEODESIC DISTANCE OF PENTADENDRITE NETWORKS

The average geodesic distance is an important index in the study of complex networks. In this paper, we investigate the weighted average distance of Pentadendrite fractal and Pentadendrite networks. To provide the formula, we use the integral of geodesic distance in terms of self-similar measure with respect to the weighted vector.

THE NON-SPECTRAL PROPERTY OF A CLASS OF PLANAR SELF-SIMILAR MEASURES

Let the self-similar measure [Formula: see text] be generated by an expanding real matrix [Formula: see text] and a digit set [Formula: see text] in space [Formula: see text]. In this paper, we only consider [Formula: see text] and the case [Formula: see text] is similar. We show that there exists an infinite orthogonal set of exponential functions in [Formula: see text] if and only if [Formula: see text] for some [Formula: see text] with [Formula: see text]. Furthermore, for the cases that [Formula: see text] does not admit any infinite orthogonal set of exponential functions, the exact cardinality of orthogonal exponential functions in [Formula: see text] is given.

Estimated Sp02/Fio2 ratio to predict mortality in patients with suspected COVID-19 in the Emergency Department: a prospective cohort study

Background This study examined whether the presence and severity of Type 1 Respiratory Failure (T1RF), as measured by the ratio of pulse oximetry to estimated fraction of inspired oxygen (SpO2/eFiO2 ratio), is a predictor of in-hospital mortality in patients presenting to the ED with suspected COVID19 infection. Methods We undertook a prospective observational cohort study of patients admitted to hospital with suspected COVID-19 in a single ED in England. We used univariate and multiple logistic regression to examine whether the presence and severity of T1RF in the ED was independently associated with in-hospital mortality. Results 180 patients with suspected COVID19 infection met the inclusion criteria for this study, of which 39 (22%) died. Severity of T1RF was associated with increased mortality with odds ratios (OR) and 95% confidence intervals of 1.58 (0.49 to 5.14), 3.60 (1.23 to 10.6) and 18.5 (5.65 to 60.8) for mild, moderate and severe T1RF, respectively. After adjusting for age, gender, pre-existing cardiovascular disease, neutrophil-lymphocyte ration (NLR) and estimated glomerular filtration rate (eGFR), the association remained, with ORs of 0.63 (0.13 to 3.03), 3.95 (0.94 to 16.6) and 45.8 (7.25 to 290). The results were consistent across a number of sensitivity analyses. Conclusions Severity of T1RF in the ED is an important prognostic factor of mortality in patients admitted with suspected COVID19 infection. Current prediction models frequently do not include this factor and should be applied with caution. Further large scale research on predictors of mortality in COVID19 infection should include SpO2/eFiO2 ratios or a similar measure of respiratory dysfunction.

A CONVEX SURFACE WITH FRACTAL CURVATURE

We construct a surface that is obtained from the octahedron by pushing out four of the faces so that the curvature is supported in a copy of the Sierpinski gasket (SG) in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.