Uniform Random Covering Problems

Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $\omega =(\omega _n)_{n\geq 1}$ uniformly distributed on the unit circle $\mathbb{T}$ and a sequence $(r_n)_{n\geq 1}$ of positive real numbers with limit $0$. We investigate the size of the random set $$\begin{align*} & {\operatorname{{{\mathcal{U}}}}} (\omega):=\{y\in \mathbb{T}: \ \forall N\gg 1, \ \exists n \leq N, \ \text{s.t.} \ | \omega_n -y | < r_N \}. \end{align*}$$Some sufficient conditions for ${\operatorname{{{\mathcal{U}}}}}(\omega )$ to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that ${\operatorname{{{\mathcal{U}}}}}(\omega )$ is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.

The Setting of Strength Parameters in Stability Analysis of Open-Pit Slope Using the Random Set Method in the Bełchatów Lignite Mine, Central Poland

The slopes of open-pit mines are often at risk of failure. To identify this hazard, stability analyses are performed. An important element of these stability analyses is the reliable selection of input parameter values for the calculations. This selection is difficult because the slopes of the open pit are disturbed by mining activities. In such conditions, rheological processes, intensified by weathering, develop in open-pit slopes. This study is aimed at setting the strength parameters for the stability analysis of open-pit slopes with a developed slide process, using the random set method. The study was performed on the example of the open pit of the Bełchatów lignite mine, central Poland. A four-stage methodology, according to the random set method, was proposed. The methodology covered the following: site investigation, sensitivity analyses, shear strength reduction (SSR) analyses using numerical calculations, and probability analyses of the factor of safety (FoS) calculation results. The setting of the input parameters took into account the peak and residual strength parameters for each lithological unit in the physical model of the open-pit slope. Samples for laboratory tests were taken from the cores of nine test boreholes. The sensitivity analysis included all peak and residual strength parameters for each lithological unit in the body. As a result of the sensitivity analysis, specific strength parameters were adopted that would have a great impact upon the results of the calculations. Selected sets of parameter values were then used for the FoS calculations. The resultant FoS values revealed the probable slide planes. The positions of the slide planes were consistent with the interpreted slide surfaces based on the control boreholes and terrain observations. Knowledge of the slide planes positions and the values of the strength parameters enabled the designing of a securing approach for this landslide, and the taking of preventive measures to reduce this risk.

A novel Acoustic Fluid Level Estimation Method Based on Evidence Fusion Mechanism

In order to reduce the missed detection error and the systematic error caused by acoustic resonance fluid level detection, liquid level estimation method based on evidence fusion mechanism is designed. It establishes a two dimensional dynamic system model of the standingwavelength. The state evidence of wavelength is obtained through the random set description of evidence, and the extension principle of random set is used to get the observation evidence ofwavelength. The evidential reasoning (ER) rule and dependent evidence fusion are used to fuse those evidence, and the estimation value of fluid level can be calculated from fused result based on pignistic expectation. The corresponding liquid level estimation experiment illustrates the validity and feasibilityof the proposed method.

Intermediate intrinsic density and randomness

Given any 1-random set X and any r in ( 0 , 1 ), we construct a set of intrinsic density r which is computable from both r and X. For almost all r, this set will be the first known example of an intrinsic density r set which cannot compute any r-Bernoulli random set. To achieve this, we shall formalize the into and within noncomputable coding methods which work well with intrinsic density.

On the least common multiple of random q-integers

For every positive integer n and for every $$\alpha \in [0, 1]$$ α ∈ [ 0 , 1 ] , let $${\mathcal {B}}(n, \alpha )$$ B ( n , α ) denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$ A ⊆ { 1 , … , n } is constructed by picking independently each element of $$\{1, \ldots , n\}$$ { 1 , … , n } with probability $$\alpha $$ α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$ A .Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$ [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$ X : = deg lcm ( [ A ] q ) , where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$ [ A ] q : = { [ k ] q : k ∈ A } . Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.

Were Ancestral Proteins Less Specific?

Some have hypothesized that ancestral proteins were, on average, less specific than their descendants. If true, this would provide a universal axis along which to organize protein evolution and suggests that reconstructed ancestral proteins may be uniquely powerful tools for protein engineering. Ancestral sequence reconstruction studies are one line of evidence used to support this hypothesis. Previously, we performed such a study, investigating the evolution of peptide-binding specificity for the paralogs S100A5 and S100A6. The modern proteins appeared more specific than their last common ancestor (ancA5/A6), as each paralog bound a subset of the peptides bound by ancA5/A6. In this study, we revisit this transition, using quantitative phage display to measure the interactions of 30,533 random peptides with human S100A5, S100A6, and ancA5/A6. This unbiased screen reveals a different picture. While S100A5 and S100A6 do indeed bind to a subset of the peptides recognized by ancA5/A6, they also acquired new peptide partners outside of the set recognized by ancA5/A6. Our previous work showed that ancA5/A6 had lower specificity than its descendants when measured against biological targets; our new work shows that ancA5/A6 has similar specificity to the modern proteins when measured against a random set of peptide targets. This demonstrates that altered biological specificity does not necessarily indicate altered intrinsic specificity, and sounds a cautionary note for using ancestral reconstruction studies with biological targets as a means to infer global evolutionary trends in specificity.

Impact of Teaching Cohesive Devices to Iranian EFL learners: Case of IELTS writing task 2 examination

The notion of “textuality” encouraged Halliday and Hasan in 1976 to present their model of discourse analysis through raising questions about whether “cohesion” was a semantic concept or a structural relation, whether a text was a structural unit or not or even if there were semantic or structural relationships within a text. Cohesion was like the glue that unified the meaning within a text through binding the textual elements. Tackling relations of meaning and references in a text was often related to cohesion. A text can be cohesive if its units were bound together with explicit or implicit relations. Cohesion was often defined as the network of lexical, grammatical, and other relations which provide links between various parts of a text. The units of a cohesive text were not just a random set of sentences. Writing section on IELTS was commonly considered one of the most difficult parts of test. The test takers cannot even understand what to do with the tasks provided. They eventually wrote without knowing the expected direction. Therefore, there should be a fast way to equip students well to successfully cope with such hindrances. This research was an action research report of 10 students to make them better understand and answer writing tasks on IELTS test by employing probing techniques as one of test taking strategies. As the result, teaching material about cohesive devices, according to criteria and references to IELTS writing task 2, could help students to understand of their logic on writing proper materials.

Falling shadow theory with applications in hoops

The falling shadow theory is applied to subhoops and filters in hoops. The notions of falling fuzzy subhoops and falling fuzzy filters in hoops are introduced, and several properties are investigated. Relationship between falling fuzzy subhoops and falling fuzzy filters are discussed, and conditions for a falling fuzzy subhoop to be a falling fuzzy filter are provided. Also conditions for a falling shadow of a random set to be a falling fuzzy filer are displayed.