DHHFL-MABAC Approach based on DHHFLDM-LCME for Multi-criteria Group Decision Making Involving Sewage Treatment Company Selection

With the continuous acceleration of urban construction and development, ecological governance has become an important part of our green life. Therefore, how to select the appropriate governance company in the complex decision-making environment is very important. A double hierarchy hesitant fuzzy linguistic multi-attributive border approximation area method (DHHFL-MABAC) is proposed in this paper, which is based on distance measure and comprehensive weight. DHHFL-MABAC method not only considers the potential loss, but also has simple calculation and stable results. The double hierarchy hesitant fuzzy linguistic distance measure based on least common multiple (DHHFLDM-LCME) is proposed, which reduces the loss of original information and makes the result more accurate. Binary contrast method is proposed and combined with entropy weight to obtain comprehensive weight, which makes the determined weight more reasonable. Finally, this method is applied to the case selected by the sewage treatment company and proved its effectiveness.

A variant of the Hardy-Ramanujan theorem

For each natural number $n$, we define $\omega^*(n)$ to be the number of primes $p$ such that $p-1$ divides $n$. We show that in contrast to the Hardy-Ramanujan theorem which asserts that the number $\omega(n)$ of prime divisors of $n$ has a normal order $\log\log n$, the function $\omega^*(n)$ does not have a normal order. We conjecture that for some positive constant $C$, $$\sum_{n\leq x} \omega^*(n)^2 \sim Cx(\log x). $$ Another conjecture related to this function emerges, which seems to be of independent interest. More precisely, we conjecture that for some constant $C>0$, as $x\to \infty$, $$\sum_{[p-1,q-1]\leq x} {1 \over [p-1, q-1]} \sim C \log x, $$ where the summation is over primes $p,q\leq x$ such that the least common multiple $[p-1,q-1]$ is less than or equal to $x$.

On the divisibility among power GCD and power LCM matrices on gcd-closed sets

Let [Formula: see text] and [Formula: see text] be positive integers and let [Formula: see text] be a set of [Formula: see text] distinct positive integers. For [Formula: see text], one defines [Formula: see text]. We denote by [Formula: see text] (respectively, [Formula: see text]) the [Formula: see text] matrix having the [Formula: see text]th power of the greatest common divisor (respectively, the least common multiple) of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry. In this paper, we show that for arbitrary positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], the [Formula: see text]th power matrices [Formula: see text] and [Formula: see text] are both divisible by the [Formula: see text]th power matrix [Formula: see text] if [Formula: see text] is a gcd-closed set (i.e. [Formula: see text] for all integers [Formula: see text] and [Formula: see text] with [Formula: see text]) such that [Formula: see text]. This confirms two conjectures of Shaofang Hong proposed in 2008.

The Specific Length of an Underground Tunnel and the Effects of Rock Block Characteristics on the Inflow Rate

The specific length of a tunnel (STL) and a new analytical model for calculating the block surface area of the rock mass are introduced. First, a method for determining the appropriate length of a tunnel for a numerical simulation is described. The length is then used to examine the correlation between the inflow rate to the tunnel and the block volume, the block surface area, and the fracture intensity (P32) through analytical and numerical modeling. The results indicate that the length of the tunnel should at least be equal to the least common multiple (LCM) of the apparent spacings of the joint sets at the wall of the tunnel to obtain the more reliable and immediate results for the inflow rate to a tunnel that is excavated in a fractured rock mass. A new analytical model was developed to calculate the block surface area and determine the essential joint set parameters, which include the dip, dip direction, and spacing. The determination of the rock block characteristics through numerical modeling requires considering the intact block for calculations. The results indicated that the inflow rate to the tunnel increased with an increase in fracture intensity and a decrease in block volume and surface area. The STL and the analytical model used for calculating the block surface area are validated through numerical simulations with 3DEC software version 7.0.

On the least common multiple of polynomial sequences at prime arguments

Cilleruelo conjectured that if [Formula: see text] is an irreducible polynomial of degree [Formula: see text] then, [Formula: see text] In this paper, we investigate the analog of prime arguments, namely, [Formula: see text] where [Formula: see text] denotes a prime and obtain nontrivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of [Formula: see text]

Fourth-Grade Primary School Students’ Misconception on Greatest Common Factor and Least Common Multiple

This study is aimed at determining students’ misconceptions on the teaching material of the greatest common factor and least common multiple. The sample of this study consisted of 124 fourth-grade elementary school students in the academic year of 2019–2020 in areas of Mataram, West Lombok, North Lombok, and East Lombok at West Nusa Tenggara Province. The instrument of this study is a test covering five questions in the form of open questions. The data are analyzed based on the students’ explanations while they answer the test. The students’ wrong answers are grouped into categories. The interview activities are carried out for students who have a misconception. Then, researchers create a table of frequencies/presentations relating to each type of students’ misconception. The results show that students experience misconceptions due to factors, including having a weak multiplication concept, having a weak prime number concept, determining the least common multiple of two numbers by multiplying the two numbers, and inability to distinguish between multiples and factors of a number. In light of the findings, a number of conclusions are obtained and several implications are put forward.