Related papers: Effective estimates for the least common multiple …
In this paper, we first prove that for any strong divisibility sequences $\boldsymbol{a} = \left(a_n\right)_{n\geq 1}$, we have the identity: $\mathrm{lcm} \left\lbrace \binom{n}{0}_{\bf{a}}, \binom{n}{1}_{\bf{a}},\dots,…
In this paper, we present a method for estimating the least common multiple of a large class of binary linear recurrence sequences. Let $P,Q,R_0$, and $R_1$ be fixed integers and let $\boldsymbol{R}=\left(R_n\right)_{n}$ be the recurrence…
We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and…
This paper is devoted to establish nontrivial effective lower bounds for the least common multiple of consecutive terms of a sequence ${(u_n)}_{n \in \mathbb{N}}$ whose general term has the form $u_n = r {[n]}_q + u_0$, where $q , r$ are…
For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathrm{lcm}(u_0,u_1,\ldots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ which…
Since the $\mathrm{Fibonacci}$ sequence has good properties, it's important in theory and applications, such as in combinatorics, cryptography, and so on. In this paper, for the generalized Fibonacci sequence…
Linear Least Squares is a very well known technique for parameter estimation, which is used even when sub-optimal, because of its very low computational requirements and the fact that exact knowledge of the noise statistics is not required.…
This paper shows a novel fuzzy approximate reasoning method based on the least common multiple (LCM). Its fundamental idea is to obtain a new fuzzy reasoning result by the extended distance measure based on LCM between the antecedent fuzzy…
This paper is devoted to studying the numbers $L_{c,m,n} := \mathrm{lcm}\{m^2+c ,(m+1)^2+c , \dots , n^2+c\}$, where $c,m,n$ are positive integers such that $m \leq n$. Precisely, we prove that $L_{c,m,n}$ is a multiple of the rational…
In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers $a$ and $b$, with…
We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size…
We study the logarithm of the least common multiple of the sequence of integers given by $1^2+1, 2^2+1,..., n^2+1$. Using a result of Homma on the distribution of roots of quadratic polynomials modulo primes we calculate the error term for…
This is a brief tutorial on the least square estimation technique that is straightforward yet effective for parameter estimation. The tutorial is focused on the linear LSEs instead of nonlinear versions, since most nonlinear LSEs can be…
This note provides very simple, efficient algorithms for computing the number of distinct longest common subsequences of two input strings and for computing the number of LCS embeddings.
Recently Greg Martin derived an interesting formula for the least common multiple of {1,2,...,n}. Here, we give an exposition of a concise proof in terms of the sine function.
We study the growth behaviour of rational linear recurrence sequences. We show that for low-order sequences, divergence is decidable in polynomial time. We also exhibit a polynomial-time algorithm which takes as input a divergent rational…
We study the typical behavior of the least common multiple of the elements of a random subset $A\subset \{1,\dots, n\}$. For example we prove that $\text{lcm}\{a:\ a\in A\}=2^{n(1+o(1))}$ for almost all subsets $A\subset\{1,\dots,n\}$.
In this paper, we prove the following identity $$ \lcm({n\brack 0}_q,{n\brack 1}_q,...,{n\brack n}_q) =\frac{\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, $$ where ${n\brack k}_q$ denotes the $q$-binomial coefficient and…
The use of realistic input models has gained popularity in the theory community. Assuming a realistic input model often precludes complicated hypothetical inputs, and the analysis yields bounds that better reflect the behaviour of…
Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${\rm lcm}_{\lceil n/2\rceil \le i\le n} \{f(i)\}\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with…