Average Value of the Euler Function on Binary Palindromes

William D. Banks; Igor E. Shparlinski

Displaying similar documents to “Average Value of the Euler Function on Binary Palindromes”

Expansions of binary recurrences in the additive base formed by the number of divisors of the factorial

Florian Luca, Augustine O. Munagi (2014)

Colloquium Mathematicae

Similarity:

We note that every positive integer N has a representation as a sum of distinct members of the sequence ${d (n!)}_{n \geq 1}$ , where d(m) is the number of divisors of m. When N is a member of a binary recurrence $u = {u ₙ}_{n \geq 1}$ satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂. ...

On $☆$ - associated comonotone functions

Ondrej Hutník, Jozef Pócs (2018)

Kybernetika

Similarity:

We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper [1]. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone right-continuous operation. More precisely, the comonotonicity of functions is equivalent not only to $+$ -associatedness of functions (as proved by Boczek and Kaluszka), but also to their $☆$ -associatedness with $☆$ being an arbitrary...

$ℱ$ -hypercyclic and disjoint $ℱ$ -hypercyclic properties of binary relations over topological spaces

Marko Kostić (2020)

Mathematica Bohemica

Similarity:

We examine various types of $ℱ$ -hypercyclic ( $ℱ$ -topologically transitive) and disjoint $ℱ$ -hypercyclic (disjoint $ℱ$ -topologically transitive) properties of binary relations over topological spaces. We pay special attention to finite structures like simple graphs, digraphs and tournaments, providing a great number of illustrative examples.

From binary cube triangulations to acute binary simplices

Brandts, Jan, van den Hooff, Jelle, Kuiper, Carlo, Steenkamp, Rik

Similarity:

Cottle’s proof that the minimal number of $0 / 1$ -simplices needed to triangulate the unit $4$ -cube equals $16$ uses a modest amount of computer generated results. In this paper we remove the need for computer aid, using some lemmas that may be useful also in a broader context. One of the $0 / 1$ -simplices involved, the so-called antipodal simplex, has acute dihedral angles. We continue with the study of such acute binary simplices and point out their possible relation to the Hadamard determinant problem. ...

On the sum of the first n values of the Euler function

R. Balasubramanian, Florian Luca, Dimbinaina Ralaivaosaona (2014)

Acta Arithmetica

Similarity:

Let ϕ(n) be the Euler function of n. We put $E (n) = \sum_{m \leq n} ϕ (m) - (3 / π ²) n ²$ and give an asymptotic formula for the second moment of E(n).

Regular elements and Green's relations in Menger algebras of terms

Klaus Denecke, Prakit Jampachon (2006)

Discussiones Mathematicae - General Algebra and Applications

Similarity:

Defining an (n+1)-ary superposition operation $S^{n}$ on the set $W_{τ} (X_{n})$ of all n-ary terms of type τ, one obtains an algebra $n - c l o n e τ : = (W_{τ} (X_{n}); S^{n}, x_{1}, . . ., x_{n})$ of type (n+1,0,...,0). The algebra n-clone τ is free in the variety of all Menger algebras ([9]). Using the operation $S^{n}$ there are different possibilities to define binary associative operations on the set $W_{τ} (X_{n})$ and on the cartesian power $W_{τ} {(X_{n})}^{n}$ . In this paper we study idempotent and regular elements as well as Green’s relations in semigroups of terms with these binary associative...

Representation functions for binary linear forms

Fang-Gang Xue (2024)

Czechoslovak Mathematical Journal

Similarity:

Let $ℤ$ be the set of integers, $ℕ_{0}$ the set of nonnegative integers and $F (x_{1}, x_{2}) = u_{1} x_{1} + u_{2} x_{2}$ be a binary linear form whose coefficients $u_{1}$ , $u_{2}$ are nonzero, relatively prime integers such that $u_{1} u_{2} \neq \pm 1$ and $u_{1} u_{2} \neq - 2$ . Let $f : ℤ \to ℕ_{0} \cup {\infty}$ be any function such that the set $f^{- 1} (0)$ has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set $A$ of integers such that $r_{A, F} (n) = f (n)$ for all integers $n$ , where $r_{A, F} (n) = | {(a, a^{'}) : n = u_{1} a + u_{2} a^{'} : a, a^{'} \in A} |$ . We add the structure of difference for the binary linear form $F (x_{1}, x_{2})$ .

A Menon-type identity using Klee's function

Arya Chandran, Neha Elizabeth Thomas, K. Vishnu Namboothiri (2022)

Czechoslovak Mathematical Journal

Similarity:

Menon’s identity is a classical identity involving gcd sums and the Euler totient function $φ$ . A natural generalization of $φ$ is the Klee’s function $Φ_{s}$ . We derive a Menon-type identity using Klee’s function and a generalization of the gcd function. This identity generalizes an identity given by Y. Li and D. Kim (2017).

Properties of unique information

Johannes Rauh, Maik Schünemann, Jürgen Jost (2021)

Kybernetika

Similarity:

We study the unique information function $U I (T : X ∖ Y)$ defined by Bertschinger et al. within the framework of information decompositions. In particular, we study uniqueness and support of the solutions to the convex optimization problem underlying the definition of $U I$ . We identify sufficient conditions for non-uniqueness of solutions with full support in terms of conditional independence constraints and in terms of the cardinalities of $T$ , $X$ and $Y$ . Our results are based on a reformulation of the first...

On a sum involving the Möbius function

I. Kiuchi, M. Minamide, Y. Tanigawa (2015)

Acta Arithmetica

Similarity:

Let $c_{q} (n)$ be the Ramanujan sum, i.e. $c_{q} (n) = \sum_{d | (q, n)} d μ (q / d)$ , where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for $\sum_{n \leq y} {(\sum_{q \leq x} c_{q} (n))}^{k}$ (k = 1,2) are obtained. As an analogous problem, we evaluate $\sum_{n \leq y} {(\sum_{n \leq x} c ̂_{q} (n))}^{k}$ (k = 1,2), where $c ̂_{q} (n) : = \sum_{d | (q, n)} d | μ (q / d) |$ .

Incidence coalgebras of interval finite posets of tame comodule type

Zbigniew Leszczyński, Daniel Simson (2015)

Colloquium Mathematicae

Similarity:

The incidence coalgebras $K^{□} I$ of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form $q^{•} : ℤ^{(I)} \to ℤ$ , where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category $K^{□} I - c o m o d$ of finite-dimensional left $K^{□} I$ -modules is equivalent to the tameness of the category $K^{□} I - C o m o d_{f c}$ of finitely copresented left $K^{□} I$ -modules. Hence, the tame-wild dichotomy for the coalgebras $K^{□} I$ is deduced. Moreover, we prove that for an interval finite...

On the number of finite algebraic structures

Erhard Aichinger, Peter Mayr, R. McKenzie (2014)

Journal of the European Mathematical Society

Similarity:

We prove that every clone of operations on a finite set $A$ , if it contains a Malcev operation, is finitely related – i.e., identical with the clone of all operations respecting $R$ for some finitary relation $R$ over $A$ . It follows that for a fixed finite set $A$ , the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra...

Lagrangians and Euler morphisms on fibered-fibered frame bundles from projectable-projectable classical linear connections

Anna Bednarska (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

We classify all $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $A$ transforming projectable-projectable torsion-free classical linear connections $\nabla$ on fibered-fibered manifolds $Y$ of dimension $(m_{1}, m_{2}, n_{1}, n_{2})$ into $r$ th order Lagrangians $A (r)$ on the fibered-fibered linear frame bundle $L^{f i b - f i b} (Y)$ on $Y$ . Moreover, we classify all $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $B$ transforming projectable-projectable torsion-free classical linear connections r on fiberedfibered manifolds $Y$ of dimension $(m_{1}, m_{2}, n_{1}, n_{2})$ into Euler morphism $B (\nabla)$ on $L^{f i b - f i b} (Y)$ . These classifications can be expanded on...

On the least common multiple of Lucas subsequences

Shigeki Akiyama, Florian Luca (2013)

Acta Arithmetica

Similarity:

We compare the growth of the least common multiple of the numbers $u_{a_{1}}, . . ., u_{a_{n}}$ and $| u_{a_{1}} \dots u_{a_{n}} |$ , where ${(u_{n})}_{n \geq 0}$ is a Lucas sequence and ${(a_{n})}_{n \geq 0}$ is some sequence of positive integers.

A symmetry problem in the calculus of variations

Graziano Crasta (2006)

Journal of the European Mathematical Society

Similarity:

We consider the integral functional $J (u) = \int_{Ω} [f (| D u |) - u] d x$ , $u \in W_{0}^{1, 1} (Ω)$ , where $Ω \subset ℝ^{n}$ , $n \geq 2$ , is a nonempty bounded connected open subset of $ℝ^{n}$ with smooth boundary, and $ℝ ∋ s \mapsto f (| s |)$ is a convex, differentiable function. We prove that if $J$ admits a minimizer in $W_{0}^{1, 1} (Ω)$ depending only on the distance from the boundary of $Ω$ , then $Ω$ must be a ball.

Repdigits in generalized Pell sequences

Jhon J. Bravo, Jose L. Herrera (2020)

Archivum Mathematicum

Similarity:

For an integer $k \geq 2$ , let ${(n)}_{n}$ be the $k -$ generalized Pell sequence which starts with $0, ..., 0, 1$ ( $k$ terms) and each term afterwards is given by the linear recurrence $n = 2 n - 1 + n - 2 + \dots + n - k$ . In this paper, we find all $k$ -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ${(P_{n}^{(2)})}_{n}$ . ...

On a special class of left-continuous uninorms

Gang Li (2018)

Kybernetika

Similarity:

This paper is devoted to the study of a class of left-continuous uninorms locally internal in the region $A (e)$ and the residual implications derived from them. It is shown that such uninorm can be represented as an ordinal sum of semigroups in the sense of Clifford. Moreover, the explicit expressions for the residual implication derived from this special class of uninorms are given. A set of axioms is presented that characterizes those binary functions $I : {[0, 1]}^{2} \to [0, 1]$ for which a uninorm $U$ of this special...