Related papers: On a sequence related to the Josephus problem

This article demonstrates a new kind of programmable logic for the representation of an integer that can be used for the programmable Josephson voltage standard. It can enable the numbers of junctions in most bits to be variable integer…

Let $q\in(1,2)$ and $x\in[0,\frac1{q-1}]$. We say that a sequence $(\varepsilon_i)_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is an expansion of $x$ in base $q$ (or a $q$-expansion) if \[ x=\sum_{i=1}^{\infty}\varepsilon_iq^{-i}. \] For any…

For three natural classes of dynamic decision problems; 1. additively separable problems, 2. discounted problems, and 3. discounted problems for a fixed discount factor; we provide necessary and sufficient conditions for one sequential…

We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope. We use this correspondence to prove combinatorial results…

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot...\cdot(n_lg)$ where $g\in G$ and $n_1,\cdots,n_l\in[1,{\hbox{\rm ord}}(g)]$, and the index $\ind(S)$ of $S$ is defined to be the…

We introduce a bijection between inequivalent minimal factorizations of the n-cycle (1 2 ... n) into a product of smaller cycles of given length, on one side, and trees of a certain structure on the other. We use this bijection to count the…

We study the complexity of the following cell connection and separation problems in segment arrangements. Given a set of straight-line segments in the plane and two points $a$ and $b$ in different cells of the induced arrangement: (i)…

We give several partial positive answers to a question of Juhasz and Szentmiklossy regarding the minimum number of discrete sets required to cover a compact space. We study the relationship between the size of discrete sets, free sequences…

Twin prime number problem is mainly the structure of the twin prime numbers and whether there are infinitely many prime twins group. In this paper, by constructing a special cluster number set(see formula(2.3)in the paper), proves that the…

This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional…

The Josephus problem is a well--studied elimination problem consisting in determining the position of the survivor after repeated applications of a deterministic rule removing one person at a time from a given group. A natural probabilistic…

For Lebesgue generic $(x_1,x_2)\in \mathbb{R}^2$, we investigate the distribution of small values of products $q\cdot \|qx_1\| \cdot \|qx_2\|$ with $q\in\mathbb{N}$, where $\|\cdot \|$ denotes the distance to the closest integer. The main…

This paper addresses the problem of sequential submodular maximization: selecting and ranking items in a sequence to optimize some composite submodular function. In contrast to most of the previous works, which assume access to the utility…

We look at extensions of formulas given by Jovovic and recently proved by Dhar on integer partitions where the smallest part occurs at least $m$ times and on integer partitions with fixed differences between the largest and smallest parts…

In this short note we present a class of conjectures on partitions of integers as summations of primes, which are extensions of Goldbach conjecture.

Having in mind applications to particle physics we develop the differential calculus over Jordan algebras and the theory of connections on Jordan modules. In particular we focus on differential calculus over the exceptional Jordan algebra…

In this paper the fractional-order Mandelbrot and Julia sets in the sense of $q$-th Caputo-like discrete fractional differences, for $q\in(0,1)$, are introduced and several properties are analytically and numerically studied. Some…

In this paper, we study uniqueness problems for an entire function that shares small functions of finite order with their difference operators. In particular, we give a generalization of results in [2,3,13].

We establish the existence and symmetry of all minimizers of a constrained variational problem involving the fractional gradient. This problem is closely connected to some fractional kinetic equations.

The Dirichlet divisor problem is used as a model to give a conjecture concerning the conditional convergence of the Dirichlet series of an L-function.