Related papers: Sprague-Grundy Value for Common Divisor Nim

The conditional gradient method (CGM) has been widely used for fast sparse approximation, having a low per iteration computational cost for structured sparse regularizers. We explore the sparsity acquiring properties of a generalized CGM…

We define the family of {\it locally path-bounded} digraphs, which is a class of infinite digraphs, and show that on this class it is relatively easy to compute an optimal strategy (winning or nonlosing); and realize a win, when possible,…

Wythoff's Game is a variation of Nim in which players may take an equal number of stones from each pile or make valid Nim moves. W. A. Wythoff proved that the set of P-Positions (losing position), $C$, for Wythoff's Game is given by $C :=…

We propose a variant of Nim, named StrNim. Whereas a position in Nim is a tuple of non-negative integers, that in StrNim is a string, a sequence of characters. In every turn, each player shrinks the string, by removing a substring repeating…

The aim of this note is to provide a Master Theorem for some discrete divide and conquer recurrences: $$X_{n}=a_n+\sum_{j=1}^m b_j X_{\lfloor{\frac{n}{m_j}}\rfloor},$$ where the $m_i$'s are integers with $m_i\ge 2$. The main novelty of this…

Nim is a well-known combinatorial game with several variants, e.g., Delete Nim and Variant Delete Nim. In Variant Delete Nim, the player deletes one of the two heaps of stones and splits the other heap on his/her turn. In this paper, we…

Let $\mathbb{K}$ be a number field of degree $n$ over $\mathbb{Q}$. Let $\widehat{\mathbb{A}}$ be the set of integers of $\mathbb{K}$ which are primitive over $\mathbb{Q}$ and $I(\mathbb{K})$ be its index. Gunji and McQuillan defined the…

\textsc{cut} is a class of partition games played on a finite number of finite piles of tokens. Each version of \textsc{cut} is specified by a cut-set $\mathcal{C}\subseteq\mathbb{N}$. A legal move consists of selecting one of the piles and…

A generalized $N$-sided die is a random variable $D$ on a sample space of $N$ equally likely outcomes taking values in the set of positive integers. We say of independent $N$ sided dice $D_i, D_j$ that $D_i$ beats $D_j$, written $D_i \to…

We solve the stochastic generalized Nash equilibrium (SGNE) problem in merely monotone games with expected value cost functions. Specifically, we present the first distributed SGNE seeking algorithm for monotone games that requires one…

We propose a new class of games, called Multi-Games (MG), in which a given number of players play a fixed number of basic games simultaneously. In each round of the MG, each player will have a specific set of weights, one for each basic…

For a partition $\lambda \vdash n$, we let $\operatorname{pd}(\lambda)$, the parity difference of $\lambda$, be the number of odd parts of $\lambda$ minus the number of even parts of $\lambda$. We prove for $c_0\in\mathbb{R}$ an asymptotic…

For any complex number $c$, let $\sigma_c\colon\mathbb N\rightarrow\mathbb C$ denote the divisor function defined by $\sigma_c(n)=\displaystyle{\sum_{d|n}d^c}$ for all $n\in\mathbb N$, and define $R(c)=\{\sigma_c(n)\in\mathbb C\colon…

In this paper, we study an impartial game called Delete Nim. In this game, there are two heaps of stones. The player chooses one of the heaps and delete the other heap. Next, she takes away one stone from the chosen heap and optionally…

We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)})^m$ of $n\geq r$ times continuously differentiable…

We define a generalization of the winding number of a piecewise $C^1$ cycle in the complex plane which has a geometric meaning also for points which lie on the cycle. The computation of this winding number relies on the Cauchy principal…

We consider a subtraction Nim with subtraction set {s_1,s_2,s_3={2,4n,4n+2}, where n is a positive integer such that n >= 3. We do not treat the case that n=1 or n=2 in this article. We show that this game satisfies the reverse-mex property…

Given a positive integer $m$, the $m$-step competition graph of a digraph $D$, denoted by $C^m(D)$, has the same vertex set as $D$ and has an edge between vertices $u$ and $v$ if and only if there exists a vertex $w$ such that there exist…

Let $d(n)$ denote the number of divisors of $n$. In this paper, we study the average value of $d(a(p))$, where $p$ is a prime and $a(p)$ is the $p$-th Fourier coefficient of a normalized Hecke eigenform of weight $k \ge 2$ for $\Gamma_0(N)$…

The main objective of this paper is to obtain some Gruss Like Inequalities using Pompeiu's mean value theorem on Conformable Fractional Calculus.