50 Years of the Golomb--Welch Conjecture

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.Comment: 28 pages, 2 figure

The Effect of Project Types and Technologies on Software Developers' Efforts

Considering intrinsic valuation of software developers as the main motive for participating in open source projects, we examine the (Nash) equilibrium effort levels of the software developers in implementing projects that follow one of the three different technologies: the summation, the weakest-link, and the best-shot. Under the summation technology, developers having higher intrinsic valuation exert more effort in open source projects but all developers in commercial projects expend the same effort. Under the weakest-link technology, regardless of the types of the projects, all developers exert the same effort at equilibrium. In open source projects, the developer with the lowest intrinsic valuation has a crucial role in determining the equilibrium effort level while, in case of commercial projects, the equilibrium effort level is bounded by the net wage. Finally, under the best-shot technology, only one developer makes serious effort and the others free ride in both open source and commercial projects.Open Source Software, Intrinsic Motivation, Software Economics, Game Theory

Isolation and exploitation of minority: Game theoretical analysis

We investigate various group-size distributions occurring in a situation where each group???s resource is exposed to appropriation by other groups. The amount of appropriation depends on the size difference between groups. Our work focuses on the cases where the entire community isolates a small group or even an individual to maximize its gain. While people???s basic motivation to form a group can be understood based on the group-size effect on multiplying a collective asset, sensitive factors that induce a asymmetric group distribution are the group efficiency and the ratio of secured assets to assets pending in a competition. We show that social rejection to a minor group may occur when the group efficiency is relatively low and their asset is severely exposed to possible appropriation

Connected cubic graphs with the maximum number of perfect matchings

It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well. In addition, it is shown that the number of perfect matchings in any cubic graph $G$ equals the expected value of a random variable defined on all $2$-colorings of edges of $G$. Finally, an improved lower bound on the maximum number of cycles in a cubic graph is provided.Comment: 20 pages, 19 figure

Neoliberal Paradox? Explaining the Unremitting Corruptions in the Deregulated Korean Economy

The recent financial scandal in Korea surrounding savings banks call into question two kinds wisdom we have about corruption: corruption is a result of political rent-seeking, and corruption in East Asia is manageable with the help of the centralized and strong state. Contrary to the expectation that the double advent of democratization and globalization would scale down, if not eradicate, such a political rent-seeking, the recent corruption seen from the scandal has been encompassing the entire regulatory regime. The profile of the corruption was also ugly along with blatant trades between regulatory favors and bribes. This paper defines the Korean scandal as a bureaucratic institutional corruption resulting from the decline of a few management mechanisms occurred throughout the political and economic liberalization during the 1990s. First, the financial scandal is understood in this paper as a policy failure involving misbehaviors while implementing regulatory policies, differently from a generic political failure where policy favors are awarded to selected businesses in return for payments, mostly illegal, at the stage of policy making. It happens with an unsettled economic reform like in Korea where the liberalized financial market meets with a few still highly discretionary government policy agencies. Second, a few fallouts of bureaucratic institutional deterioration or abuse have collectively driven the financial regulatory regimes down to corruption. Most fatal, this study suggests, is the disruption of deferred compensation to policy agents

Rigidity and Circular slices

Let $n, m\ge 2$. Let $\Gamma<\text{SO}^\circ(n+1,1)$ be a Zariski dense convex cocompact subgroup and $\Lambda \subset \mathbb{S}^n$ its limit set. Sullivan proved in 1979 that the $\delta$-dimensional Hausdorff measure satisfies $0<\mathcal{H}^{\delta}(\Lambda)<\infty$ where $\delta$ is the Hausdorff dimension of $\Lambda$. Suppose that the ordinary set $\Omega=\mathbb{S}^n-\Lambda$ has at least two components. For a Zariski dense convex cocompact faithful representation $\rho : \Gamma \to \text{SO}^\circ(m+1,1)$ and its boundary map $f:\Lambda\to \mathbb{S}^{m}$, we present a criterion on when $\rho$ is algebraic (i.e., given by a conjugation by a M\"obius transformation on $\mathbb{S}^n$) in terms of the Hausdorff measure of all circular slices of $\Lambda$ that are mapped into circles, or more generally, into proper spheres of $\mathbb{S}^m$. More precisely, letting \Lambda_f:= \bigcup \left\{ C \cap \Lambda : \begin{matrix} C \subset \mathbb{S}^n \mbox{ is a circle such that} \\ f(C \cap \Lambda) \mbox{ is contained in a proper sphere } \mbox{of $\mathbb{S}^m$} \end{matrix} \right\}, we prove the following dichotomy: $$\text{either}\quad \Lambda_f= \Lambda \quad \text{ or } \quad \mathcal{H}^{\delta}(\Lambda_f) =0,$$ and in the former case, we have $n=m$ and $\rho$ is a conjugation by a M\"obius transformation on $\mathbb{S}^n$. Our proof uses ergodic theory and higher rank conformal measure theory for Anosov subgroups of higher rank semisimple Lie groups.Comment: 15 pages, 2 figures, New title and introductio

Rigidity of Kleinian groups via self-joinings

Let $\Gamma<\text{PSL}_2(\mathbb{C})\simeq \text{Isom}^+(\mathbb{H}^3)$ be a finitely generated non-Fuchsian Kleinian group whose ordinary set $\Omega=\mathbb{S}^2-\Lambda$ has at least two components. Let $\rho : \Gamma \to \text{PSL}_2(\mathbb{C})$ be a faithful discrete non-Fuchsian representation with boundary map $f:\Lambda\to \mathbb{S}^2$ on the limit set. In this paper, we obtain a new rigidity theorem: if $f$ maps every circular slice of $\Lambda$ into a circle, then $\rho$ is a conjugation by some $g\in \text{M\"ob}(\mathbb{S}^2)$ and $f=g|_\Lambda$. Moreover, unless $\rho$ is a conjugation, the set of circles $C\subset \mathbb{S}^2$ such that $f(C\cap \Lambda)$ is contained in a circle has empty interior in the space of all circles meeting $\Lambda$. This answers a question asked by McMullen on the rigidity of maps $\Lambda\to \mathbb{S}^2$ sending vertices of every tetrahedron of zero-volume to vertices of a tetrahedron of zero-volume. The novelty of our proof is a new viewpoint of relating the rigidity of $\Gamma$ with the higher rank dynamics of the self-joining $(\text{id} \times \rho)(\Gamma)<\text{PSL}_2(\mathbb{C})\times \text{PSL}_2(\mathbb{C})$

Ergodic dichotomy for subspace flows in higher rank

In this paper, we obtain an ergodic dichotomy for {\it directional} flows, more generally, subspace flows, for a class of discrete subgroups of a connected semisimple real algebraic group $G$, called transverse subgroups. The class of transverse subgroups of $G$ includes all discrete subgroups of rank one Lie groups, Anosov subgroups and their relative versions. Let $\Gamma$ be a Zariski dense $\theta$-transverse subgroup for a subset $\theta$ of simple roots. Let $L_\theta=A_\theta S_\theta$ be the Levi subgroup associated with $\theta$ where $A_\theta$ is the central maximal real split torus and $S_\theta$ is the product of a semisimple subgroup and a compact torus. There is a canonical $\Gamma$-invariant subspace $\tilde \Omega_{\theta}$ of $G/S_\theta$ on which $\Gamma$ acts properly discontinuously. Setting $\Omega_\theta=\Gamma \backslash \tilde \Omega_\theta$, we consider the subspace flow given by $A_W=\exp W$ for any linear subspace $W< \frak a_\theta$. Our main theorem is a Hopf-Tsuji-Sullivan type dichotomy for the ergodicity of $(\Omega_\theta, A_W, \mathsf m)$ with respect to a Bowen-Margulis-Sullivan measure $\mathsf m$ satisfying a certain hypothesis. As an application, we obtain the codimension dichotomy for a $\theta$-Anosov subgroup $\Gamma <G$: for any subspace $W<\mathfrak{a}_\theta$ containing a vector $u$ in the interior of the $\theta$-limit cone of $\Gamma$, we have $\operatorname{codim} W \le 2$ if and only if the $A_W$-action on $(\Omega_\theta, \mathsf{m}_u)$ is ergodic where $\mathsf{m}_u$ is the Bowen-Margulis-Sullivan measure associated with $u$.Comment: 48 pages, 1 figur