Notes and Comments: Stock of a Closely Held Corporation in Decedent\u27s Estate—Post-Mortem Considerations

When pre-death estate planning is absent or ineffective, the executor of an estate consisting primarily of stock of a closely held corporation faces a problem in deriving the liquidity necessary to pay estate taxes. The problem is aggravated if the beneficiaries wish to maintain the current balance of control in the closely held corporation. As a practical guide to executors in this situation, the author discusses the provisions of the Internal Revenue Code which can help mitigate the problem

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that holds for all $n$ node graphs (as a function of $f$, $k$ and $n$)? This question was first studied in the context of geometric graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more recently been considered in general undirected graphs [Chechik et al. STOC '09, Dinitz and Krauthgamer PODC '11]. In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor $k$ is fixed. Specifically, we prove that every (undirected, possibly weighted) $n$-node graph $G$ has a $(2k-1)$-spanner resilient to $f$ vertex faults with $O_k(f^{1 - 1/k} n^{1 + 1/k})$ edges, and this is fully optimal (unless the famous Erdos Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating $dist_{G \setminus F}(s, t)$ similarly can beat the space usage of our spanner in the worst case. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case $k=2$ (and hence we close the EFT problem for $3$-approximations), but it falls to $\Omega(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k})$ for $k \ge 3$. We leave it as an open problem to close this gap.Comment: To appear in SODA 201

Optimal indicators of socioeconomic status for health research

Objectives: This paper examines the relationship between various measures of SES and mortality for a representative sample of individuals. ; Methods: Data are from the Panel Study of Income Dynamics. Sample includes 3,734 individuals aged 45 and above who participated in the 1984 interview. Mortality was tracked between 1984 and 1994 and is related to SES indicators using Cox event-history regression models. ; Results: Wealth has the strongest associations with subsequent mortality, and these associations differ little by age and sex. Other economic measures, especially family-size-adjusted household income, have significant associations with mortality, particularly for nonelderly women. ; Conclusions: By and large, the economic components of SES have associations with mortality that are at least as strong as, and often stronger than, more conventional components (e.g., completed schooling, occupation).Income distribution

The greedy basis equals the theta basis

We prove the equality of two canonical bases of a rank 2 cluster algebra, the greedy basis of Lee-Li-Zelevinsky and the theta basis of Gross-Hacking-Keel-Kontsevich.Comment: 17 page

Preserving Distances in Very Faulty Graphs

Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively. Since real-world networks are prone to failures, it makes sense to study fault-tolerant versions of the above structures. This turns out to be a surprisingly difficult task. For every small but arbitrary set of edge or vertex failures, the preservers and spanners need to contain replacement paths around the faulted set. Unfortunately, the complexity of the interaction between replacement paths blows up significantly, even from 1 to 2 faults, and the structure of optimal preservers and spanners is poorly understood. In particular, no nontrivial bounds for preservers and additive spanners are known when the number of faults is bigger than 2. Even the answer to the following innocent question is completely unknown: what is the worst-case size of a preserver for a single pair of nodes in the presence of f edge faults? There are no super-linear lower bounds, nor subquadratic upper bounds for f>2. In this paper we make substantial progress on this and other fundamental questions: - We present the first truly sub-quadratic size fault-tolerant single-pair preserver in unweighted (possibly directed) graphs: for any n node graph and any fixed number f of faults, O~(fn^{2-1/2^f}) size suffices. Our result also generalizes to the single-source (all targets) case, and can be used to build new fault-tolerant additive spanners (for all pairs). - The size of the above single-pair preserver grows to O(n^2) for increasing f. We show that this is necessary even in undirected unweighted graphs, and even if you allow for a small additive error: If you aim at size O(n^{2-eps}) for eps>0, then the additive error has to be Omega(eps f). This surprisingly matches known upper bounds in the literature. - For weighted graphs, we provide matching upper and lower bounds for the single pair case. Namely, the size of the preserver is Theta(n^2) for f > 1 in both directed and undirected graphs, while for f=1 the size is Theta(n) in undirected graphs. For directed graphs, we have a superlinear upper bound and a matching lower bound. Most of our lower bounds extend to the distance oracle setting, where rather than a subgraph we ask for any compact data structure

A simple method to identify kinases that regulate embryonic stem cell pluripotency by high-throughput inhibitor screening

Embryonic stem cells (ESCs) can self-renew or differentiate into all cell types, a phenomenon known as pluripotency. Distinct pluripotent states have been described, termed "naïve" and "primed" pluripotency. The mechanisms that control naïve-primed transition are poorly understood. In particular, we remain poorly informed about protein kinases that specify naïve and primed pluripotent states, despite increasing availability of high-quality tool compounds to probe kinase function. Here, we describe a scalable platform to perform targeted small molecule screens for kinase regulators of the naïve-primed pluripotent transition in mouse ESCs. This approach utilizes simple cell culture conditions and standard reagents, materials and equipment to uncover and validate kinase inhibitors with hitherto unappreciated effects on pluripotency. We discuss potential applications for this technology, including screening of other small molecule collections such as increasingly sophisticated kinase inhibitors and emerging libraries of epigenetic tool compounds