On Rational Choice and the Representation of Decision Problems

In economic theory, an agent chooses from available alternatives—modeled as a set. In decisions in the field or in the lab, however, agents do not have access to the set of alternatives at once. Instead, alternatives are represented by the outside world in a structured way. Online search results are lists of items, wine menus are often lists of lists (grouped by type or country), and online shopping often involves filtering items which can be viewed as navigating a tree. Representations constrain how an agent can choose. At the same time, an agent can also leverage representations when choosing, simplifying their choice process. For instance, in the case of a list he or she can use the order in which alternatives are represented to make their choice. In this paper, we model representations and decision procedures operating on them. We show that choice procedures are related to classical choice functions by a canonical mapping. Using this mapping, we can ask whether properties of choice functions can be lifted onto the choice procedures which induce them. We focus on the obvious benchmark: rational choice. We fully characterize choice procedures which can be rationalized by a strict preference relation for general representations including lists, list of lists, trees and others. Our framework can thereby be used as the basis for new tests of rational behavior. Classical choice theory operates on very limited information, typically budgets or menus and final choices. This is in stark contrast to the vast amount of data that specifically web companies collect about their users’ choice process. Our framework offers a way to integrate such data into economic choice models.

Reversed Lempel–Ziv Factorization with Suffix Trees

We present linear-time algorithms computing the reversed Lempel–Ziv factorization [Kolpakov and Kucherov, TCS’09] within the space bounds of two different suffix tree representations. We can adapt these algorithms to compute the longest previous non-overlapping reverse factor table [Crochemore et al., JDA’12] within the same space but pay a multiplicative logarithmic time penalty.

TreeSummarizedExperiment: a S4 class for data with hierarchical structure

Data organized into hierarchical structures (e.g., phylogenies or cell types) arises in several biological fields. It is therefore of interest to have data containers that store the hierarchical structure together with the biological profile data, and provide functions to easily access or manipulate data at different resolutions. Here, we present TreeSummarizedExperiment, a R/S4 class that extends the commonly used SingleCellExperiment class by incorporating tree representations of rows and/or columns (represented by objects of the phylo class). It follows the convention of the SummarizedExperiment class, while providing links between the assays and the nodes of a tree to allow data manipulation at arbitrary levels of the tree. The package is designed to be extensible, allowing new functions on the tree (phylo) to be contributed. As the work is based on the SingleCellExperiment class and the phylo class, both of which are popular classes used in many R packages, it is expected to be able to interact seamlessly with many other tools.

Non-Overlapping LZ77 Factorization and LZ78 Substring Compression Queries with Suffix Trees

We present algorithms computing the non-overlapping Lempel–Ziv-77 factorization and the longest previous non-overlapping factor table within small space in linear or near-linear time with the help of modern suffix tree representations fitting into limited space. With similar techniques, we show how to answer substring compression queries for the Lempel–Ziv-78 factorization with a possible logarithmic multiplicative slowdown depending on the used suffix tree representation.

Some Remarks on the Entanglement Number

Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement.Quanta 2020; 9: 22–36.

Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

TreeSummarizedExperiment: a S4 class for data with hierarchical structure

Data organized into hierarchical structures (e.g., phylogenies or cell types) arises in several biological fields. It is therefore of interest to have data containers that store the hierarchical structure together with the biological profile data, and provide functions to easily access or manipulate data at different resolutions. Here, we present TreeSummarizedExperiment, a R/S4 class that extends the commonly used SingleCellExperiment class by incorporating tree representations of rows and/or columns (represented by objects of the phylo class). It follows the convention of the SummarizedExperiment class, while providing links between the assays and the nodes of a tree to allow data manipulation at arbitrary levels of the tree. The package is designed to be extensible, allowing new functions on the tree (phylo) to be contributed. As the work is based on the SingleCellExperiment class and the phylo class, both of which are popular classes used in many R packages, it is expected to be able to interact seamlessly with many other tools.