Super-Accuracy Calculation for the Half Width of a Voigt Profile

A simple approximation scheme to describe the half width of the Voigt profile as a function of the relative contributions of Gaussian and Lorentzian broadening is presented. The proposed approximation scheme is highly accurate and provides an accuracy better than 10−17 for arbitrary αL/αG ratios. In particular, the accuracy reaches an astonishing 10−34 (quadruple precision) in the domain 0 ≤ αL/αG ≤ 0.2371 ∪ αL/αG ≥ 33.8786.

Rothe–Legendre pseudospectral method for a semilinear pseudoparabolic equation with nonclassical boundary condition

A semilinear pseudoparabolic equation with nonlocal integral boundary conditions is studied in the present paper. Using Rothe method, which is based on backward Euler finitedifference schema, we designed a suitable semidiscretization in time to approximate the original problem by a sequence of standard elliptic problems. The questions of convergence of the approximation scheme as well as the existence and uniqueness of the solution are investigated. Moreover, the Legendre pseudospectral method is employed to discretize the time-discrete approximation scheme in the space direction. The main advantage of the proposed approach lies in the fact that the full-discretization schema leads to a symmetric linear algebraic system, which may be useful for theoretical and practical reasons. Finally, numerical experiments are included to illustrate the effectiveness and robustness of the presented algorithm.

An Entropic Approach to Classical Density Functional Theory

The classical Density Functional Theory (DFT) is introduced as an application of entropic inference for inhomogeneous fluids in thermal equilibrium. It is shown that entropic inference reproduces the variational principle of DFT when information about the expected density of particles is imposed. This process introduces a family of trial density-parametrized probability distributions and, consequently, a trial entropy from which the preferred one is found using the method of Maximum Entropy (MaxEnt). As an application, the DFT model for slowly varying density is provided, and its approximation scheme is discussed.

Maximizing the Number of Visible Labels on a Rotating Map

For a map that can be rotated, we consider the following problem. There are a number of feature points on the map, each having a geometric object as a label. The goal is to find the largest subset of these labels such that when the map is rotated and the labels remain vertical, no two labels in the subset intersect. We show that, even if the labels are vertical bars of zero width, this problem remains NP-hard, and present a polynomial approximation scheme for solving it. We also introduce a new variant of the problem for vertical labels of zero width, in which any label that does not appear in the output must be coalesced with a label that does. Coalescing a subset of the labels means to choose a representative among them and set its label height to the sum of the individual label heights.

Omnichannel Assortment Optimization Under the Multinomial Logit Model with a Features Tree

Problem definition: We consider the assortment optimization problem of a retailer that operates a physical store and an online store. The products that can be offered are described by their features. Customers purchase among the products that are offered in their preferred store. However, customers who purchase from the online store can first test out products offered in the physical store. These customers revise their preferences for online products based on the features that are shared with the in-store products. The full assortment is offered online, and the goal is to select an assortment for the physical store to maximize the retailer’s total expected revenue. Academic/practical relevance: The physical store’s assortment affects preferences for online products. Unlike traditional assortment optimization, the physical store’s assortment influences revenue from both stores. Methodology: We introduce a features tree to organize products by features. The nonleaf vertices on the tree correspond to features, and the leaf vertices correspond to products. The ancestors of a leaf correspond to features of the product. Customers choose among the products within their store’s assortment according to the multinomial logit model. We consider two settings; either all customers purchase online after viewing products in the physical store, or we have a mix of customers purchasing from each store. Results: When all customers purchase online, we give an efficient algorithm to find the optimal assortment to display in the physical store. With a mix of customers, the problem becomes NP-hard, and we give a fully polynomial-time approximation scheme. We numerically demonstrate that we can closely approximate the case where products have arbitrary combinations of features without a tree structure and that our fully polynomial-time approximation scheme performs remarkably well. Managerial implications: We characterize conditions under which it is optimal to display expensive products with underrated features and expose inexpensive products with overrated features.

The Complexity of Counting Problems Over Incomplete Databases

We study the complexity of various fundamental counting problems that arise in the context of incomplete databases, i.e., relational databases that can contain unknown values in the form of labeled nulls. Specifically, we assume that the domains of these unknown values are finite and, for a Boolean query q , we consider the following two problems: Given as input an incomplete database D , (a) return the number of completions of D that satisfy q ; or (b) return the number of valuations of the nulls of D yielding a completion that satisfies q . We obtain dichotomies between #P-hardness and polynomial-time computability for these problems when q is a self-join–free conjunctive query and study the impact on the complexity of the following two restrictions: (1) every null occurs at most once in D (what is called Codd tables ); and (2) the domain of each null is the same. Roughly speaking, we show that counting completions is much harder than counting valuations: For instance, while the latter is always in #P, we prove that the former is not in #P under some widely believed theoretical complexity assumption. Moreover, we find that both (1) and (2) can reduce the complexity of our problems. We also study the approximability of these problems and show that, while counting valuations always has a fully polynomial-time randomized approximation scheme (FPRAS), in most cases counting completions does not. Finally, we consider more expressive query languages and situate our problems with respect to known complexity classes.