Wild division algebras over Laurent series fields

In this paper we study some special classes of division algebras over a Laurent series field with arbitrary residue field. We call the algebras from these classes as splittable and good splittable division algebras. It is shown that these classes contain the group of tame division algebras. For the class of good division algebras a decomposition theorem is given. This theorem is a generalization of the decomposition theorems for tame division algebras given by Jacob and Wadsworth. For both clases we introduce a notion of a $\delta$-map and develop a technique of $\delta$-maps for division algebras from these classes. Using this technique we reprove several old well known results of Saltman and get the positive answer on the period-index conjecture of M.Artin: the exponent of $A$ is equal to its index for any division algebra $A$ over a $C_2$-field $F$, when F\eq F_1((t_2)), where $F_1$ is a $C_1$-field. The paper includes also some other results about splittable division algebras, which, we hope, will be useful for the further investigation of wild division algebras.Comment: 32 page

Two-person cake-cutting: the optimal number of cuts

A cake is a metaphor for a heterogeneous, divisible good. When two players divide such a good, there is always a perfect division—one that is efficient (Pareto-optimal), envy-free, and equitable—which can be effected with a finite number of cuts under certain mild conditions; this is not always the case when there are more than two players (Brams, Jones, and Klamler, 2011b). We not only establish the existence of such a division but also provide an algorithm for determining where and how many cuts must be made, relating it to an algorithm, “Adjusted Winner” (Brams and Taylor, 1996, 1999), that yields a perfect division of multiple homogenous goods.Cake-cutting; fair division; envy-freeness; adjusted winner; heterogeneous good

N-Person cake-cutting: there may be no perfect division

A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envy-free, and equitable. We give an example of a cake in which it is impossible to divide it among three players such that these three properties are satisfied, however many cuts are made. It turns out that two of the three properties can be satisfied by a 3-cut and a 4-cut division, which raises the question of whether the 3-cut division, which is not efficient, or the 4-cut division, which is not envy-free, is more desirable (a 2-cut division can at best satisfy either envy-freeness or equitability but not both). We prove that no perfect division exists for an extension of the example for three or more players.Cake-cutting; fair division; efficiency; envy-freeness; equitability; heterogeneous good

Preferences over the Fair Division of Goods: Information, Good, and Sample Effects in a Health Context

Greater recognition by economists of the influential role that concern for distributional equity exerts on decision making in a variety of economic contexts has spurred interest in empirical research on the public judgments of fair distribution. Using a stated-preference experimental design, this paper contributes to the growing literature on fair division by investigating the empirical support for each of five distributional principles — equal division among recipients, Rawlsian maximin, total benefit maximization, equal benefit for recipients, and allocation according to relative need among recipients — in the division of a fixed bundle of a good across settings that differ with respect to the good being allocated (a health care good — pills, and non-health care but still health-affecting good — apples) and the way that alternative possible divisions of the good are described (quantitative information only, verbal information only, and both). It also offers new evidence on sample effects (university sample vs. community samples) and how the aggregate ranking of principles is affected by alternative vote-scoring methods. We find important information effects. When presented with quantitative information only, support for the division to equalize benefit across recipients is consistent with that found in previous research; changing to verbal descriptions causes a notable shift in support among principles, especially between equal division of the goods and total benefit maximization. The judgments made when presented with both quantitative and verbal information match more closely those made with quantitative-only descriptions rather than verbal-only descriptions, suggesting that the quantitative information dominates. The information effects we observe are consistent with a lack of understanding among participants as to the relationship between the principles and the associated quantitative allocations. We also find modest good effects in the expected direction: the fair division of pills is tied more closely to benefit-related criterion than is the fair division of apples (even though both produce health benefits). We find evidence of only small differences between the university and community samples and important sex-information interactions.Distributive Justice; Equity; Resource Allocation; Health Care

Preferences over the Fair Division of Goods: Information, Good, and Sample Effects in a Health Context

Greater recognition by economists of the influential role that concern for distributional equity exerts on decision making in a variety of economic contexts has spurred interest in empirical research on the public judgments of fair distribution. Using a stated-preference experimental design, this paper contributes to the growing literature on fair division by investigating the empirical support for each of five distributional principles — equal division among recipients, Rawlsian maximin, total benefit maximization, equal benefit for recipients, and allocation according to relative need among recipients — in the division of a fixed bundle of a good across settings that differ with respect to the good being allocated (a health care good — pills, and non-health care but still health-affecting good — apples) and the way that alternative possible divisions of the good are described (quantitative information only, verbal information only, and both). It also offers new evidence on sample effects (university sample vs. community samples) and how the aggregate ranking of principles is affected by alternative vote-scoring methods. We find important information effects. When presented with quantitative information only, support for the division to equalize benefit across recipients is consistent with that found in previous research; changing to verbal descriptions causes a notable shift in support among principles, especially between equal division of the goods and total benefit maximization. The judgments made when presented with both quantitative and verbal information match more closely those made with quantitative-only descriptions rather than verbal-only descriptions, suggesting that the quantitative information dominates. The information effects we observe are consistent with a lack of understanding among participants as to the relationship between the principles and the associated quantitative allocations. We also find modest good effects in the expected direction: the fair division of pills is tied more closely to benefit-related criterion than is the fair division of apples (even though both produce health benefits). We find evidence of only small differences between the university and community samples and important sex-information interactions.Distributive justice, equity, resource allocation, health care

Cooperation in Dividing the Cake

This paper defines models of cooperation among players partition- ing a completely divisible good (such as a cake or a piece of land). The novelty of our approach lies in the players' ability to form coalitions before the actual division of the good with the aim to maximize the average utility of the coalition. A social welfare function which takes into account coalitions drives the division. In addition, we derive a cooperative game which measures the performance of each coalition. This game is compared with the game in which players start cooper- ating only after the good has been portioned and has been allocated among the players. We show that a modified version of the game played before the division outperforms the game played after the di- vision.fair division;cooperative games;maximin partition

Soft-Information Assisted Near-Optimum Nonlinear Detection for BLAST-type Space Division Multiplexing OFDM Systems

In this contribution, a nonlinear hybrid detection scheme based on a novel soft-information assisted Genetic Algorithm (GA) is proposed for a Turbo Convolutional (TC) coded Space Division Multiplexing (SDM) aided Orthogonal Frequency Division Multiplexing (OFDM) system. Our numerical results show that the performance of the currently known GA-assisted system can be improved by about 2dB with the aid of the GA’s population-based soft solution, approaching the optimum performance of the soft-information assisted Maximum Likelihood (ML) detection, while exhibiting a lower complexity, especially in high-throughput scenarios. Furthermore, the proposed scheme is capable of achieving a good performance even in the so-called overloaded systems, where the number of transmit antennas is higher than the number of receiver antennas. Index Terms—Genetic algorithm, orthogonal frequency division multiplexing, soft information, space division multiplexing

Photobase Generator Enabled Pitch Division: A Progress Report

Pitch division lithography (PDL) with a photobase generator (PBG) allows printing of grating images with twice the pitch of a mask. The proof-of-concept has been published in the previous paper[1, 2] and demonstrated by others[1]. Forty five nm half-pitch (HP) patterns were produced using a 90nm HP mask, but the image had line edge roughness (LER) that does not meet requirements. Efforts have been made to understand and improve the LER in this process. Challenges were summarized toward low LER and good performing pitch division. Simulations and analysis showed the necessity for an optical image that is uniform in the z direction in order for pitch division to be successful. Two-stage PBGs were designed for enhancement of resist chemical contrast. New pitch division resists with polymer-bound PAGs and PBGs, and various PBGs were tested. This paper focuses on analysis of the LER problems and efforts to improve patterning performance in pitch division lithography.Chemical Engineerin