On the degree of distortions under second-degree price discrimination

This paper highlights the limitations and applicability of results developed by Chao & Nahata (2015) for nonlinear pricing. Although Chao and Nahata appear to provide necessary and sufficient conditions for general utility functions, we show that one of their results leads only to a restatement of two constraints, and another result may not be valid when consumers can freely dispose of the good. Their model allows for the possibility that higher quantities will have a lower price than smaller quantities. We provide conditions under free disposal that preclude this anomaly. Our analysis suggests that further research on violations of the single-crossing condition should be encouraged.

Small one-dimensional Euclidean preference profiles

We characterize one-dimensional Euclidean preference profiles with a small number of alternatives and voters. We show that every single-peaked preference profile with two voters is one-dimensional Euclidean, and that every preference profile with up to five alternatives is one-dimensional Euclidean if and only if it is both single-peaked and single-crossing. By the work of Chen et al. (Social Choice and Welfare 48(2):409–432, 2017), we thus obtain that the smallest single-peaked and single-crossing preference profiles that are not one-dimensional Euclidean consist of three voters and six alternatives.

IMPROVEMENT OF BROWN CARPATHIAN CATTLE BREED BY THE METHOD OF "SINGLE CROSSING" OF LEBEDYN BREED

The Lebedyn breed of the combined direction of productivity was used for improvement of economically useful signs, especially milk productivity of cows of Brown Carpathian breed. Both breeds belong to the dual-purpose direction of productivity, they are adapted to the natural and climatic zones of breeding and "belong" to a related group of brown breeds. The largest related group of the Brown Carpathian breed are the descendants of the bull Rupora 6507, to which the "single crossing" of the Lebedyn breed of the dual-purpose direction of productivity was carried out. In addition, the following servicing bulls of the Lebedyn breed were used to improve the economically useful characteristics of the Brown Carpathian breed: Tuman 779 Shafran 2012, Henii 958, Kokos 923 (related group of Elbrus 1871) Zhdanyi 035, Limonad 2188, (line of Narzan 937) Landysh 2012 (related group of Rolik 113) Minus 1353 (line of Fordzon-Mylyi 290). The Lebedyn breed has played an important role in strengthening the productive characteristics of Brown Carpathian breed and expanding its genealogical structure.

Chiral smoothings of knots

Can smoothing a single crossing in a diagram for a knot convert it into a diagram of the knot's mirror image? Zeković found such a smoothing for the torus knot T(2, 5), and Moore–Vazquez proved that such smoothings do not exist for other torus knots T(2, m) with m odd and square free. The existence of such a smoothing implies that K # K bounds a Mobius band in B4. We use Casson–Gordon theory to provide new obstructions to the existence of such chiral smoothings. In particular, we remove the constraint that m be square free in the Moore–Vazquez theorem, with the exception of m = 9, which remains an open case. Heegaard Floer theory provides further obstructions; these do not give new information in the case of torus knots of the form T(2, m), but they do provide strong constraints for other families of torus knots. A more general question asks, for each pair of knots K and J, what is the minimum number of smoothings that are required to convert a diagram of K into one for J. The methods presented here can be applied to provide lower bounds on this number.

Stable roommates with narcissistic, single-peaked, and single-crossing preferences

The classical Stable Roommates problem is to decide whether there exists a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their respectively assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates that are, compared to those in the literature, more efficient when the input preferences are complete and have some structural property, such as being narcissistic, single-peaked, and single-crossing. However, when the preferences are incomplete and have ties, we show that being single-peaked and single-crossing does not reduce the computational complexity—Stable Roommates remains NP-complete.

Pareto efficient income taxation without single-crossing

We provide a full characterization of a two-type optimal nonlinear income tax model where the single-crossing condition is violated due to an assumption that agents differ both in terms of market abilities and in terms of their needs for a work-related good. We set up a Pareto-efficient tax problem and analyze the entire second-best Pareto-frontier, highlighting several non-standard results, such as the possibility of income re-ranking relative to the laissez-faire and gaps in the Pareto-frontier.

Genus One Cobordisms Between Torus Knots

We determine the pairs of torus knots that have a genus one cobordism between them, with one notable exception. This is done by combining obstructions using $\nu ^+$ from the Heegaard Floer knot complex and explicit constructions of cobordisms. As an application, we determine the pairs of torus knots related by a single crossing change. Also, we determine the pairs of Thurston–Bennequin number maximizing Legendrian torus knots that have a genus one exact Lagrangian cobordism, with one exception.