Envy-free rights assignments and self-oriented preferences

Very recently a new solution to Sen's “Impossibility of a Paretian liberal” has been suggested where the focus is on the rights assignments per se (Austen-Smith, 1979). It was shown that the concept of fairness, when applied to rights, admits the existence of social decision functions which satisfy Sen's original conditions. Unfortunately this result collapses when individuals have rights over more than one pair of alternatives.In order to obtain possibility results for this more general case the present paper proposes to restrict individuals' preference orderings. It is proved that envy-free collective choice rules exist if individual preferences are self-oriented and if, in addition, people attach primary importance to their own private sphere alternatives. These restrictions are quite severe, but they may be justified if one values the absence of envy in rights allocations very highly.     

Least manipulable Envy-free rules in economies with indivisibilities

We consider envy-free and budget-balanced allocation rules for problems where a number of indivisible objects and a fixed amount of money is allocated among a group of agents. In finite economies, we identify under classical preferences each agent’s maximal gain from manipulation. Using this result we find the envy-free and budget-balanced allocation rules which are least manipulable for each preference profile in terms of any agent’s maximal gain. If preferences are quasi-linear, then we can find an envy-free and budget-balanced allocation rule such that for any problem, the maximal utility gain from manipulation is equalized among all agents.     

Envy-free matchings with one-sided preferences and matroid constraints

In this paper, we consider the problem of checking the existence of an envy-free matching in a many-to-one matching model with one-sided preferences and matroid constraints. For this problem, we propose a polynomial-time algorithm which is a generalization of the algorithm proposed by Gan, Suksompong, and Voudouris for the one-to-one setting. Furthermore, we consider a stronger variant of envy-freeness.     

Covering by discrete and closed discrete sets

Say that a cardinal number κ is small relative to the space X if κ<Δ(X), where Δ(X) is the least cardinality of a non-empty open set in X. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.     

Dynamics of discrete screw dislocations via discrete gradient flow

We present variational approaches (developed in [3,4,11]) to the study of statics and dynamics of screw dislocations in crystals. We model the crystal as a cubic lattice and we give the asymptotic Γ-convergence expansion of the elastic energy induced by a finite family of screw dislocations as the lattice spacing goes to zero. We show that the effective energy associated to the presence of a finite system of screw dislocations coincides with the renormalized energy, studied within the Ginzburg-Landau framework and ruling the interactions between the dislocations. As a byproduct of this analysis, we show the existence of many metastable configurations of dislocations pinned by energy barries. Using the minimizing movement approach á la De Giorgi, we introduce a discrete-in-time variational dynamics, referred to as Discrete Gradient Flow, which allows to overcome these energy barriers. More precisely, we show that lettting first the lattice spacing and then the time step of minimizing movements tend to zero, dislocations move accordingly with the gradient flow of the renormalized energy. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The discrete picard condition for discrete ill-posed problems

We investigate the approximation properties of regularized solutions to discrete ill-posed least squares problems. A necessary condition for obtaining good regularized solutions is that the Fourier coefficients of the right-hand side, when expressed in terms of the generalized SVD associated with the regularization problem, on the average decay to zero faster than the generalized singular values. This is the discrete Picard condition. We illustrate the importance of this condition theoretically as well as experimentally.This work was carried out during a visit to Dept. of Mathematics, UCLA, and was supported by the Danish Natural Science Foundation, by the National Science Foundation under contract NSF-DMS87-14612, and by the Army Research Office under contract No. DAAL03-88-K-0085.     

Dimensions of division rings

Letk be a field. WriteD(G) for the quotient division ring of the group ringkG of a torsion-free, polycyclic-by-finite groupG, andD(g) for the quotient ring of the enveloping algebra of a finite-dimensional Lie algebrag overk. In this note we show that the Hirsch numberh(G) and dim _k g are invariants for the respective division rings, by calculating the Krull and global dimensions ofD(G)? _k D(G) andD(g)? _k D(g).     

Distributive division groupoids

There are given two equivalent methods to construct BANACH lattices of compact operators. All known examples of such lattices are included.     

Instability of discrete point sets]{Instability of discrete point sets

Summary Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>X$ be a discrete subset of Euclidean $d$-space. We allow subsequently continuous movements of single elements, whenever the minimum distance to other elements does not decrease. We discuss the question, if it is possible to move all elements of $X$ in this way, for example after removing a finite subset $Y$ from $X$. Although it is not possible in general, we show the existence of such finite subsets $Y$ for many discrete sets $X$, including all lattices. We define the \textit{instability degree} of $X$ as the minimum cardinality of such a subset $Y$ and show that the maximum instability degree among lattices is attained by perfect lattices. Moreover, we discuss the $3$-dimensional case in detail.     

Graded division algebras

Mathematische Zeitschrift -     

Cyclic division algebras

A general example of cyclic division algebra is given, based on a construction of Brauer, yielding examples of division algebras of arbitrary prime exponent without proper central subalgebras, and also noncrossed products of arbitrary exponent. This research was supported in part by the U.S.-Israel Binational Science Foundation. An erratum to this article is available at .     

Sharp discrete inequalities and applications to discrete variational problems

Some new discrete inequalities involving monotonic or convex functions are obtained. While these are interesting inequalities in their own right, they can be applied to solving certain types of discrete variational problems effectively.     

On division subrings normalized by almost subnormal subgroups in division rings

Periodica Mathematica Hungarica - Let D be a division ring with infinite center, K a proper division subring of D and N an almost subnormal subgroup of the multiplicative group $$D^*$$ of D. The...     

Dynamics of discrete automata

This review deals with the theory and practice of discrete automata, especially with transition processes and hazardous races in such automata.Translated from Itogi Nauki i Tekhniki. Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 14, pp. 81–122, 1977.