Some remarks on one-sided regularity

A ring is said to be right (resp., left) regular-duo if every right (resp., left) regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of (one-sided) regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.     

Optimal bounds for the no-show paradox via SAT solving

One of the most important desirable properties in social choice theory is Condorcet-consistency, which requires that a voting rule should return an alternative that is preferred to any other alternative by some majority of voters. Another desirable property is participation, which requires that no voter should be worse off by joining an electorate. A seminal result by Moulin (1988) has shown that Condorcet-consistency and participation are incompatible whenever there are at least 4 alternatives and 25 voters. We leverage SAT solving to obtain an elegant human-readable proof of Moulin’s result that requires only 12 voters. Moreover, the SAT solver is able to construct a Condorcet-consistent voting rule that satisfies participation as well as a number of other desirable properties for up to 11 voters, proving the optimality of the above bound. We also obtain tight results for set-valued and probabilistic voting rules, which complement and significantly improve existing theorems.     

Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity

An absolute valued algebra is a non-zero real algebra that is equipped with a multiplicative norm. We classify all finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity, up to algebra isomorphism. This completes earlier results of Ramírez Álvarez and Rochdi which, in our self-contained presentation, are recovered from the wider context of composition k-algebras with an LR-bijective idempotent.     

Hopf powers and orders for some bismash products

The exponent of a finite group G can be viewed as a Hopf algebraic invariant of the group algebra H=kG: it is the least integer n for which the nth Hopf power endomorphism [n] of H is trivial. The exponent of a group scheme G as studied by Gabriel and Tate and Oort can be defined in the same way using the coordinate Hopf algebra H=O(G).The power map and the corresponding notion of exponent have been studied for a general finite-dimensional Hopf algebra beginning with work of Kashina. Several positive results, suggested by analogy to the group case, were proved by Kashina and by Etingof and Gelaki.Given these positive results, there was some hope that the Hopf order of an individual element of a Hopf algebra might also be a well-behaved notion, with some properties analogous to well-known facts on the orders of elements of a finite group.In fact we prove that such analogous properties do hold for Hopf algebras satisfying the usual rule for iterated powers; for example, such a Hopf algebra H has an element of order n if and only if n divides the exponent of H. However, in general such properties are not true. We will give examples where the behavior of Hopf powers, Hopf orders, and related notions is rather strange, unexpected, and seemingly hard to predict. We will see this using computer algebra calculations in Drinfeld doubles of finite groups, and more generally in bismash products constructed from factorizable groups.     

Scheduling products with subassemblies and changeover time

We revisit the problem, previously studied by Coffman et al, of scheduling products with two subassemblies on a common resource, where changeovers consume time, under the objective of flow-time minimization. We derive some previously unidentified structural properties that could be important to researchers working on similar batch scheduling problems. We show that there exists a series of base schedules from which optimal schedules can be easily derived. As these base schedules build on each other, they are easy to construct as well. We also show that the structure of these base schedules is such that batch sizes decrease over time in a well-defined manner. These insights about the general form of the schedules might also be important to practitioners wanting some intuition about the schedule structure that they are implementing.     

Qualitative and infinitesimal robustness of tail-dependent statistical functionals

The main goal of this article is to introduce a new notion of qualitative robustness that applies also to tail-dependent statistical functionals and that allows us to compare statistical functionals in regards to their degree of robustness. By means of new versions of the celebrated Hampel theorem, we show that this degree of robustness can be characterized in terms of certain continuity properties of the statistical functional. The proofs of these results rely on strong uniform Glivenko-Cantelli theorems in fine topologies, which are of independent interest. We also investigate the sensitivity of tail-dependent statistical functionals w.r.t. infinitesimal contaminations, and we introduce a new notion of infinitesimal robustness. The theoretical results are illustrated by means of several examples including general L- and V-functionals.     

Hyperbolicity of the invariant sets for the real polynomial maps

In this paper, the conditions under which there exits a uniformly hyperbolic invariant set for the map f_a(x) = ag(x) are studied, where a is a real parameter, and g(x) is a monic real-coefficient polynomial. It is shown that for certain parameter regions, the map has a uniformly hyperbolic invariant set on which it is topologically conjugate to the one-sided subshift of finite type for A, where ∣a∣ is sufficiently large, A is an eventually positive transition matrix, and g has at least two different real zeros or only one real zero. Further, it is proved that there exists an invariant set on which the map is topologically semiconjugate to the one-sided subshift of finite type for a particular irreducible transition matrix under certain conditions, and one type of these maps is not hyperbolic on the invariant set.     

On ideal and subalgebra coefficients in semigroup algebras

Let k[S] be a semigroup algebra with coefficients in a commutative field k, and let U be a one-sided ideal in k[S] or a k-subalgebra of k[S], It is proven that there exists a smallest subfield k′ ≤ k such that U as a one-sided ideal resp. as a k-algebra can be generated by elements in k′[S]. By means of an example it is shown that the straightforward extension of this result to finitely generated commutative k-algebras is not valid.     

Regular and irregular solutions for a class of elliptic systems in the critical dimension

We study regularity properties of weak solutions in the Sobolev space ${W^{1,n}_0}$ to inhomogeneous elliptic systems under a natural growth condition and on bounded Lipschitz domains in ${\mathbb{R}^n}$ , i. e. we investigate weak solutions in the limiting situation of the Sobolev embedding. Several counterexamples of irregular solutions are constructed in cases, where additional structure conditions might have led to regularity. Among others we present both bounded irregular and unbounded weak solutions to elliptic systems obeying a one-sided condition, and we further construct unbounded extremals of two-dimensional variational problems. These counterexamples do not exclude the existence of a regular solution. In fact, we establish the existence of regular solutions—under standard assumptions on the principal part and the aforementioned one-sided condition on the inhomogeneity. This extends previous works for n = 2 to more general cases, including arbitrary dimensions. Moreover, this result is achieved by a simplified proof invoking modern techniques.     

Multi-Criteria Optimization for Filter Design of L1 Adaptive Control

This paper considers the problem of optimal trade-off between robustness, in terms of time-delay margin, and performance of the $\mathcal {L}_{1}$ adaptive controller. Although the architectures of the $\mathcal {L}_{1}$ adaptive control theory can be systematically tuned to trade-off performance for robustness, there is no practical methodology as of today for the design of the underlying filter toward obtaining the optimal performance. The only available results in this direction were given in the form of linear matrix inequalities (LMIs), which are numerically tractable but tend to yield conservative results similar to what one would obtain by applying methods from robust control theory. This paper presents design schemes that are based on search for the design parameters of the filter. We consider filters of certain structures to satisfy the design specifications and investigate their properties over the space of the design parameters. To handle uncertain variables in the design specifications, greedy randomized algorithms are adopted to analyze and synthesize the system performance and robustness in the presence of uncertainties. In addition, we compute approximate sets of allowable design parameters in both analytical and numerical ways. Illustrative examples are provided to demonstrate that these methods can achieve the desired performance and robustness specifications with reasonable degree of conservatism.     

On Two-sided Locally Convex H*-algebras

We introduce (left, right, two-sided) locally convex H*-algebras, and we give conditions under which an one-sided locally convex H*-algebra turns to be a two-sided one (actually, a locally convex H*-algebra). We also give an example of a proper right locally convex H*-algebra with a (right) involution, which is not a left involution and an example of a proper two-sided locally convex H*-algebra, which is not a locally convex H*-algebra. Moreover, we connect (via an Arens-Michael decomposition) a two-sided locally m-convex H*-algebra with the classical (Banach) two-sided H*-algebras. Further, we present conditions so that the left, right involutions be continuous, and we see when a twosided locally convex H*-algebra is a dual one. Finally, we present some properties of invariant ideals which play an important rôle in structure theory of two-sided locally convex H*-algebras.     

Frame Fundamental Sensor Modeling and Stability of One-Sided Frame Perturbation

We demonstrate that for all linear devices and/or sensors, signal requisition and reconstruction is naturally a mathematical frame expansion and reconstruction issue, whereas the measurement is carried out via a sequence generated by the exact physical response function (PRF) of the device, termed sensory frame {h _n }. The signal reconstruction, on the other hand, will be carried out using the dual frame $\{\tilde{h}^{a}_{n}\}$ of an estimated sensory frame {h _n ^a }. This consequently results in a one-sided perturbation to a frame expansion, which resides in each and every signal and image reconstruction problem. We show that the stability of such a one-sided frame perturbation exits. Examples of image reconstructions in de-blurring are demonstrated.     

On supplementation and generalized projective modules

Discrete (quasi) modules form an important class in module theory, they are studied extensively by many authors. The decomposition theorem for quasidiscrete modules plays an important rule in the better understanding of such modules. In fact, every quasidiscrete module is a direct sum of hollow submodules. Here we introduce some new concepts (weak quasidiscrete, and S ₁- and S ₂-supplemented modules) which generalize the concept of quasidiscrete module. We show that some of the properties of quasidiscrete modules still hold in the class of weak quasidiscrete modules. We also obtain some properties of weak quasidiscrete modules, which are similar to the properties known for quasidiscrete modules. We introduce the concept of generalized relative projectivity (relative S-projectivemodules), and use it to characterize direct sums of hollowmodules. In fact, relative S-projectivity is an essential condition for direct sums of hollow modules to be weak quasidiscrete modules.     

Vector-valued Littlewood-Paley-Stein theory for semigroups

We develop a generalized Littlewood-Paley theory for semigroups acting on L^p-spaces of functions with values in uniformly convex or smooth Banach spaces. We characterize, in the vector-valued setting, the validity of the one-sided inequalities concerning the generalized Littlewood-Paley-Stein g-function associated with a subordinated Poisson symmetric diffusion semigroup by the martingale cotype and type properties of the underlying Banach space. We show that in the case of the usual Poisson semigroup and the Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on Rⁿ, this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calderón-Zygmund singular integral operators.     

Resource allocation games: a compromise stable extension of bankruptcy games

This paper presents an extension of the traditional bankruptcy problem. In a resource allocation problem there is a common-pool resource, which needs to be divided among agents. Each agent is characterized by a claim on this pool and an individual linear monetary reward function for assigned resources. Analyzing these problems a new class of transferable utility games is introduced, called resource allocation games. These games are based on the bankruptcy model, as introduced by O’Neill (Math Soc Sci 2:345–371, 1982). It is shown that the properties of totally balancedness and compromise stability can be extended to resource allocation games, although the property of convexity is not maintained in general. Moreover, an explicit expression for the nucleolus of these games is provided.     

Normalizers of irreducible subfactors

We consider normalizers of an infinite index irreducible inclusion N⊆M of II₁ factors. Unlike the finite index setting, an inclusion uNu^∗⊆N can be strict, forcing us to also investigate the semigroup of one-sided normalizers. We relate these one-sided normalizers of N in M to projections in the basic construction and show that every trace one projection in the relative commutant N^′∩〈M,eN〉 is of the form u^∗eNu for some unitary u∈M with uNu^∗⊆N generalizing the finite index situation considered by Pimsner and Popa. We use this to show that each normalizer of a tensor product of irreducible subfactors is a tensor product of normalizers modulo a unitary. We also examine normalizers of infinite index irreducible subfactors arising from subgroup-group inclusions H⊆G. Here the one-sided normalizers arise from appropriate group elements modulo a unitary from L(H). We are also able to identify the finite trace L(H)-bimodules in ?²(G) as double cosets which are also finite unions of left cosets.     

Avoiding unfairness of Owen allocations in linear production processes

This paper deals with cooperation situations in linear production problems in which a set of goods are to be produced from a set of resources so that a certain benefit function is maximized, assuming that resources not used in the production plan have no value by themselves. The Owen set is a well-known solution rule for the class of linear production processes. Despite their stability properties, Owen allocations might give null payoff to players that are necessary for optimal production plans. This paper shows that, in general, the aforementioned drawback cannot be avoided allowing only allocations within the core of the cooperative game associated to the original linear production process, and therefore a new solution set named EOwen is introduced. For any player whose resources are needed in at least one optimal production plan, the EOwen set contains at least one allocation that assigns a strictly positive payoff to such player.     

One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws

In this paper, the one-sided exact boundary null controllability of entropy solutions is studied for a class of general strictly hyperbolic systems of conservation laws, whose negative (or positive) characteristic families are all linearly degenerate. The authors first prove the well-posedness of semi-global solutions constructed as the limit of ε-approximate front tracking solutions to the mixed initial-boundary value problem with general nonlinear boundary conditions and they establish various properties of both the ε-approximate front tracking solutions and such solutions. By means of essential modifications of the strategy suggested by the first author in [17] originally for the local exact boundary controllability in the framework of classical solutions, the one-sided local exact boundary null controllability of entropy solutions can then be realized via boundary controls acting on one side of the boundary, where the incoming characteristics are all linearly degenerate.     

Self-enforcing coalitions with power accumulation

Agents endowed with power compete for a divisible resource by forming coalitions with other agents. The coalition with the greatest power wins the resource and divides it among its members. The agents’ power increases according to their share of the resource.We study two models of coalition formation where winning agents accumulate power and losing agents may participate in further coalition formation processes. An axiomatic approach is provided by focusing on variations of two main axioms: self-enforcement, which requires that no further deviation happens after a coalition has formed, and rationality, which requires that agents pick the coalition that gives them their highest payoff. For these alternative models, we determine the existence of stable coalitions that are self-enforcing and rational for two traditional sharing rules. The models presented in this paper illustrate how power accumulation, the sharing rule, and whether losing agents participate in future coalition formation processes, shape the way coalitions will be stable throughout time.