The Index and Spectrum of Lie Poset Algebras of Types B, C, and D

We define posets of types B, C, and D. These posets encode the matrix forms of certain Lie algebras which lie between the algebras of upper-triangular and diagonal matrices. Our primary concern is the index and spectral theories of such type-B, C, and D Lie poset algebras. For an important restricted class, we develop combinatorial index formulas and, in particular, characterize posets corresponding to Frobenius Lie algebras. In this latter case we show that the spectrum is binary; that is, consists of an equal number of 0's and 1's. Interestingly, type-B, C, and D Lie poset algebras can be related to Reiner's notion of a parset.

Attempting—and Failing—to Balance Fairness and Efficiency in the Arbitral System

The Supreme Court has actively expanded the Federal Arbitration Act into realms not originally contemplated by Congress. This harms consumers who are parties to pre-dispute, binding arbitration agreements. If consumers sign a contract containing an arbitration agreement, they may be required to arbitrate everything within the agreement’s scope, including their statutory rights. Simultaneously, the Court has restricted class action arbitration—a device on which consumers have relied when they are forced to arbitrate. The Court’s expansion of arbitration and restriction of class action arbitration has led many to distrust and advocate for changing the arbitral system. Arbitration institutions have directly reacted to the concerns about arbitration by promulgating more rules, procedures, and safeguards to make arbitration fairer for consumers. However, adding rules and procedures is probably not enough to make arbitration proceedings truly fair, and doing so creates a system that is so court-like that arbitration loses its chief benefits—affordability and efficiency. Thus, if the Court continues with its expansive arbitration jurisprudence and its anti-class action arbitration jurisprudence, institutional reaction is an unlikely solution to address arbitration’s fairness concerns.

Equilibrium Inefficiency and Computation in Cost-Sharing Games in Real-Time Scheduling Systems

We study cost-sharing games in real-time scheduling systems where the server’s activation cost in every time slot is a function of its load. We focus on monomial cost functions and consider both the case when the degree is less than one (inducing positive congestion effect for the jobs) and when it is greater than one (inducing negative congestion effect for the jobs). For the former case, we provide tight bounds on the price of anarchy, and show that the price of anarchy grows to infinity as a polynomial of the number of jobs in the game. For the latter, we observe that existing results provide constant and tight (asymptotically in the degree of the monomial) bounds on the price of anarchy. We then turn to analyze payment mechanism with arbitrary cost-sharing, that is, when the strategy of a player includes also its payment. We show that our mechanism reduces the price of anarchy of games with n jobs and unit server costs from Θ(n) to 2. We also show that, for a restricted class of instances, a similar improvement is achieved for monomial server costs. This is not the case, however, for unrestricted instances of monomial costs, for which we prove that the price of anarchy remains super-constant for our mechanism. For systems with load-independent activation costs, we show that our mechanism can produce an optimal solution which is stable against coordinated deviations.

Unifying Decidable Entailments in Separation Logic with Inductive Definitions

The entailment problem $$\upvarphi \models \uppsi $$ φ ⊧ ψ in Separation Logic [12, 15], between separated conjunctions of equational ($$x \approx y$$ x ≈ y and $$x \not \approx y$$ x ≉ y ), spatial ($$x \mapsto (y_1,\ldots ,y_\upkappa )$$ x ↦ ( y 1 , … , y κ ) ) and predicate ($$p(x_1,\ldots ,x_n)$$ p ( x 1 , … , x n ) ) atoms, interpreted by a finite set of inductive rules, is undecidable in general. Certain restrictions on the set of inductive definitions lead to decidable classes of entailment problems. Currently, there are two such decidable classes, based on two restrictions, called establishment [10, 13, 14] and restrictedness [8], respectively. Both classes are shown to be in $$\mathsf {2\text {EXPTIME}}$$ 2 EXPTIME by the independent proofs from [14] and [8], respectively, and a many-one reduction of established to restricted entailment problems has been given [8]. In this paper, we strictly generalize the restricted class, by distinguishing the conditions that apply only to the left- ($$\upvarphi $$ φ ) and the right- ($$\uppsi $$ ψ ) hand side of entailments, respectively. We provide a many-one reduction of this generalized class, called safe, to the established class. Together with the reduction of established to restricted entailment problems, this new reduction closes the loop and shows that the three classes of entailment problems (respectively established, restricted and safe) form a single, unified, $$\mathsf {2\text {EXPTIME}}$$ 2 EXPTIME -complete class.

A kapitalista társadalom struktúrája

In this paper, I discuss the social structure of modern capitalist society in a new conception based on the theory of institutional sociology. In the first part of the paper, I briefly outline the social structure of modern capitalist society. Taking social relations into account in terms of certain types of social capital and social relationships, I differentiate the following social classes in the modern capitalist society: (1) authority class, (2) strong tolerated class, (3) supported class, (4) medial tolerated class, (5) patronized class, (6) restricted class, (7) less weak tolerated class, (8) less exposed class, (9) very weak tolerated class and (10) very exposed class. In the second part of the paper, I analyse the social structure or the social classes composing the social structure in more detail. In the third part, I point out further aspects that ought to be considered in the empirical research of the social structure of capitalist society

History of the Albanian present suffix ‑i/‑ën from Proto-Indo-European to the modern dialects

In standard Albanian, there is a restricted class of three intransitive verbs (hip-, ec-, ik-) that shows a suffix ‑i/‑ën in the conjugation of the present singular. In dialects and in Old Albanian, however, this suffix is more prolific. This paper studies the history of this suffix and its function by comparing its use in a wide range of Albanian dialects. Based on the dialectal data a Proto- Albanian inchoative/imperfective function inherited from Proto-Indo-European can be reconstructed for the suffix. This function is still present in the Arbëresh dialect of San Costantino Albanese. As the suffix lost its semantic component, it became a marker for the imperfect in Tosk and southern Geg, while in northern Geg the suffix is used as a present marker in verbs where ambiguity with the imperfect could arise. The verbs hip-, ec-, and ik- were originally transitive verbs which were intransitivized by the suffix ‑i/‑ën. As their original transitive meaning was lost, the suffix was obligatory in these verbs in Proto-Albanian already, a situation that is continued in almost all present dialects.

Lower bounds on approximating some variations of vertex coloring problem over restricted graph classes

A [Formula: see text] vertex coloring of a graph [Formula: see text] partitions the vertex set into [Formula: see text] color classes (or independent sets). In minimum vertex coloring problem, the aim is to minimize the number of colors used in a given graph. Here, we consider three variations of vertex coloring problem in which (i) each vertex in [Formula: see text] dominates a color class, (ii) each color class is dominated by a vertex and (iii) each vertex is dominating a color class and each color class is dominated by a vertex. These minimization problems are known as Min-Dominator-Coloring, Min-CD-Coloring and Min-Domination-Coloring, respectively. In this paper, we present approximation hardness results for these problems for some restricted class of graphs.

Quantum Relative Entropy of Tagging and Thermodynamics

Thermodynamics establishes a relation between the work that can be obtained in a transformation of a physical system and its relative entropy with respect to the equilibrium state. It also describes how the bits of an informational reservoir can be traded for work using Heat Engines. Therefore, an indirect relation between the relative entropy and the informational bits is implied. From a different perspective, we define procedures to store information about the state of a physical system into a sequence of tagging qubits. Our labeling operations provide reversible ways of trading the relative entropy gained from the observation of a physical system for adequately initialized qubits, which are used to hold that information. After taking into account all the qubits involved, we reproduce the relations mentioned above between relative entropies of physical systems and the bits of information reservoirs. Some of them hold only under a restricted class of coding bases. The reason for it is that quantum states do not necessarily commute. However, we prove that it is always possible to find a basis (equivalent to the total angular momentum one) for which Thermodynamics and our labeling system yield the same relation.