Finite Set Of Types Research Articles

Spectral properties of substitutions on compact alphabets

AbstractWe consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows one to construct examples for which the Koopman operator on the associated function space has specific spectral components. For abelian bijective substitutions, we provide a dichotomy result regarding the spectral type of the diffraction. We also provide the first example of a substitution that has countably infinite Lebesgue spectral components and countably infinite singular continuous components. Lastly, we give a non‐constant length substitution on a countably infinite alphabet that gives rise to substitutive Delone sets of infinite type. This extends the spectral theory of substitutions on finite alphabets and Delone sets of finite type with inflation symmetry.

Optimal large population Tullock contests

Abstract We consider Tullock contests where contestants can be divided into a finite set of types according to their strategy cost function. Solving such contests is intractable if the number of players is finite but large and there are nonlinearities and asymmetries present. But by approximating the finite player contest with a large population model that can be solved in closed form, we can approximate equilibrium behavior in the finite player model. We then characterize the optimal bias parameters of the large population contest and interpret them as approximations of optimal bias parameters in finite player contests. We also identify conditions under which those parameters are increasing or decreasing according to the cost parameters. The parameters are biased in favor of high-cost agents if the cost functions are strictly convex and the likelihood of success is sufficiently responsive to strategy.

High-Throughput Bin Packing: Scheduling Jobs With Random Resource Demands in Clusters

We consider a natural scheduling problem which arises in many distributed computing frameworks. Jobs with diverse resource demands (e.g. memory requirements) arrive over time and must be served by a cluster of servers. To improve throughput and delay, the scheduler can pack as many jobs as possible in each server, however the sum of the jobs' resource demands cannot exceed the server's capacity. Motivated by the increasing complexity of workloads in shared clusters, we consider a setting where jobs' resource demands belong to a very large set of diverse types, or in the extreme case even infinitely many types, i.e. resource demands are drawn from a general unknown distribution over a possibly continuous support. The application of classical scheduling approaches that crucially rely on a predefined finite set of types is discouraging in this high (or infinite) type setting. We first characterize a fundamental limit on the maximum throughput in such setting. We then develop oblivious scheduling algorithms, based on Best-Fit and Universal Partitioning, that have low complexity and can achieve at least 1/2 and 2/3 of the maximum throughput respectively, without the knowledge of the resource demand distribution. Extensive simulation results, using both synthetic and real traffic traces, are presented to verify the performance of our algorithms.

Dominant strategy implementation in a large population public goods game

We consider implementation of the efficient state in a large population public goods game. Agents are divided into a finite set of types. The planner asks agents to report types, which generates a reported type distribution. Based on reported types and distribution, the planner calculates the efficient strategy level and a Pigouvian transfer for each type of agent. We show that this direct mechanism satisfies incentive compatibility in strictly dominant strategies, strong budget balance and ex-post individual rationality.

Equivariant Cohomotopy implies orientifold tadpole cancellation

There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory. Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments, lacking a microscopic underpinning in M-brane charge quantization. We provide a framework for answering these questions, crucial not only for mathematical consistency but also for phenomenological accuracy of string theory, by formulating the M-theory C-field on flat M-orientifolds in the generalized cohomology theory called Equivariant Cohomotopy. This builds on our previous results for smooth but curved spacetimes, showing in that setting that charge quantization in twisted Cohomotopy rigorously implies a list of expected anomaly cancellation conditions. Here we further expand this list by proving that brane charge quantization in unstable equivariant Cohomotopy implies the anomaly cancellation conditions for M-branes and D-branes on flat orbi-orientifolds. For this we (a) use an unstable refinement of the equivariant Hopf-tom Dieck theorem to derive local/twisted tadpole cancellation, and in addition (b) the lift to super-differential cohomology to establish global/untwisted tadpole cancellation. Throughout, we use (c) the unstable Pontrjagin–Thom theorem to identify the brane/O-plane configurations encoded in equivariant Cohomotopy and (d) the Boardman homomorphism to equivariant K-theory to identify Chan–Paton representations of D-brane charge. We find that unstable equivariant Cohomotopy, but not its image in equivariant K-theory, distinguishes D-brane charge from the finite set of types of O-plane charges.

On dynamic consistency in ambiguous games

I consider static, incomplete information games where players may not be ambiguity neutral. Every player is one of a finite set of types, and each knows her own type but not that of the other players. Ex ante, players differ only in their taste for outcomes. If every player is dynamically consistent with respect to her own information structure and respects Consequentialism, then players act as if expected utility for uncertainty about types.

Bargaining in dynamic markets

We study non-stationary markets in which traders are randomly matched to bargain over the price of a heterogeneous good or the terms of a partnership. The economy consists of a continuum of players drawn from a finite set of types. Players exogenously enter the market over time and exit upon trading. At every date, matching probabilities for each pair of types are determined by the endogenous distribution of trader types in the market. The balance of bargaining power at any stage depends on variations in potential gains from trade, the inflows of new traders, the structure of agreements at future dates, and the induced frequency of trading opportunities. We establish that an equilibrium always exists. Moreover, all equilibria that lead to the same evolution of the economy are payoff equivalent. However, we show that multiple self-fulfilling expectations about the trajectory of the economy, generating distinct equilibrium dynamics and payoffs, may coexist.

Space-filling curves of self-similar sets (I): iterated function systems with order structures

This paper is the first part of a series which provides a systematic treatment of the space-filling curves of self-similar sets. In the present paper, we introduce a notion of linear graph-directed IFS (linear GIFS in short). We show that to construct a space-filling curve of a self-similar set, it amounts to exploring its linear GIFS structures. Compared to the previous methods, such as the L-system or recurrent set method, the linear GIFS approach is simpler, more rigorous and leads to further studies on this topic. We also propose a new algorithm for the beautiful visualization of space-filling curves.In a series of papers Dai et al (2015 arXiv:1511.05411 [math.GN]), Rao and Zhang (2015) and Rao and Zhang (2015), we investigate for a given self-similar set how to get ‘substitution rules’ for constructing space-filling curves, which was obscure in the literature. We solve the problem for self-similar sets of finite type, which covers most of the known results on constructions of space-filling curves.

Moment Asymptotics for Multitype Branching Random Walks in Random Environment

We study a discrete-time multitype branching random walk on a finite space with finite set of types. Particles move in space according to a Markov chain whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman–Kac formula. We choose Weibull-type distributions with parameter \(1/\rho _{ij}\) for the upper tail of the mean number of \(j\) type particles produced by an \(i\) type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.

Recognition, prediction, and planning for assisted teleoperation of freeform tasks

The approach of inferring user's intended task and optimizing low-level robot motions has promise for making robot teleoperation interfaces more intuitive and responsive. But most existing methods assume a finite set of candidate tasks, which limits a robot's functionality. This paper proposes the notion of freeform tasks that encode an infinite number of possible goals (e.g., desired target positions) within a finite set of types (e.g., reach, orient, pick up). It also presents two technical contributions to help make freeform UIs possible. First, an intent predictor estimates the user's desired task, and accepts freeform tasks that include both discrete types and continuous parameters. Second, a cooperative motion planner continuously updates the robot's trajectories to achieve the inferred tasks by repeatedly solving optimal control problems. The planner is designed to respond interactively to changes in the indicated task, avoid collisions in cluttered environments, handle time-varying objective functions, and achieve high-quality motions using a hybrid of numerical and sampling-based techniques. The system is applied to the problem of controlling a 6D robot manipulator using 2D mouse input in the context of two tasks: static target reaching and dynamic trajectory tracking. Simulations suggest that it enables the robot to reach intended targets faster and to track intended trajectories more closely than comparable techniques.

Isotropic foliations of coadjoint orbits from the Iwasawa decomposition

Let G be a noncompact real semisimple Lie group. The set of regular coadjoint orbits of G can be partitioned according to a finite set of types. We show that on each regular orbit, the Iwasawa decomposition induces a left-invariant foliation which is isotropic with respect to the Kirillov symplectic form. Moreover, the leaves are affine subspaces of the dual of the Lie algebra, and the dimension of the leaves depends only on the type of the orbit. When G is a split real form, the foliations induced from the Iwasawa decomposition are actually Lagrangian fibrations with a global transverse Lagrangian section.

Equilibrium price formation in markets with differentially informed agents

We consider a pure exchange economy with a finite set of types of agents which have incomplete and asymmetric information on the states of nature. Our aim was to describe the equilibrium price formation and analyze how the lack of information may affect the allocation of resources. To do so, we adapt to an asymmetric information scenario a variant of the Shapley–Shubik game introduced by Dubey and Geanakoplos (J Math Econ 39:391–400, 2003 ).

Partitions of large Rado graphs

Let κ be a cardinal which is measurable after generically adding $${\beth_{\kappa+\omega}}$$ many Cohen subsets to κ and let $${\mathcal G= ( \kappa,E )}$$ be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value $${r_m^+}$$ such that the set [κ] m can be partitioned into classes $${\langle{C_i:i<r^{+} _m}\rangle}$$ such that for any coloring of any of the classes C i in fewer than κ colors, there is a copy $${\mathcal{G} ^\ast}$$ of $${\mathcal G}$$ in $${\mathcal G}$$ such that $${[\mathcal{G} ^\ast ] ^m\cap C_i}$$ is monochromatic. It follows that $${\mathcal{G}\rightarrow (\mathcal{G} ) ^m_{<\kappa/r_m^+}}$$ , that is, for any coloring of $${[ \mathcal {G} ] ^m}$$ with fewer than κ colors there is a copy $${\mathcal{G} ^{\prime}}$$ of $${\mathcal{G}}$$ such that $${[\mathcal{G} ^{\prime} ] ^{m}}$$ has at most $${r_m^+}$$ colors. On the other hand, we show that there are colorings of $${\mathcal{G}}$$ such that if $${\mathcal{G} ^{\prime}}$$ is any copy of $${\mathcal{G}}$$ then $${C_i\cap [\mathcal{G} ^{\prime} ] ^m\not=\emptyset}$$ for all $${i<r^{+} _m}$$ , and hence $${\mathcal{G}\nrightarrow [ \mathcal{G} ] ^{m} _{r^{+} _m}}$$ . We characterize $${r_m^+}$$ as the cardinality of a certain finite set of types and obtain an upper and a lower bound on its value. In particular, $${r_2^+=2}$$ and for m > 2 we have $${r_m^+ > r_m}$$ where r m is the corresponding number of types for the countable Rado graph.

Algorithmic analysis of polygonal hybrid systems, part I: Reachability

In this work we are concerned with the formal verification of two-dimensional non-deterministic hybrid systems, namely polygonal differential inclusion systems (SPDIs). SPDIs are a class of non-deterministic systems that correspond to piecewise constant differential inclusions on the plane, for which we study the reachability problem. Our contribution is the development of an algorithm for solving exactly the reachability problem of SPDIs. We extend the geometric approach due to Maler and Pnueli [O. Maler, A. Pnueli. Reachability analysis of planar multi-linear systems. in: C. Courcoubetis (Ed.), CAV’93, in: LNCS, vol. 697, Springer-Verlag, 1993, pp. 194–209] to non-deterministic systems, based on the combination of three techniques: the representation of the two-dimensional continuous-time dynamics as a one-dimensional discrete-time system (using Poincaré maps), the characterization of the set of qualitative behaviors of the latter as a finite set of types of signatures, and acceleration used to explore reachability according to each of these types.

On Equilibrium in Pure Strategies in Games with Many Players

Motivated by issues of imitation, learning and evolution, we introduce a framework of non-cooperative games, allowing both countable sets of pure actions and player types and player types and demonstrate that for all games with sufficiently many players, every mixed strategy Nash equilibrium can be used to construct a Nash-equilibrium in pure strategies that is equivalent. Our framework introduces and exploits a distinction between crowding attributes of players (their external effects on others) and their taste attributes (their payoff functions and any other attributes that are not directly relevant to other players). The set of crowding attributes is assumed to be compact; this is not required, however, for taste attributes. We stress that for studying issues such as conformity, the case of a finite set of types and actions, while illuminating, cannot yield completely satisfactory results. Our main theorems are based on a new mathematical result, in the spirit of the Shapley-Folkman Theorem but applicable to a countable (not necessarily finite dimensional) strategy space.

A'Campo-Gusein-Zade diagrams as partially ordered sets

The real analogues of many results about complex monodromies of singularities can be formulated and proved in terms of partial orderings on A'Campo-Gusein-Zade diagrams, the real versions of Coxeter-Dynkin diagrams of singularities. In this paper it is proved that the only diagrams among the A'Campo-Gusein-Zade diagrams of singularities that determine partially ordered sets of finite type (in the sense of representations of a quiver) are the diagrams of simple singularities. To encode the real decompositions of a singularity the analogue of Vasilev invariants turn out to be surjections of a partially ordered set onto a chain. Formulae are proved for Arnold (mod 2)-invariants of plane curves in terms of the corresponding A'Campo-Gusein-Zade diagrams. We define, in the context of higher Bruhat orders, higher partially ordered sets and we describe their connection with the higher M-Morsifications An. We also consider certain previously known results about real singularities from the point of view of partially ordered sets.

Functors and dyadic sets of finite type

Functors and dyadic sets of finite type

Geometric Models for Quasicrystals II. Local Rules Under Isometries

The atomic structures of quasicrystalline materials exhibit long range order under translations. It is believed that such materials have atomic structures which approximately obey local rules restricting the location of nearby atoms. These local constraints are typically invariant under rotations, and it is of interest to establish conditions under which such local rules can nevertheless enforce order under translations in any structure that satisfies them. A set of local rules $ \cal L $ in $ {\frak R}^n $ is a finite collection of discrete sets {Y i } containing 0, each of which is contained in the ball of radius ρ around 0 in $ {\frak R}^n $ . A set X satisfies the local rules $ \cal L $ under isometries if the ρ -neighborhood of each $ {\bf x} \in X $ is isometric to an element of $ \cal L $ . This paper gives sufficient conditions on a set of local rules $ \cal L $ such that if X satisfies $ \cal L $ under isometries, then X has a weak long-range order under translations, in the sense that X is a Delone set of finite type. A set X is a Delone set of finite type if it is a Delone set whose interpoint distance set X-X is a discrete closed set. We show for each minimal Delone set of finite type X that there exists a set of local rules $ \cal L $ such that X satisfies $ \cal L $ under isometries and all other Y that satisfy $ \cal L $ under isometries are Delone sets of finite type. A set of perfect local rules (under isometries or under translations, respectively) is a set of local rules $ \cal L $ such that all structures X that satisfy $ \cal L $ are in the same local isomorphism class (under isometries or under translations, respectively). If a Delone set of finite type has a set of perfect local rules under translations, then it has a set of perfect local rules under isometries, and conversely.

Geometric Models for Quasicrystals I. Delone Sets of Finite Type

This paper studies three classes of discrete sets X in $\Bbb R$ n which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X-X . A finitely generated Delone set is one such that the abelian group [X-X] generated by X-X is finitely generated, so that [X-X] is a lattice or a quasilattice. For such sets the abelian group [X] is finitely generated, and by choosing a basis of [X] one obtains a homomorphism $\varphi : [X] \rightarrow {\Bbb Z}^s$ . A Delone set of finite type is a Delone set X such that X-X is a discrete closed set. A Meyer set is a Delone set X such that X-X is a Delone set. Delone sets of finite type form a natural class for modeling quasicrystalline structures, because the property of being a Delone set of finite type is determined by ``local rules.'' That is, a Delone set X is of finite type if and only if it has a finite number of neighborhoods of radius 2R , up to translation, where R is the relative denseness constant of X . Delone sets of finite type are also characterized as those finitely generated Delone sets such that the map ϕ satisfies the Lipschitz-type condition ||ϕ (x) - ϕ (x')|| < C ||x - x'|| for x, x' ∈X , where the norms || . . . || are Euclidean norms on $\Bbb R$ s and $\Bbb R$ n , respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map $\tilde{L} : {\Bbb R}^n \rightarrow {\Bbb R}^s$ and a constant C such that ||ϕ (x) - $\tilde{L}$ (x)|| $\leq C$ for all x∈X . Suppose that X is a Delone set with an inflation symmetry, which is a real number η > 1 such that $\eta X \subseteq X$ . If X is a finitely generated Delone set, then η must be an algebraic integer; if X is a Delone set of finite type, then in addition all algebraic conjugates | η ' | $\leq$ η; and if X is a Meyer set, then all algebraic conjugates | η ' | $\leq$ 1.

Hausdorff measure of sets of finite type of one-sided symbolic space

Some estimation formulae and a computation formula for the Hausdorff measure of the sets of finite type of the one-sided symbolic space are given.