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A principal wants to procure a discrete number of homogeneous units from a finite set of agents. Each agent has an increasing and convex cost function, whose exact shape is unknown to the principal. Utility is quasilinear in money. We characterize the set of mechanisms that are strategy-proof and robust to collusion. The latter is captured by reallocation-proofness (group strategy-proofness) if the colluding agents can (cannot) make side deals among each other. We show that in every reallocation-proof mechanism, the principal offers the agents a fixed price per unit. While group-strategy-proof mechanisms can be more complex, they cannot guarantee that the principal will receive exactly her desired quantity. We fully characterize the set of group-strategy-proof mechanisms with a uniform price.

In 2002 Mauduit and Sarkozy started to study finite sequences of \begin{document}$ k $\end{document} symbols \begin{document}$ E_{N} = \left(e_{1},e_{2},\cdots,e_{N}\right)\in \mathcal{A}^{N}, $\end{document} where \begin{document}$ \mathcal{A} = \left\{a_{1},a_{2},\cdots,a_{k}\right\}(k\in \mathbb{N},k\geq 2) $\end{document} is a finite set of \begin{document}$ k $\end{document} symbols. Later many pseudorandom sequences of \begin{document}$ k $\end{document} symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the \begin{document}$ k $\end{document} -ary Sidel'nikov sequences with length \begin{document}$ q-1 $\end{document} by using the estimates for certain character sums with exponential function, where \begin{document}$ q $\end{document} is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size \begin{document}$ \phi(q-1) $\end{document} of \begin{document}$ k $\end{document} -ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that \begin{document}$ k = o(q^{\frac{1}{4}}) $\end{document} , where \begin{document}$ \phi $\end{document} denotes Euler's totient function.

A guidance law parameter identification model based on Gated Recurrent Unit (GRU) neural network is established. The scenario of the model is that an incoming missile (called missile) attacks a target aircraft (called aircraft) using Proportional Mavigation (PN) guidance law. The parameter identification is viewed as a regression problem in this paper rather than a classification problem, which means the assumption that the parameter is in a finite set of possible results is discarded. To increase the training speed of the neural network and obtain the nonlinear mapping relationship between kinematic information and the guidance law parameter of the incoming missile, an output processing method called Multiple-Model Mechanism (MMM) is proposed. Compared with a conventional GRU neural network, the model established in this paper can deal with data of any length through an encoding layer in front of the input layer. The effectiveness of the proposed Multiple-Model Mechanism and the performance of the guidance law parameter identification model are demonstrated using numerical simulation.

We define and analyze the notion of variational Wardrop equilibrium for nonatomic aggregative games with an infinity of players types. These equilibria are characterized through an infinite-dimensional variational inequality. We show, under monotonicity conditions, a convergence theorem enables to approximate such an equilibrium with arbitrary precision. To this end, we introduce a sequence of nonatomic games with a finite number of players types, which approximates the initial game. We show the existence of a symmetric Wardrop equilibrium in each of these games. We prove that those symmetric equilibria converge to an equilibrium of the infinite game, and that they can be computed as solutions of finite-dimensional variational inequalities. The model is illustrated through an example from smart grids: the description of a large population of electricity consumers by a parametric distribution gives a nonatomic game with an infinity of different players types, with actions subject to coupling constraints., arXiv admin note: substantial text overlap with arXiv:1806.06230

In this article, we study the optimal control problem arising from the mean-field game formulation of the collective decision-making in honeybee swarms. A population of homogeneous players (the honeybees) has to reach consensus on one of two options. We consider three states: the first two represent the available options (or strategies), and the third one represents the uncommitted state. We formulate the continuous-time discrete-state mean-field game model. The contributions of this article are the following: 1) we propose an optimal control model where players have to control their transition rates to minimize a running cost and a terminal cost, in the presence of an adversarial disturbance; 2) we develop a formulation of the micro–macro model in the form of an initial-terminal value problem with switched dynamics; 3) we study the existence of stationary solutions and the mean-field Nash equilibrium for the resulting switched system; 4) we show that under certain assumptions on the parameters, the game may admit periodic solutions; and 5) we analyze the resulting microscopic dynamics in a structured environment where a finite number of players interact through a network topology.

This article proposes a new discrete collision handling method (DCH method) that can be used for resolving tanglements in clothing simulation, based on the existing continuous collision handling methods (CCH methods). The proposed method performs intersection analysis of the clothing mesh at every time step, and stores the result in the form of coloring the vertices, edges, and triangles. Referring to the coloring, the method resolves the tanglements in the out-to-in manner by applying the proposed operations, the triangle shrinkage and vertex pull, to the triangles and vertices around the intersection path. We take note of the CCH methods that use some small tolerance value to defend the round-off errors for the purpose of preventing false negatives. This work gives a second thought to that tolerance value, and proposes a new DCH method which uses the tolerance value for the resolution purpose. Under certain conditions, the method turns out to guarantee resolution of the tanglements in a finite number of time steps.

When a set of self-interested users shares multiple resources in a distributed system, we face the problem of allocating resources, called the load balancing problem. In particular, load balancing is defined as allocating the load to the servers of the distributed system such that jobs' response time is minimized, and the utilization of servers is improved. In this paper, the load balancing problem in a distributed system consists of a finite set of servers, and a finite set of users is studied. The load balancing problem considered here is a bi-objective problem with two highly probable conflicting objectives: (i) minimizing jobs' response time (ii) providing the fair utilization of servers. In order to satisfy these two objectives simultaneously, both the objectives are considered in an integrated manner. Next, the load balancing problem is formulated as a noncooperative game; and to solve the game (i.e., to find the Nash equilibrium), a distributed load balancing algorithm (DLBA) is proposed. An experimental study is carried out to ascertain the efficacy of the proposed DLBA. Further, we compare DBLA with three existing load balancing approaches to evaluate its comparative effectiveness. The experimental results validate the effectiveness of the DLBA over the existing approaches.

We study bargaining models in discrete time with a finite number of players, stochastic selection of the proposing player, endogenously determined sets and orders of responders, and a finite set of feasible alternatives. The standard optimality conditions and system of recursive equations may not be sufficient for the existence of a subgame perfect equilibrium in stationary strategies (SSPE) in case of costless delay. We present a characterization of SSPE that is valid for both costly and costless delay. We address the relationship between an SSPE under costless delay and the limit of SSPEs under vanishing costly delay. An SSPE always exists when delay is costly, but not necessarily so under costless delay, even when mixed strategies are allowed for. This is surprising as a quasi SSPE, a solution to the optimality conditions and the system of recursive equations, always exists. The problem is caused by the potential singularity of the system of recursive equations, which is intimately related to the possibility of perpetual disagreement in the bargaining process.

Diurnal fluctuations in volatility are a well-documented stylized fact of intraday price data. This warrants an investigation how this intraday periodicity (IP) affects both finite sample as well as asymptotic properties of several popular realized estimators of daily integrated volatility which are based on functionals of a finite number of intraday returns. It turns out that most of the estimators considered in this study exhibit a finite-sample bias due to IP, which can however get negligible when the number of intraday returns diverges to infinity. The appropriate correction factors for this bias are derived based on estimates of the IP. The adequacy of the new corrections is evaluated by means of a Monte Carlo simulation study and an empirical example.

This paper proposes the problem of joint state estimation and mode recognition for nonlinear stochastic systems with unknown sensor mode. The considered sensor mode is represented by a random finite set, whose elements can be one specific mode or a set of certain modes. A set-valued mode recognition-based Bayesian estimation framework is proposed to propagate the posterior density of the state conditioned on sensor modes and measurements, where the mode is recognized based on the maximum correntropy criterion. Furthermore, a mode-separability metric is proposed to discern the reliability of mode recognition, and utilized to derive two distinct implementation schemes, including state estimation based on separable and inseparable modes. Simulation results of fault detection and target tracking are provided to demonstrate the superiority of the proposed method in terms of state estimation accuracy and mode recognition effectiveness.

In this study, first, a comprehensive model is introduced to model actuator faults, and then a novel fault-tolerant control (FTC) strategy is proposed to compensate the loss of actuator's effectiveness in networked control systems (NCSs). A Markov chain is exploited to represent networked-induced random delays, and data packet dropouts as well as disorders to address the stochastic characteristic of the network issues. Accordingly, the resulting closed-loop system lies in the framework of Markovian jump systems (MJSs). Moreover, partly unknown transition probabilities are considered in the current study since the identification of the exact value of transition probabilities of the Markov chain is difficult or even impractical due to the complex structure of the network. Sufficient conditions for the stochastic stability are derived by means of the solutions of a finite set of linear matrix inequalities (LMIs) to design a novel robust FTC through the output feedback technique, which requires only the outputs. A numerical example and an engineering benchmark system are presented to verify the capability of the proposed method in practical applications.

Let $\mathbf{H}$ be the cartesian product of a family of finite abelian groups indexed by a finite set $\Omega$. A given poset (i.e., partially ordered set) $\mathbf{P}=(\Omega,\preccurlyeq_{\mathbf{P}})$ gives rise to a poset metric on $\mathbf{H}$, which further leads to a partition $\mathcal{Q}(\mathbf{H},\mathbf{P})$ of $\mathbf{H}$. We prove that if $\mathcal{Q}(\mathbf{H},\mathbf{P})$ is Fourier-reflexive, then its dual partition $\Lambda$ coincides with the partition of $\hat{\mathbf{H}}$ induced by $\mathbf{\overline{P}}$, the dual poset of $\mathbf{P}$, and moreover, $\mathbf{P}$ is necessarily hierarchical. This result establishes a conjecture proposed by Gluesing-Luerssen in \cite{4}. We also show that with some other assumptions, $\Lambda$ is finer than the partition of $\hat{\mathbf{H}}$ induced by $\mathbf{\overline{P}}$. In addition, we give some necessary and sufficient conditions for $\mathbf{P}$ to be hierarchical, and for the case that $\mathbf{P}$ is hierarchical, we give an explicit criterion for determining whether two codewords in $\hat{\mathbf{H}}$ belong to the same block of $\Lambda$. We prove these results by relating the involved partitions with certain family of polynomials, a generalized version of which is also proposed and studied to generalize the aforementioned results.

Given a finite number of stochastic systems, the goal of our problem is to dynamically allocate a finite sampling budget to maximize the probability of selecting the “best” system. Systems are encoded with the probability distributions that govern sample observations, which are unknown and only assumed to belong to a broad family of distributions that need not admit any parametric representation. The best system is defined as the one with the highest quantile value. The objective of maximizing the probability of selecting this best system is not analytically tractable. In lieu of that, we use the rate function for the probability of error relying on large deviations theory. Our point of departure is an algorithm that naively combines sequential estimation and myopic optimization. This algorithm is shown to be asymptotically optimal; however, it exhibits poor finite-time performance and does not lead itself to implementation in settings with a large number of systems. To address this, we propose practically implementable variants that retain the asymptotic performance of the former while dramatically improving its finite-time performance.

The purpose of the present paper is to give an effective version of the noncritical $p$-tame Belyi theorem. That is to say, we compute explicitly an upper bound of the minimal degree of tamely ramified Belyi maps in positive characteristic which are unramified at a prescribed finite set of points., Comment: 16 pages

Voronoi constellations (VCs) are finite sets of vectors of a coding lattice enclosed by the translated Voronoi region of a shaping lattice, which is a sublattice of the coding lattice. In conventional VCs, the shaping lattice is a scaled-up version of the coding lattice. In this paper, we design low-complexity VCs with a cubic coding lattice of up to 32 dimensions, in which pseudo-Gray labeling is applied to minimize the bit error rate. The designed VCs have considerable shaping gains of up to 1.03 dB and finer choices of spectral efficiencies in practice. A mutual information estimation method and a log-likelihood approximation method based on importance sampling for very large constellations are proposed and applied to the designed VCs. With error-control coding, the proposed VCs can have higher achievable information rates than the conventional scaled VCs because of their inherently good pseudo-Gray labeling feature, with a lower decoding complexity., Comment: 10 pages, 6 figures

The plants of nano air vehicles (NAVs) are generally unstable, adversely coupled, and uncertain. Besides, the autopilot hardware of a NAV has limited sensing and computational capabilities. Hence, these vehicles need a single controller referred to as robust simultaneously stabilizing decoupling (RSSD) output feedback controller that achieves simultaneous stabilization (SS), desired decoupling, robustness, and performance for a finite set of unstable multi-input-multioutput adversely coupled uncertain plants. To synthesize an RSSD output feedback controller, a new method that is based on a central plant is proposed in this article. Given a finite set of plants for SS, we considered a plant in this set that has the smallest maximum v-gap metric as the central plant. Following this, the sufficient condition for the existence of a simultaneous stabilizing controller associated with such a plant is described. The decoupling feature is then appended to this controller using the properties of the eigenstructure assignment method. Afterward, the sufficient conditions for the existence of an RSSD output feedback controller are obtained. Using these sufficient conditions, a new optimization problem for the synthesis of an RSSD output feedback controller is formulated. To solve this optimization problem, a new genetic algorithm-based offline iterative algorithm is developed. The effectiveness of this iterative algorithm is then demonstrated by generating an RSSD controller for a fixed-wing NAV. The performance of this controller is validated through numerical and hardware-in-the-loop simulations.

This paper addresses the joint calibration problem of SPX options and VIX options or futures. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints, in the spirit of [Guo, Loeper, and Wang, Math. Finance, 32 (2021)]. We introduce a PDE formulation along with its dual counterpart. The solution, a calibrated diffusion process, can be represented via the solutions of Hamilton--Jacobi--Bellman equations arising from the dual formulation. The method is tested on both simulated data and market data. Numerical examples show that the model can be accurately calibrated to SPX options, VIX options, and VIX futures simultaneously.

Markov switching (MS-)GARCH(1,1) models allow for structural changes in volatility dynamics between a finite number of regimes. Since the regimes are not observed, computation of the likelihood requires integrating over an exponentially increasing number of regime paths, which is intractable. An existing smooth likelihood estimation procedure for sequential Monte Carlo (SMC), that is currently limited to hidden Markov models with a one-dimensional state variable, is modified to enable likelihood estimation and maximisation for MS-GARCH(1,1) models, a model which requires two dimensions, volatility and regime, to evolve its hidden state process. Furthermore, the modified SMC procedure is shown to be easily adapted to fitting MS-GARCH(1,1) models even when there are missing observations. The proposed methodology is validated with simulated data and is also illustrated with analysis of two financial time series, the daily returns on the S&P 500 index and on the Henry Hub natural gas spot price, with the latter series containing a gap caused by shutdown in response to hurricane Rita in 2005.

In this letter, we propose a Koopman operator based approach to describe the nonlinear dynamics of a quadrotor on $SE(3)$ in terms of an infinite-dimensional linear system which evolves in the space of observables (lifted space) and which is more appropriate for control design purposes. The major challenge when using the Koopman operator is the characterization of a set of observables that can span the lifted space. Most of the existing methods either start from a set of dictionary functions and then search for a subset that best fits the underlying nonlinear dynamics or they rely on machine learning algorithms to learn these observables. Instead of guessing or learning the observables, in this work we derive them in a systematic way for the quadrotor dynamics on $SE(3)$ . In addition, we prove that the proposed sequence of observables converges pointwise to the zero function, which allows us to select only a finite set of observables to form (an approximation of) the lifted space. Our theoretical analysis is also confirmed by numerical simulations which demonstrate that by increasing the dimension of the lifted space, the derived linear state space model can approximate the nonlinear quadrotor dynamics more accurately.

In this letter, we address the design of a finite-dimensional control law based on sampled-data for a class of linear continuous-time fractional-order systems. Using the proposed finite-dimensional controller which only requires a finite number of previous control inputs to compute the current control signal, input-to-state stability (ISS) is established for the corresponding closed-loop systems by considering the hereditary and infinite memory properties of the fractional-order systems. A sufficient condition for determining the control signal storage space is also provided to guarantee certain control performance. Numerical simulations illustrate the effectiveness of the proposed design methods and validate the theoretical results.

The paper describes a novel method of sampled-data in space (spatial variable) nonlinear control of scalar semilinear parabolic and hyperbolic systems with unknown parameters, distributed disturbances and finite number of measurements along the spatial variable. Differently from recent results based on piecewise constant control laws, the proposed one is used piecewise nonlinear functions choosing by designer for providing some properties in the closed-loop system. In particular, we propose several types of functions providing reduced control. The gain design in the control law is found as a solution of linear matrix inequalities with minimum ultimate bound guarantee. The simulations confirm theoretical results and show the efficiency of the proposed control scheme compared with some existing ones.

We consider the problem of tracking for a discrete-time plant with unknown plant parameters; we assume knowledge of an upper bound on the plant order, and for each admissible order we assume knowledge of a compact set in which the plant parameters lie. We carry out parameter estimation of an associated auxiliary model; indeed, for each admissible dimension, we cover the set of admissible parameters by a finite number of compact and convex sets and use an original-projection-algorithm-based estimator for each set. At each point in time, we employ a switching algorithm to determine which model and parameter estimates are used in the pole-placement-based control law. We prove that this adaptive controller guarantees desirable linear-like closed-loop behavior: exponential stability, a bounded noise gain in every p-norm, a convolution bound on the effect of the exogenous inputs, as well as exponential tracking for certain classes of reference and noise signals; this linear-like behavior is leveraged to immediately show tolerance to a degree of plant time-variations and unmodelled dynamics.

We consider the output-feedback state consensus problem for a homogeneous multi-agent system consisting of one leader agent and $N$ follower agents. The dynamics of each of these agents is governed by a single-input single-output regular linear system (RLS), with the input to the leader agent being zero at all times. The transfer function of this RLS has all its poles in the closed left-half of the complex-plane, with only a finite number of them lying on the imaginary axis. Each follower agent can access the relative output of all its neighboring agents and some of the follower agents can access the relative output of the leader. The communication graph associated with the exchange of relative outputs is directed. Under this setting, we first establish that a controller solves the above leader-follower consensus problem if and only if it can simultaneously stabilize $N$ regular linear systems. We then adapt a recently developed frequency-domain technique to construct a stable finite-dimensional output-feedback controller which solves the simultaneous stabilization problem, and hence the consensus problem. We demonstrate the efficacy of our controller design technique using a numerical example.

Formal language theory is defined as computational theory of languages. It provides theoretical aspects of computer science and helps to determine infinite languages in finite number of ways. So, algorithms can be constructed for relevant problems. Then again, decide either a string is existed in language or not under the language. A formal language is an abstraction of the common features of programming languages. Furthermore, a formal language comprises a set of symbols and then some rules of creation by way of these symbols may be combined into entities which is called as sentences. Additionally, a formal language is the grouping the strings accepted by the rules of construction. Even though, many of the formal languages easier than programming languages that they have lot of similar essential properties. It is a great deal to study about programming languages from formal languages. Mathematical structure of languages and its basic properties are the natural languages. It is an integrative area of science.