Norm Of Vector Research Articles

Introducing and solving the hesitant fuzzy system AX = B

Abstract In this paper the solution for hesitant fuzzy system as AX = B is introduced where A is an n × n known hesitant fuzzy matrix, B is an n × 1 known hesitant fuzzy vector and X is an n × 1 unknown hesitant fuzzy vector. First, L ∞-norm and L 1-norm of a hesitant fuzzy vector are introduced. Then, the concepts of hesitant fuzzy zero, ’almost equal’ and ’less than’ and ’equal’ are defined for two hesitant fuzzy numbers. Finally, using a minimization problem; the hesitant fuzzy system is solved. At the end, some numerical examples are presented to show the effectiveness of the proposed method.

Distributed robust fault‐tolerant consensus tracking control for multi‐agent systems with exogenous disturbances under switching topologies

AbstractIn this article, the consensus tracking problem for multi‐agent systems (MASs) under switching communication topologies is studied. In the objective system, the unknown nonlinear function, exogenous disturbances, and actuator faults are considered. For this problem, a new fully distributed fault‐tolerant control method is developed via utilizing adaptive mechanisms. A sufficient condition is obtained to guarantee the stability of the considered closed‐loop system by topology‐based average dwell time technique combined with Lyapunov stability theory. Unlike the traditional method to estimate the weight matrix, the neural network is used to estimate the norm of weight vectors. Thus, the difficulty that the unknown nonlinear function cannot be compensated when outage or stuck faults occur in some actuators is solved. Compared with the existing controllers for nonlinear MASs, the proposed controller does not require any knowledge about the terms relevant to the global information associated with the network. Furthermore, the proposed controller can compensate for the leader's unknown bounded input and actuator faults. Besides, the closed‐loop system is proven to be stable by the developed controller, and all followers can track the state of the leader. Finally, a numerical simulation is given to validate the effectiveness of the proposed protocol.

Unifying framework for information processing in stochastically driven dynamical systems

A dynamical system can be regarded as an information processing apparatus that encodes input streams from the external environment to its state and processes them through state transitions. The information processing capacity (IPC) is an excellent tool that comprehensively evaluates these processed inputs, providing details of unknown information processing in black box systems; however, this measure can be applied to only time-invariant systems. This paper extends the applicable range to time-variant systems and further reveals that the IPC is equivalent to coefficients of polynomial chaos (PC) expansion in more general dynamical systems. To achieve this objective, we tackle three issues. First, we establish a connection between the IPC for time-invariant systems and PC expansion, which is a type of polynomial expansion using orthogonal functions of input history as bases. We prove that the IPC corresponds to the squared norm of the coefficient vector of the basis in the PC expansion. Second, we show that an input following an arbitrary distribution can be used for the IPC, removing previous restrictions to specific input distributions. Third, we extend the conventional orthogonal bases to functions of both time and input history and propose the IPC for time-variant systems. To show the significance of our approach, we demonstrate that our measure can reveal information representations in not only machine learning networks but also a real, cultured neural network. Our generalized measure paves the way for unveiling the information processing capabilities of a wide variety of physical dynamics which has been left behind in nature.

On a discrete game problem with non-convex control vectograms

In a normed space of finite dimension, a discrete game problem with fixed duration is considered. The terminal set is determined by the condition that the norm of the phase vector belongs to a segment with positive ends. In this paper, a set defined by this condition is called a ring. At each moment, the vectogram of the first player's controls is a certain ring. The controls of the second player at each moment are taken from balls with given radii. The goal of the first player is to lead a phase vector to the terminal set at a fixed time. The goal of the second player is the opposite. In this paper, necessary and sufficient termination conditions are found, and optimal controls of the players are constructed.

A parametric level set method for the optimization of composite structures with curvilinear fibers

This paper presents a parametric level set method for the optimization of composite structures with curvilinear fibers. We use iso-contours of a scalar function, also called the level set function, to represent the fibers, and the level set function is constructed by a set of radial basis functions with compact support. The orientation of fiber at any arbitrary point is defined by the orientation of the tangent vector of iso-contour. In order to prevent gaps and overlaps between neighboring fiber tows from arising during the manufacturing of composite structures, a constraint is formulated that the norm of gradient vector of the level set function is equal to 1 almost everywhere. Furthermore, these gradient constraints are aggregated into a single one by using the p-norm method to improve the efficiency of optimization. The compliance minimization problem is considered, and the coefficients of radial basis functions are taken as the design variables. The results of optimizations proved that the proposed method can provide an optimized composite structure with equally spaced curvilinear fibers.

Prior Assumptions for Leak Localisation in Water Distribution Networks with Uncertainties

Hydraulic model-based leak (burst) localisation in water distribution networks is a challenging problem due to a limited number of hydraulic measurements, a wide range of leak properties, and model and data uncertainties. In this study, prior assumptions are investigated to improve the leak localisation in the presence of uncertainties. For example, ell _2-regularisation relies on the assumption that the Euclidean norm of the leak coefficient vector should be minimised. This approach is compared with a method based on the sensitivity matrix, which assumes the existence of only a single leak. The results show that while the sensitivity matrix method often yields a better leak location estimate in single leak scenarios, the ell _2-regularisation successfully identifies a search area for pinpointing the accurate leak location. Furthermore, it is shown that the additional error introduced by a quadratic approximation of the Hazen-Williams formula for the solution of the localisation problem is negligible given the uncertainties in Hazen-Williams resistance coefficients in operational water network models.

Team Composition Revisited: A Team Member Attribute Alignment Approach

Research methods for studying team composition tend to employ either a variable-centered or person-centered approach. The variable-centered approach allows scholars to consider how patterns of attributes between team members influence teams, while the person-centered approach allows scholars to consider how variation in multiple attributes within team members influences subgroup formation and its effects. Team composition theory, however, is becoming increasingly sophisticated, assuming variation on multiple attributes both within and between team members—for example, in predicting how a team functions differently when its most assertive members are also optimistic rather than pessimistic. To support this new theory, we propose an attribute alignment approach, which complements the variable-centered and person-centered approaches by modeling teams as matrices of their members and their members’ attributes. We first demonstrate how to calculate attribute alignment by determining the vector norm and vector angle between team members’ attributes. Then, we demonstrate how the alignment of team member personality attributes (neuroticism and agreeableness) affects team relationship conflict. Finally, we discuss the potential of using the attribute alignment approach to enrich broader team research.

When does ‖f(A)‖ = f(‖A‖) hold true?

Let a strongly stable norm on the set of complex n-by-n matrices be given, which means that for all and all . Furthermore, let be a power series with nonnegative coefficients and radius of convergence R>0. If , we additionally suppose that . We aim to characterize those A with , which fulfil . We first show how to reduce the discussion of f to Neumann series. For matrix norms induced by uniformly convex vector norms, like the -norms, , it follows from known results of the Daugavet equation that holds true if, and only if, is an eigenvalue of A, provided that for some . Under adapted assumptions on the we prove that this equivalence remains true for the - and the -norm, for unitarily invariant matrix norms and for the numerical radius. We conjecture this equivalence to be valid for all strongly stable norms if for all .

OPTIMIZATION OF TIMETABLE AT THE UNIVERSITY

Context. In this paper, we consider a well-known university scheduling problem. Such tasks are solved several times a year in every educational institution. The problem of constructing an optimal schedule remains open despite numerous studies in this area. This is due to the complexity of the corresponding optimization problem, in particular, its significant dimension. This complicates its numerical solution with existing optimization methods. Scheduling models also need improvement. Thus, schedule optimization is a complex computational problem and requires the development of new methods for solving it. 
 Objective. Improvement of optimization models for timetabling at the university and the use of new effective methods to solve them.
 Method. We use the exact quadratic regularization method to solve timetabling optimization problems. Exact quadratic regularization allows transforming complex optimization models with Boolean variables into the problem of maximizing the vector norm on a convex set. We use the efficient direct dual interior point method and dichotomy method to solve this problem. This method has shown significantly better results in solving many complex multimodal problems. This is confirmed by many comparative computational experiments. The exact quadratic regularization method is even more effective in solving timetabling problems. This optimization method is used for the first time for this class of problems, so it required the development of adequate algorithmic support.
 Results. We propose a new, simpler timetabling optimization model that can be easily implemented software in Excel with the OpenSolver, RoskSolver, and others. We give a small example of building a schedule and describe step-by-step instructions for obtaining the optimal solution. 
 Conclusions. An efficient new technology developed for university timetable, which is simple to implement and does not require the development of special software. The efficiency of the technology is ensured by the use of a new method of exact quadratic regularization.

Effect of Incidence Angle on the Spatial-Average of Incident Power Density Definition to Correlate Skin Temperature Rise for Millimeter Wave Exposures

This article reports on an intercomparison study on the effect of the incidence angle on the spatial average of incident power density (PD) and resultant temperature rise using computational and thermographic measurement approaches. Two definitions of the spatial average of incident PD—the peak spatial-average normal component of the Poynting vector and peak spatial-average norm of the Poynting vector—were compared. First, an intercomparison of incident PD and temperature rise in a layered skin model was conducted for a 4 × 4 dipole array antenna. The variations caused by antenna type, antenna-body distance, and skin model to these definitions were then discussed. The results revealed that both definitions are in good agreement and correlate with the peak temperature rise for small or moderate incidence angles. The heating factor was enhanced for transverse-magnetic-like polarized waves for peak spatial-average normal component of the Poynting vector for large-angle incidences because of the Brewster effect. The normal incidence scenario was confirmed to be essential for considering the peak skin temperature rise.

On a Vector-Valued Measure of Multivariate Skewness

The canonical skewness vector is an analytically simple function of the third-order, standardized moments of a random vector. Statistical applications of this skewness measure include semiparametric modeling, independent component analysis, model-based clustering, and multivariate normality testing. This paper investigates some properties of the canonical skewness vector with respect to representations, transformations, and norm. In particular, the paper shows its connections with tensor contraction, scalar measures of multivariate kurtosis and Mardia’s skewness, the best-known scalar measure of multivariate skewness. A simulation study empirically compares the powers of tests for multivariate normality based on the squared norm of the canonical skewness vector and on Mardia’s skewness. An example with financial data illustrates the statistical applications of the canonical skewness vector.

Fuzzy fault-tolerant containment control for multi-agent systems with unknown nonlinear dynamics

This paper studies the containment control problem for a class of nonlinear multi-agent systems (MASs) with actuator faults (AFs) and external disturbance under switching communication topologies. To address this problem, a new fuzzy fault-tolerant containment control method is developed via utilizing adaptive mechanisms. Furthermore, a sufficient condition is obtained to guarantee the stability of the considered closed-loop system by the dwell time technique combined with Lyapunov stability theory. Unlike the traditional method to estimate the weight matrix, the fuzzy logic system is used to estimate the norm of weight vectors. Thus, the difficulty that the unknown nonlinear function cannot be compensated for when the actuator produces outage or stuck fault is solved. Compared with the existing controllers for nonlinear MASs, the proposed controller is more suitable for the considered problem under the influence of AFs that are detrimental to the operation of each agent system. Besides which, the closed-loop system is proven to be stable by using the developed controller, and all followers converge asymptotically to the convex hull formed by the leaders. Finally, an example based on a reduced-order aircraft model is presented to verify the effectiveness of the designed control scheme.

A sparse approach for high-dimensional data with heavy-tailed noise

High-dimensional data have commonly emerged in diverse fields, such as economics, finance, genetics, medicine, machine learning, and so on. In this paper, we consider the sparse quantile regression problem of high-dimensional data with heavy-tailed noise, especially when the number of regressors is much larger than the sample size. We bring the spirit of L p -norm support vector regression into quantile regression and propose a robust L p -norm support vector quantile regression for high-dimensional data with heavy-tailed noise. The proposed method achieves robustness against heavy-tailed noise due to its use of the pinball loss function. Furthermore, L p -norm support vector quantile regression ensures that the most representative variables are selected automatically by using a sparse parameter. We use a simulation study to test the variable selection performance of L p -norm support vector quantile regression, where the number of explanatory variables greatly exceeds the sample size. The simulation study confirms that L p -norm support vector quantile regression is not only robust against heavy-tailed noise but also selects representative variables. We further apply the proposed method to solve the variable selection problem of index construction, which also confirms the robustness and sparseness of L p -norm support vector quantile regression.

Additive logistic processes in option pricing

In option pricing, it is customary to first specify a stochastic underlying model and then extract valuation equations from it. However, it is possible to reverse this paradigm: starting from an arbitrage-free option valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper, we start from two simple arbitrage-free valuation equations, inspired by the log-sum-exponential function and an ell ^{p} vector norm. Such expressions lead respectively to logistic and Dagum (or “log-skew-logistic”) risk-neutral distributions for the underlying security price. We proceed to exhibit supporting martingale processes of additive type for underlying securities having as time marginals two such distributions. By construction, these processes produce closed-form valuation equations which are even simpler than those of the Bachelier and Samuelson–Black–Scholes models. Additive logistic processes provide parsimonious and simple option pricing models capturing various important stylised facts at the minimum price of a single market observable input.

Estimation of frequency and phasor using enhanced nonlinear least error squares method for synchrophasor application

Nonlinear least error squares (LES) method reported in the literature, has been used for frequency estimation of power signal having harmonic contents. However, anticipation of achieving high accuracy and increased range of frequency estimate with existing solution technique leads to computational burden on FPGA processor. In this paper, a novel algorithm is proposed to enhance the accuracy of nonlinear LES technique without imposing any computational burden for the requirement of synchrophasor application. This is accomplished by finding the minimum value of squared Euclidian norm of error vector obtained by nonlinear LES method only at the selected frequency intervals. The behavior of algorithm with respect to window length of signal is analyzed through simulation. Also the proposed algorithm is employed for measurement of phasors. The performance of the proposed algorithm is verified and demonstrated for the different tests mentioned in synchrophasor standard IEEE C37.118.1 and its amendment IEEE C37.118.1a-2014. The enhanced nonlinear LES is implemented on NI cRIO 9082 using Xilinx Spartan-6 LX150 FPGA platform for real-time measurement.

Linear-size universal discretization of geometric center-based problems in fixed dimensions

Many geometric optimization problems can be reduced to finding points in space (centers) minimizing an objective function which continuously depends on the distances from the centers to given input points. Examples are $k$-Means, Geometric $k$-Median/Center, Continuous Facility Location, $m$-Variance, etc. We prove that, for any fixed $\varepsilon>0$, every set of $n$ input points in fixed-dimensional space with the metric induced by any vector norm admits a set of $O(n)$ candidate centers which can be computed in almost linear time and which contains a $(1+\varepsilon)$-approximation of each point of space with respect to the distances to all the input points. It gives a universal approximation-preserving reduction of geometric center-based problems with arbitrary continuity-type objective functions to their discrete versions where the centers are selected from a fairly small set of candidates. The existence of such a linear-size set of candidates is also shown for any metric space of fixed doubling dimension.

A new method for global optimization

This paper presents a new method for global optimization. We use exact quadratic regularization for the transformation of the multimodal problems to a problem of a maximum norm vector on a convex set. Quadratic regularization often allows you to convert a multimodal problem into a unimodal problem. For this, we use the shift of the feasible region along the bisector of the positive orthant. We use only local search (primal-dual interior point method) and a dichotomy method for search of a global extremum in the multimodal problems. The comparative numerical experiments have shown that this method is very efficient and promising.

Adaptive neural network control for nonstrict-feedback uncertain nonlinear systems with input delay and asymmetric time-varying state constraints

This paper is devoted to adaptive neural network control issue for a class of nonstrict-feedback uncertain systems with input delay and asymmetric time-varying state constraints. State-related external disturbances are involved into the system, and the upper bounds of disturbances are assumed as functions of state variables instead of constants. Additionally, during the approximations of unknown functions by neural networks, the online computation burdens are declined sharply, since the norms of neural network weight vectors are only estimated. In the process of dealing with input delay, an auxiliary function is applied such that the conditions for time delay are more general than the ones in existing literature. A novel adaptive neural network controller is designed by constructing the asymmetric barrier Lyapunov function, which guarantees that the output of system has a good tracking performance and the state variables never violate the asymmetric time-varying constraints. Finally, numerical simulations are presented to verify the proposed adaptive control scheme.

Reconfigurable Slip Vectoring Control in Four In-Wheel Drive Electric Vehicles

Controllability, maneuverability, fault-tolerance/isolation and safety are significantly enhanced in electric vehicles (EV) equipped with the redundant actuator configuration of four-in-wheel electric motors (4IWM). A highly reconfigurable architecture is proposed and illustrated for the adaptive, nonmodel-based control of 4IWM-EVs. Given the longitudinal force, yaw-moment requests and the reconfiguration matrix, each IWM is given a slip reference according to a Slip Vectoring (SV) allocation strategy, which minimizes the overall slip vector norm. The distributed electric propulsion and the slip vector reference allow for a decentralized online estimation of the four-wheel torque-loads, which are uncertain depending on loading and road conditions. This allows for the allocation of four different torques depending on individual wheel conditions and to determine in which region (linear/nonsaturated or nonlinear/saturated) of the torque/slip characteristics each wheel is operating. Consequently, the 4IWMs can be equalized or reconfigured, including actuator fault-isolation as a special case, so that they are enforced to operate within the linear tire region. The initial driving-mode selection can be automatically adjusted and restored among eighteen configurations to meet the safety requirements of linear torque/slip behavior. Three CarSim realistic simulations illustrate the equalization algorithm, the quick fault-isolation capabilities and the importance of a continuous differential action in a critical double-lane-change maneuver.

Low-shot transfer with attention for highly imbalanced cursive character recognition

Recognition of ancient Korean–Chinese cursive character (Hanja) is a challenging problem mainly because of large number of classes, damaged cursive characters, various hand-writing styles, and similar confusable characters. They also suffer from lack of training data and class imbalance issues. To address these problems, we propose a unified Regularized Low-shot Attention Transfer with Imbalance τ-Normalizing (RELATIN) framework. This handles the problem with instance-poor classes using a novel low-shot regularizer that encourages the norm of the weight vectors for classes with few samples to be aligned to those of many-shot classes. To overcome the class imbalance problem, we incorporate a decoupled classifier to rectify the decision boundaries via classifier weight-scaling into the proposed low-shot regularizer framework. To address the limited training data issue, the proposed framework performs Jensen–Shannon divergence based data augmentation and incorporate an attention module that aligns the most attentive features of the pretrained network to a target network. We verify the proposed RELATIN framework using highly-imbalanced ancient cursive handwritten character datasets. The results suggest that (i) the extreme class imbalance has a detrimental effect on classification performance; (ii) the proposed low-shot regularizer aligns the norm of the classifier in favor of classes with few samples; (iii) weight-scaling of decoupled classifier for addressing class imbalance appeared to be dominant in all the other baseline conditions; (iv) further addition of the attention module attempts to select more representative features maps from base pretrained model; (v) the proposed (RELATIN) framework results in superior representations to address extreme class imbalance issue.