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This paper studies finite sample minimax optimal randomization schemes and estimation schemes in estimating parameters including the average treatment effect, when treatment effects are heterogeneous. A randomization scheme is a distribution over a group of permutations of a given treatment assignment vector. An estimation scheme is a joint distribution over assignment vectors, linear estimators, and permutations of assignment vectors. The key element in the minimax problem is that the worst case is over a class of distributions of the data which is invariant to a group of permutations. First, I show that given any assignment vector and any estimator, the uniform distribution over the same group of permutations, namely the complete randomization scheme, is minimax optimal. Second, under further assumptions on the class of distributions and the objective function, I show the minimax optimal estimation scheme involves completely randomizing an assignment vector, while the optimal estimator is the difference-in-means under complete invariance and a weighted average of within-block differences under a block structure, and the number of treated units is determined by the Neyman allocation.

In cascaded H-bridge (CHB) multilevel inverters (MLIs), phase-shifted pulse width modulation (PSPWM) and level-shifted pulse width modulation (LSPWM) are the two prominent techniques to attain multilevel and better performance. But both of them have certain drawbacks like in PSPWM the switching loss of all the semiconductors is high, and in LSPWM the power-sharing between the cascaded modules is not identical. These two drawbacks have been addressed in this article by proposing a novel modulation technique, named the phase opposed disposed phase-shifted PWM (PODPSPWM). In this technique, the level-shifted carriers have been rotated in such a way that there should not be any active power loss and simultaneously the carriers are phase-shifted as per the PSPWM technique. A detailed mathematical analysis of switching loss for all three methods has been presented. Moreover, the power-sharing and harmonic spectrum of the proposed PODPSPWM over other PWMs have been discussed. This novel modulation technique has been simulated in the MATLAB/Simulink software to validate the superiority of the operation. The 1.5-kW experimental prototype is developed to validate the performance of the proposed PODPSPWM modulation. 2022 IEEE. This work was supported by NPRP-EP through the Qatar National Research Fund (a member of Qatar Foundation) under Grant X-033- 2-007. Scopus

Current works on image emotion distribution learning typically extract visual representations from the holistic image or explore emotion-related regions in the image from a global-wise perspective. However, different regions of an image contribute differently to the arousal of each emotion. Existing works do not deeply explore corresponding emotion-aware regions of each emotion in the image, nor do they fully capture the relationship between each emotion-aware region and the emotion labels. In this paper, we propose a novel attention based emotion distribution learning method, which can explore the emotion-related regions of images from the perspective of each emotion category, and can conduct region relationship learning. Specifically, we introduce a semantic guided attention detection network to generate class-wise attention maps for each emotion and a global-wise attention map for the holistic image. Meanwhile, an emotional graph-based network is adopted to capture the correlation between each region and the emotion distribution. Experiments on several benchmark datasets demonstrate the superiority of the proposed method compared to related works.

We study the list decodability of different ensembles of codes over the real alphabet under the assumption of an omniscient adversary. It is a well-known result that when the source and the adversary have power constraints $ P $ and $ N $ respectively, the list decoding capacity is equal to $ \frac{1}{2}\log\frac{P}{N} $. Random spherical codes achieve constant list sizes, and the goal of the present paper is to obtain a better understanding of the smallest achievable list size as a function of the gap to capacity. We show a reduction from arbitrary codes to spherical codes, and derive a lower bound on the list size of typical random spherical codes. We also give an upper bound on the list size achievable using nested Construction-A lattices and infinite Construction-A lattices. We then define and study a class of infinite constellations that generalize Construction-A lattices and prove upper and lower bounds for the same. Other goodness properties such as packing goodness and AWGN goodness of infinite constellations are proved along the way. Finally, we consider random lattices sampled from the Haar distribution and show that if a certain number-theoretic conjecture is true, then the list size grows as a polynomial function of the gap-to-capacity., Comment: Significantly revised

A major gap between few-shot and many-shot learning is the data distribution empirically oserved by the model during training. In few-shot learning, the learned model can easily become over-fitted based on the biased distribution formed by only a few training examples, while the ground-truth data distribution is more accurately uncovered in many-shot learning to learn a well-generalized model. In this paper, we propose to calibrate the distribution of these few-sample classes to be more unbiased to alleviate such an over-fitting problem. The distribution calibration is achieved by transferring statistics from the classes with sufficient examples to those few-sample classes. After calibration, an adequate number of examples can be sampled from the calibrated distribution to expand the inputs to the classifier. Specifically, we assume every dimension in the feature representation from the same class follows a Gaussian distribution so that the mean and the variance of the distribution can borrow from that of similar classes whose statistics are better estimated with an adequate number of samples. Extensive experiments on three datasets, miniImageNet, tieredImageNet, and CUB, show that a simple linear classifier trained using the features sampled from our calibrated distribution can outperform the state-of-the-art accuracy by a large margin. Besides the favorable performance, the proposed method also exhibits high flexibility by showing consistent accuracy improvement when it is built on top of any off-the-shelf pretrained feature extractors and classification models without extra learnable parameters. The visualization of these generated features demonstrates that our calibrated distribution is an accurate estimation thus the generalization ability gain is convincing. We also establish a generalization error bound for the proposed distribution-calibration-based few-shot learning, which consists of the distribution assumption error, the distribution approximation error, and the estimation error. This generalization error bound theoretically justifies the effectiveness of the proposed method.

Time/space separation-based modeling methods have been widely researched for estimating lithium-ion battery (LIB) thermal dynamics. However, these methods have been developed in an offline environment and may not perform well in real-time application since the battery systems in electric vehicles (EVs) are usually subject to external disturbances. Furthermore, the onboard measurements of temperature are often corrupted by significant error. To address these problems, we present a reduced model-based observer design for online temperature distribution estimation in LIBs. First, an extreme learning machine (ELM) based offline spatiotemporal model is constructed to approximate the thermal dynamics of LIB. Second, an adaptive reduced order observer is designed based on the offline model developed in the previous step. The offline model is then updated with the estimation results of the observer. As the performance of the estimator is highly related to the placement of sensors, a genetic algorithm (GA) based integrated optimization strategy is also developed to determine the optimal sensor location for online estimation. Finally, the whole temperature distribution is estimated in real time using the observer, the measured voltage, current and the limited available temperature data. Simulation results on a battery thermal system verify the effectiveness of this developed model.

Ring and meshed dc distribution grids have attracted increasing interest lately. Their multi-point and multi-terminal structure facilitates flexible operational control and reliable power supply across the network. However, the systems implemented with existing power converters lack sufficient control freedoms for proper power flow adjustment; therefore, it is necessary to develop additional dc power flow controllers (DCPFCs) to be added to these grids for improved power flow control capability. Specific to multi-terminal dc networks built with modular multilevel converters (MMCs), this paper proposes an embedded DCPFC (e-DCPFC) directly integrated in the MMC topology. Unlike previously proposed solutions, the e-DCPFC configuration is derived from the conventional MMC and does not need any isolation transformers or external power supply. Its other main features include modular construction, convenience for multiport extension and bipolar regulating capacity. In this paper, the topology, working principle, and the coordinated control strategy for e-DCPFC and its host MMC are elaborated. Experimental results from a lab prototype verify the proposed topology and its performance under typical working conditions.

Person reidentification (Re-ID) aims at recognizing the same identity across different camera views. However, the cross resolution of images [high resolution (HR) and low resolution (LR)] is unavoidable in a realistic scenario due to the various distances among cameras and pedestrians of interest, thus leading to cross-resolution person Re-ID problems. Recently, most cross-resolution person Re-ID methods focus on solving the resolution mismatch problem, while the distribution mismatch between HR and LR images is another factor that significantly impacts the person Re-ID performance. In this article, we propose a dually distribution pulling network (DDPN) to tackle the distribution mismatch problem. DDPN is composed of two modules, that is: 1) super-resolution module and 2) person Re-ID module. They attempt to pull the distribution of LR images closer to the distribution of HR images from image and feature aspects, respectively, through optimizing the maximum mean discrepancy losses. Extensive experiments have been conducted on three benchmark datasets and the results demonstrate the effectiveness of DDPN. Remarkably, DDPN shows a great advantage when compared to the state-of-the-art methods, for instance, we achieve rank-1 accuracy of 76.9% on VR-Market1501, which outperforms the best existing cross-resolution person Re-ID method by 10%.

A basic assumption of the classic reinsurance model is that the distribution of the loss is precisely known. In practice, only partial information is available for the loss distribution due to the lack of data and estimation error. We study a distributionally robust reinsurance problem by minimizing the maximum Value-at-Risk (or the worst-case VaR) of the total retained loss of the insurer for all loss distributions with known mean and variance. Our model handles typical stop-loss reinsurance contracts. We show that a three-point distribution achieves the worst-case VaR of the total retained loss of the insurer, from which the closed-form solutions of the worst-case distribution and optimal deductible are obtained. Moreover, we show that the worst-case Conditional Value-at-Risk of the total retained loss of the insurer is equal to the worst-case VaR, and thus the optimal deductible is the same in both cases.

O objetivo deste trabalho é apresentar uma metodologia para fornecer uma solução para o Problema de Roteamento de Veículos Green Bi-objetivo (Bi-objective Green Vehicle Routing Problem, BGVRP). A metodologia, ilustrada por meio de um estudo de caso (problema de distribuição de jornais) e instâncias da literatura, foi dividida em três etapas: Etapa 1, tratamento dos dados; Etapa 2, abordagens meta-heurísticas (híbridas ou não híbridas), utilizadas comparativamente; Etapa 3, análise dos resultados, com comparação dos algoritmos. Uma otimização de 19,9% foi alcançada para a Função Objetivo 1 (FO1; minimização das emissões de CO2) e, consequentemente, o mesmo percentual para a minimização da distância total, e 87,5% para a Função Objetivo 2 (FO2; minimização da diferença na demanda). Abordagens meta-heurísticas híbridas alcançaram resultados superiores para o estudo de caso e as instâncias. Desta forma, o procedimento aqui apresentado poderá trazer benefícios para a sociedade já que considera questões ambientais, além do balanceamento do trabalho entre os roteiros, garantindo economia e satisfação para possíveis usuários.

Many lipid membranes of eukaryotic cells are asymmetric, which means the two leaflets differ in at least one physical property, such as lipid composition or lateral stress. Maintaining this asymmetry is helped by the fact that ordinary phospholipids rarely transition between leaflets, but cholesterol is an exception: its flip-flop times are in the microsecond range, so that its distribution between leaflets is determined by a chemical equilibrium. In particular, preferential partitioning can draw cholesterol into a more saturated leaflet, and phospholipid number asymmetry can force it out of a compressed leaflet. Combining highly coarse-grained membrane simulations with theoretical modeling, we investigate how these two driving forces play against each other until cholesterol's chemical potential is equilibrated. The theory includes two coupled elastic sheets and a Flory-Huggins mixing free energy with a χ parameter. We obtain a relationship between χ and the interaction strength between cholesterol and lipids in either of the two leaflets, and we find that it depends, albeit weakly, on lipid number asymmetry. The differential stress measurements under various asymmetry conditions agree with our theoretical predictions. Using the two kinds of asymmetries in combination, we find that it is possible to counteract the phospholipid number bias, and the resultant stress in the membrane, via the control of cholesterol mixing in the leaflets.

Let X k 1 k = 1 ∞ , X k 2 k = 1 ∞ , … , X k d k = 1 ∞ be d independent sequences of Bernoulli random variables with success-parameters p 1 , p 2 , … , p d respectively, where d ≥ 2 is a positive integer, and 0 p j 1 for all j = 1 , 2 , … , d . Let S j ( n ) = ∑ i = 1 n X i j = X 1 j + X 2 j + ⋯ + X n j , n = 1 , 2 , … . We declare a “rencontre” at time n , or, equivalently, say that n is a “rencontre time,” if S 1 ( n ) = S 2 ( n ) = ⋯ = S d ( n ) . We motivate and study the distribution of the first (provided it is finite) rencontre time.

The performance analysis for the MIMO-FSO systems employing EGC technology over Lognormal-Rician turbulence channels with pointing errors is important. However, the results so far are greatly limited since the PDF of Lognormal-Rician turbulence channels is not analytic, even for the SISO systems. In this paper, we propose a new method to approximate the sum of lognormal-Rician turbulence channels with Rayleigh pointing errors. Based on the developed formula, the approximate closed-form expressions of ergodic capacity, outage probability, and bit-error rate are derived. Numerical results demonstrate the accuracy of the proposed approach.

In this paper, we stabilize a discrete-time Markov process evolving on a compact subset of $\mathbb{R}^d$ to an arbitrary target distribution that has an $L^\infty$ density and does not necessarily have a connected support on the state space. We address this problem by stabilizing the corresponding Kolmogorov forward equation, the \textit{mean-field model} of the system, using a density-dependent transition kernel as the control parameter. Our main application of interest is controlling the distribution of a multi-agent system in which each agent evolves according to this discrete-time Markov process. To prevent agent state transitions at the equilibrium distribution, which would potentially waste energy, we show that the Markov process can be constructed in such a way that the operator that pushes forward measures is the identity at the target distribution. In order to achieve this, the transition kernel is defined as a function of the current agent distribution, resulting in a nonlinear Markov process. Moreover, we design the transition kernel to be \textit{decentralized} in the sense that it depends only on the local density measured by each agent. We prove the existence of such

The optimal strategy estimation of random evolutionary Boolean games (REBGs) is discussed in this article. First, using the minimum mean square error criterion, the optimal strategy estimator is proposed for REBGs. Then, a matrix approach is developed to calculate the optimal strategy estimator by the aid of a semitensor product of matrices, which includes the prediction matrix, updating distribution, and strategy iterative formula. Finally, an elucidative example is included to show the obtained results are valid.

This article proposes a novel gain-scheduling control method to improve vehicle lateral stability based on a variable-parameter-dependent approach. A time-varying velocity-dependent model to describe the vehicle lateral dynamic characteristics is constructed under cornering stiffness uncertainties and controller saturation. A novel Lyapunov function with variable-parameter-dependent Lyapunov matrix is established to develop the conditions. Using the proposed Lyapunov function, a general condition of the variable-parameter-dependent saturated robust yaw moment controller is provided in terms of time-varying parameter-dependent matrix inequalities. In order to obtain a tractable solution, a condition is further developed with finite linear matrix inequalities. Moreover, an optimal distribution method is adopted to generate the desired yaw moment based on torque allocation. These torques are computed by optimizing the distribution errors and tire workloads. Simulations results under both J-turn and double-lane changing scenarios are used to illustrate the merits of the proposed method. The control synthesis approach is also applicable to other applications involving time-varying parameters.

In this paper we obtain explicit estimates and existence results on the number of F q -rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the geometric properties of the varieties defined by the systems involved. We apply these results to a generalization of Waring's problem and the distribution of solutions of congruences modulo a prime number.

The distribution of abrasive particles in fluids is an important research topic in the study of abrasive waterjet cutting processes. However, it is impossible to obtain the accurate distribution law and influencing factors by performing only experiments; therefore, it is necessary to study abrasive waterjet cutting processes with the help of numerical models. The existing numerical models usually adopt the form of artificial settings for the distribution of abrasive particles in fluid. This method cannot accurately simulate the random distribution of particles. In this paper, the random algorithm method is used to simulate the impact azimuth and the random distribution of abrasive particles in water. The smoothed particle hydrodynamics (SPH) method is used to simulate the distribution of abrasive particles and the process of jet impingement. The influence of the particle distribution on the simulation results is studied. Comparisons show that the dent formed by the jet impinging on the target with random abrasive particles is similar to the dent from the actual cut, and the contour distribution of the dent is more uniform than that of the cut. The simulation results obtained by the SPH method are accurate.

Aerial coverage for Gas Distribution Map (GDM) using Unmanned Aerial Vehicles (UAVs) with olfaction capabilities has become a key for effectiveness and cost improvements. In this work, the main contribution is the application of Coverage Path Planning (CPP) approaches adapted to survey gas plumes using an autonomous fleet of UAVs. Indeed, the mission is carried out in four steps. First, a gas plume model was integrated to simulate the environment. Then, based on the sensor’s footprint model and the Voronoï diagram, an exact cellular decomposition method is proposed to generate and distribute measurement points or Points Of Interests (POIs) effectively. To take measurements from all these POIs, two planning approaches were proposed in the third step. One is based on the separation of missions between UAVs using the K-means algorithm principle and modeling each path as a Traveling Salesman Problem (TSP). While the second approach models the global mission for all UAVs as a Vehicle Routing Problem (VRP). The resolution of the two mathematical models is achieved using Genetic Algorithms (GAs) and Simulated Annealing (SA) algorithms, respectively. Eventually, the GDM is reconstituted based on triangulation-based cubic interpolation. The evaluation of the proposed simulation techniques based on comparative analyses shows their efficiency, speed and flexibility. The proposed methods take into account the UAV’s flight autonomy and the desired coverage quality, and they provide solutions for a variety of coverage applications, particularly for GDM.

The least-square support vector machine (LS-SVM) has been deeply studied in the machine-learning field and widely applied on a great deal of occasions. A disadvantage is that it is less efficient in dealing with the non-Gaussian noise. In this article, a novel probabilistic LS-SVM is proposed to enhance the modeling reliability even data contaminated by the non-Gaussian noise. The stochastic effect of noise on the kernel function and the regularization parameter is first analyzed and estimated. On the basis of this, a new objective function is constructed under a probabilistic sense. A probabilistic inference method is then developed to construct the distribution of the model parameter, including distribution estimation of both the kernel function and the regularization parameter from data. Using this distribution information, a solving strategy is then developed for this new objective function. Different from the original LS-SVM that uses a deterministic scenario approach to gain the model, the proposed method builds the distribution relation between the model and noise and makes use of this distribution information in the process of modeling; thus, it is more robust for modeling of noise data. The effectiveness of the proposed probabilistic LS-SVM is demonstrated by using both artificial and real cases.

The past few years have seen several works exploring learning economic solutions from data; these include optimal auction design, function optimization, stable payoffs in cooperative games and more. In this work, we provide a unified learning-theoretic methodology for modeling such problems, and establish tools for determining whether a given solution concept can be efficiently learned from data. Our learning theoretic framework generalizes a notion of function space dimension — the graph dimension — adapting it to the solution concept learning domain. We identify sufficient conditions for efficient solution learnability, and show that results in existing works can be immediately derived using our methodology. Finally, we apply our methods in other economic domains, yielding learning variants of competitive equilibria and Condorcet winners.

Understanding the dominant physical processes that cause fault reactivation due to fluid injection is vital to develop strategies to avoid and mitigate injection-induced seismicity. Injection-induced seismicity is a risk for several industries, including hydraulic fracturing, geothermal stimulation, oilfield waste disposal and carbon capture and storage, with hydraulic fracturing having been associated with some of the highest magnitude induced earthquakes (M > 5). As such, strict regulatory schemes have been implemented globally to limit the felt seismicity associated with operations. In the UK, a very strict “traffic light” system is currently in place. These procedures were employed several times during injection at the PNR-1z well at Preston New Road, Lancashire, UK, from October to December 2018. As injection proceeded, it became apparent to the operator that stages were interacting with a seismogenic planar structure, interpreted as a fault zone, with several ML > 0.5 events occurring. Microseismicity was clustered along this planar structure in a fashion that could not readily be explained through pore pressure diffusion or hydraulic fracture growth. Instead, we investigate the role of static elastic stress transfer created by the tensile opening of hydraulic fractures. We find that the spatial distributions of microseismicity are strongly correlated with areas that receive positive Mohr-Coulomb stress changes from the tensile fracture opening, while areas that receive negative Mohr-Coulomb stress change are quiescent. We conclude that the stressing due to tensile hydraulic fracture opening plays a significant role in controlling the spatiotemporal distribution of induced seismicity.

This work explore the possibility to determine the spray characteristics at early stage of the atomization process. This approach has been firstly proposed in [7] propose to characterize the liquid-gas surface in term of surface curvature distribution allowing the determination of which part of the surface is already representative of the final spray. This innovative approach is based on numerical simulation but could be extended in principle on experimental imagery techniques for instance. In this work, the methodology is extended to include surface velocity measurements to provide the joint distribution of the spray in term of diameter and velocity. The configuration is representative of an aeronautic injector used in the European project CHAiRLIFT and focus on the prefilming airblast atomization. Such an atomization process involves a wide range of length scales. The diameter of the aeronautic atomizer could be three orders of magnitude higher than the diameter of the smallest droplet. Therefore, performing a CFD simulation is neither a straightforward nor a cheap process computationally speaking. Thus, a workflow, which divides the atomizer in less complex processes, is presented in this work to make the simulation affordable in an industrial perspective. The objective is to compute the properties that determine the spray behavior for the following combustion process. The OpenFOAM library is used to perform the simulation. In addition to the numerical strategy to capture first steps of the atomization process the focus of our presentation will be on the post processing of the simulation to extract the main features of the spray. The results presented concern the drop size distribution of the spray and the joint velocity distributions of the droplets. The post processing methodology is based on an analysis of the surface density joint distribution for curvatures and velocity. We believe this methodology can be used to enrich the spray injection models for further reactive spray flame simulations.

This paper introduces a new mathematical procedure for the calculation of power cable ampacities in the case of water-cooled circuits. Both, internal and external cooling installations are considered. The algorithm permits a non-iterative calculation of the water temperature along the cooling length of the cables, evaluation of the heat absorbed and emitted by the cooling pipes as well as the effect of the water flow on the cable conductor temperature. Numerical examples show the effect of the direct and the indirect cooling on the ampacity of a 230 kV XLPE cable circuit.

This article proposes a second-order conic programming (SOCP) approach to solve distributionally robust two-stage linear programs over 1-Wasserstein balls. We start from the case with distribution uncertainty only in the objective function and then explore the case with distribution uncertainty only in constraints. The former program is exactly reformulated as a tractable SOCP problem, whereas the latter one is proved to be generally NP-hard as it involves a norm maximization problem over a polyhedron. However, it reduces to an SOCP problem if the extreme points of the polyhedron are given as a prior. This motivates the design of a constraint generation algorithm with provable convergence to approximately solve the NP-hard problem. Moreover, the least favorable distribution achieving the worst case cost is given as an ``empirical'' distribution by simply perturbing each original sample for both cases. Finally, experiments illustrate the advantages of the proposed model in terms of the out-of-sample performance and computational complexity.

A statistical model is a class of probability distributions assumed to contain the true distribution generating the data. In parametric models, the distributions are indexed by a finite-dimensional parameter characterizing the scientific question of interest. Semiparametric models describe the distributions in terms of a finite-dimensional parameter and an infinite-dimensional component, offering more flexibility. Ordinarily, the statistical model represents distributions for the full data intended to be collected. When elements of these full data are missing, the goal is to make valid inference on the full-data-model parameter using the observed data. In a series of fundamental works, Robins, Rotnitzky, and colleagues derived the class of observed-data estimators under a semiparametric model assuming that the missingness mechanism is at random, which leads to practical, robust methodology for many familiar data-analytic challenges. This article reviews semiparametric theory and the key steps in this derivation.

The construction of prediction sets (or regions) for future failure times based on a type-II censored sample from the exponential distribution is investigated. Based on the distribution of the sum of independent exponential variables with different parameters, we obtain the distribution of the required pivotal quantities in order to find the prediction regions. Balanced prediction sets are first derived. A constrained optimization problem is then formulated and solved to determine the prediction region with minimal area. A Monte Carlo simulation study is carried out to compare the performance of the proposed prediction sets. Two real data examples are provided and analyzed to illustrate the methods presented. Finally, some applications and extensions of the results are discussed.

We prove that a single-layer neural network trained with the Q-learning algorithm converges in distribution to a random ordinary differential equation as the size of the model and the number of training steps become large. Analysis of the limit differential equation shows that it has a unique stationary solution that is the solution of the Bellman equation, thus giving the optimal control for the problem. In addition, we study the convergence of the limit differential equation to the stationary solution. As a by-product of our analysis, we obtain the limiting behavior of single-layer neural networks when trained on independent and identically distributed data with stochastic gradient descent under the widely used Xavier initialization.

A distribution service restoration algorithm as a fundamental resilient paradigm for system operators provides an optimally coordinated, resilient solution to enhance the restoration performance. The restoration problem is formulated to coordinate distribution generators and controllable switches optimally. A model-based control scheme is usually designed to solve this problem, relying on a precise model and resulting in low scalability. To tackle these limitations, this work proposes a graph-reinforcement learning framework for the restoration problem. We link the power system topology with a graph convolutional network, which captures the complex mechanism of network restoration in power networks and understands the mutual interactions among controllable devices. Latent features over graphical power networks produced by graph convolutional layers are exploited to learn the control policy for network restoration using deep reinforcement learning. The solution scalability is guaranteed by modeling distributed generators as agents in a multi-agent environment and a proper pre-training paradigm. Comparative studies on IEEE 123-node and 8500-node test systems demonstrate the performance of the proposed solution.

Asymptotic integration theory gives a collection of results which provide a thorough description of the asymptotic growth and zero distribution of solutions of (*) f ′ ′ + P ( z ) f = 0 , where P ( z ) is a polynomial. These results have been used by several authors to find interesting properties of solutions of (*). That said, many people have remarked that the proofs and discussion concerning asymptotic integration theory that are, for example, in E. Hille’s 1969 book Lectures on Ordinary Differential Equations are difficult to follow. The main purpose of this paper is to make this theory more understandable and accessible by giving complete explanations of the reasoning used to prove the theory and by writing full and clear statements of the results. A considerable part of the presentation and explanation of the material is different from that in Hille’s book.

Many realistic mobile social networks can be characterized by evolving bipartite graphs [1], [2], in which dynamically added elements are divided into two entities and connected by links between these two entities, such as users and items in recommendation networks, followees and followers in Twitter networks, authors and scientific topics in scholarly networks, etc. However, given the fact that connections between two entities are often weighted, how to mathematically model such weighted evolving bipartite relationships, along with quantitative characterizations, remains unexplored. Motivated by this, we develop a novel Evolving Bipartite Model (EBM), which, based on empirically validated power-law distribution on multiple realistic mobile social networks, discloses that the distribution of total weights of incoming and outgoing edges in networks is determined by the weighting scale and bounded by certain ceilings and floors. Based on these theoretical results, for evolving bipartite networks whose degree follows power-law distribution, their overall weights of vertices can be predicted by EBM. To illustrate, in recommendation networks, the evaluation of items, i.e., total rating scores, can be estimated through the given bounds; in Twitter social networks, the influence of users can be roughly measured; in scholarly networks

As a three-way approximation of fuzzy sets, shadowed sets has attracted extensive attention in recent years. A fundamental issue in the process of constructing shadowed sets is the interpretation and determination of the threshold pair (α,β), and the uncertainty consistency, that is, the consistency of fuzzy entropy. However, there may be a large fuzzy entropy loss between a fuzzy set and its corresponding game theoretic shadowed sets (GTSS), and the GTSS model is also accompanied by a large time cost when the precision of (α,β) is improved. Therefore, the fuzzy-entropy-based game theoretic shadowed sets (FeGTSS) is proposed in this paper from the perspective of fuzzy entropy loss. First, based on the compromise principle of game theory, the fuzzy entropy loss of shadowed sets is analyzed. Second, in the process of calculating (α,β), the optimal game strategy is searched based on the dichotomy algorithm. Third, the FeGTSS model is extended and discussed based on the analysis of different data distribution types. Finally, the rationality and validity of the FeGTSS model are illustrated through instances and experimental analysis.