Multiple Orders Research Articles

WITHDRAWN: An improved genetic algorithm for multi-compartment vehicle selection with time window

The main problem to be solved in this article is how to choose a mixed fleet of single and multi-compartment vehicles when delivering multiple orders, and when considering the time-dependence, time window, and deterioration loss, to minimize the total distribution cost of the distribution center, namely the time dependence of multi-compartment vehicle selection problem TDMCVRPTWs. By establishing the heuristic algorithm pre self-organized clustering improved genetic algorithm (SOMGA), and based on the framework of SOMGA algorithm, We build a pre self-organizing clustering improved genetic algorithm of vehicle type selection (SOMGAs), SOMGAs to solve the hybrid fleet that can make the distribution cost the lowest. In order to verify the rationality of SOMGA algorithm, this paper will check the calculation through Solomon’s 25, 50 and 100 customer sets with time windows, and compare the results with HPSO improved particle swarm optimization algorithm. In addition, the optimal values of ACS, HAC, HVANS and HABC are compared. Through analysis, SOMGA algorithm has strong algorithm advantages. In order to verify the feasibility of the pre-self-organization cluster genetic algorithm of vehicle type selection (SOMGAs) based on the framework of SOMGA algorithm, through the Solomon 50 customer, 100 customer set, the hybrid fleet can significantly reduce the distribution cost. And in the process of vehicle class selection, the vehicle path of each vehicle is optimized. Finally, it is concluded that SOMGAs algorithm can solve a reasonable mixed fleet in 80% of cases, reducing the total distribution cost by 0% to 24%. • A new algorithm for solving the multi-compartment vehicle routing problem is proposed. • It provides a new method for studying hybrid fleet. • Various traffic environmental factors and food environmental factors are considered.

BigSSL: Exploring the Frontier of Large-Scale Semi-Supervised Learning for Automatic Speech Recognition

We summarize the results of a host of efforts using giant automatic speech recognition (ASR) models pre-trained using large, diverse unlabeled datasets containing approximately a million hours of audio. We find that the combination of pre-training, self-training and scaling up model size greatly increases data efficiency, even for extremely large tasks with tens of thousands of hours of labeled data. In particular, on an ASR task with 34k hours of labeled data, by fine-tuning an 8 billion parameter pre-trained Conformer model we can match state-of-the-art (SoTA) performance with only 3% of the training data and significantly improve SoTA with the full training set. We also report on the universal benefits gained from using big pre-trained and self-trained models for a large set of downstream tasks that cover a wide range of speech domains and span multiple orders of magnitudes of dataset sizes, including obtaining SoTA performance on many public benchmarks. In addition, we utilize the learned representation of pre-trained networks to achieve SoTA results on non-ASR tasks.

The jumping dielectric breakdown behavior induced by crack propagation in ferroelectric materials: A phase field study

Ferroelectric materials often experience fracture and dielectric breakdown under large mechanical and/or electrical loads. These two failure behaviors can affect each other and may lead to new phenomena. This study predicts a new type of jumping dielectric breakdown during crack propagation in ferroelectric materials using a phase field model involving multiple order parameters. The driving forces for the evolution of polarization, fracture, and dielectric breakdown were analysed based on the concept of configurational forces. The jumping dielectric breakdown was attributed to the competition between crack propagation and breakdown initiation at the moving crack tip. The breakdown pattern is related to the polarization domain structure and is significantly influenced by the applied electric field and crack boundary conditions. Furthermore, the jumping breakdown associated with possible mechanical degradation of the material can change the driving and resistance forces for crack propagation. This study introduces a novel dielectric breakdown behavior during crack propagation and provides new insights into the coupling failure of ferroelectric materials.

A review of synchronization problems in parts-to-picker warehouses

Triggered by the great success of e-commerce, today’s warehouses more and more evolve to fully-automated fulfillment factories. Many of them follow the parts-to-picker paradigm and employ shelf-lifting mobile robots or conveyors to deliver stock keeping units (SKUs) to stationary pickers operating in picking workstations. This paper aims to structure and review the family of synchronization problems that arise in this environment: If multiple orders demanding the same SKU can be serviced jointly, then a more efficient picking process and a relief of the bin supply system can be achieved. This paper classifies the family of slightly varying synchronization problems arising with different workstation setups in alternative warehouses. This classification scheme is applied to analyze computational complexity, to systematically quantify the gains of alternative workstation setups, and to benchmark the performance gains of synchronization with those of other well-established decision tasks. Our results show that the right workstation setup can greatly improve throughput performance, so that the gains of synchronization can outreach those promised by other well-researched decision tasks.

Beyond pluralism: a descriptive approach to non-state law

ABSTRACT The concept of legal pluralism has been used widely in legal scholarship to draw attention to the existence of multiple legal orders. Scholars have relied upon it to avoid the ideology of legal centralism, to counter colonialism, and to highlight the neglect of Indigenous laws. These are ameliorative approaches, which aim to expand the concept of law for particular purposes. But it is not clear that they help to explain what law is and does. In this article, I contrast these normative and critical projects with analytic and descriptive approaches to the concept of law. Descriptive conceptual analysis, using empirical examples to enhance the insights of legal theorists, can better shed light on what is distinctive about law as a social form. The article illustrates this approach with a number of historical and anthropological examples. This leads us, I suggest, to form rather than function, that is, to the quality of legalism. The creation and use of general rules and abstract categories, that is, usefully distinguishes law, as a category that encompasses both state and non-state forms, from the wider class of norms encompassed by the expansive theories of the legal pluralists.

Investigate exact reliability under limited time and space of a multistate online food delivery network

In recent years, especially during the outbreak of COVID-19, there have witnessed the emergence of online food delivery (OFD) services that allow customers to place orders via Internet-connected devices to obtain door-to-door delivered meals. In this service, on-time delivery to the right customer with high-quality fresh meals is deemed the primary factor. This paper investigates the reliability of OFD through its ability to meet customer needs in a specific context. During the delivery process, the courier receives the food at the store, packs it into a delivery box, and then delivers it to the customer. Note that, the delivery boxes are limited in space. Customers prefer having their orders delivered within a given time threshold. Thus, reliability is defined as the probability to complete non-integer and multiple orders under a limited time and space. Besides, the OFD system belongs to the gig economy, which allows couriers to select their shifts and working regions. Hence, the capacity of each region, which indicates the number of available couriers, is regarded as multistate. We, thus, construct the OFD system as a multistate online food delivery network (MOFN). Under the limitation of time and space, all feasible minimal capacity vectors are determined to compute the reliability. With the proposed investigation, managers can understand their MOFN’s ability to meet specific customer demands. Accordingly, they can make appropriate adjustments for better performance.

Continuously tuneable acoustic metasurface for high order transmitted acoustic vortices

Existed tuneable acoustic metasurfaces (AMSs) can only transfer high order orbital angular momentum by converting the incident wave into reflected acoustic vortices (AVs). Here, a continuously tuneable AMS is designed and fabricated to produce transmitted AVs with multiple orders. The proposed AMS is composed of multiple layers of reconfigurable sector-shaped resonator-based units. A new design mechanism of the tuneable AMS is proposed to ensure that the discrete precision of the phase shift is not affected by the number of the layer. The theoretical, numerical and experimental results prove that the proposed AMS is able to continuously generate high-order transmitted AVs, which is promising in applications of real-time acoustic rotational manipulations such as acoustic microfluidics, cell manipulation, acoustic tweezers and acoustic communication.

Theory-Guided Exploration of the Sr2Nb2O7 System for Increased Dielectric and Piezoelectric Properties and Synthesis of Vanadium-Alloyed Sr2Nb2O7

Ab initio methods provide a powerful tool in the search for novel polar materials. In particular, there has been a surge to identify lead-free piezoelectric materials to replace PbZr0.52Ti0.48O3. This study examines a computational strategy to identify increased piezoelectric and dielectric responses of alloy systems based on the linear interpolation of force constants, Born effective charges, and internal strain tensors from their end-point compounds. We choose the ferroelectric layered perovskite Sr2Nb2O7 as a parent structure and employ this alloying strategy for 19 potential cation substitutions, targeting thermodynamically metastable alloys with high piezoelectric response. From this screening, we identify Sr2Nb2–2xV2xO7 as a promising polar system. We conduct large-unit-cell calculations of Sr2Nb2–2xV2xO7 at x = 0.0625, 0.125 for multiple cation orderings and find a significant 184% enhanced piezoelectric response. The solid solution system is synthesized as single-crystalline thin-film heterostructures using pulsed-laser deposition, and an enhanced dielectric response is observed at x = 0.05 and at x = 0.1. We present the Sr2Nb2–2xV2xO7 alloy system designed through high-throughput computational screening methods with a large calculated piezoelectric response and experimentally verified increased dielectric response. Our methodology is provided as a high-throughput screening tool for novel materials with enhanced polarizability and alloy systems with potential morphotropic phase boundaries.

Decentering Hegemony and “Open” Orders: Fifteenth-Century Melaka in a World of Orders

Abstract I explain the differences between centered and decentered systems (world orders) by adopting a relational approach to international relations. I argue that centric systems have three characteristics: most meaningful social relations are oriented toward one center; this center establishes the practices governing social relations; and the center recognizes no peers. Conversely, if these attributes are not the chief hallmarks of a social system, then it is decentered. More specifically, I analyze fifteenth-century Melaka to show that simultaneous participation in multiple world orders is a distinct response to hegemony that decenters it. Melaka recognized Ming hegemony but decentered it by also participating in the Persian cosmopolis, and by emerging as a (mini-)center in maritime Southeast Asia. My analysis has two main implications. First, I make the case for “open” orders—multiple, (partially) overlapping orders as viable and long-lasting systems—in contrast to the realist and liberal logics of orders of exclusion/inclusion. Second, this has implications for contemporary Southeast Asian “choices” in the context of the US–China rivalry as these states are likely to “openly” participate in the American and Chinese world orders simultaneously instead of choosing between them. My relational perspective rooted in global history argues against treating historical East Asia as “closed” for such a view is prone to both Sinocentrism and Eurocentrism. Instead, I emphasize the pivotal role played by Melaka—that pursued the strategy of attraction to manage its unequal world—and make the case for “open” orders without flattening power hierarchies or cultural differences.

Passive sound detection of the helicopter in the far-field with a spectral coherence decomposition method

Helicopter detection is an important part of the safety and security during flight. Most of the aerodynamic noise of modern helicopters comes from the main rotor, the main rotor aerodynamic noise thus can be used for helicopter detection. However, the Signal-to-Noise Ratio (SNR) of the measurement noise has always been quite small while the helicopter is flying over long distance away from the microphone. The current passive detection methods are not robust enough in the presence of substantial interference. A robust passive sound detection method is proposed to detect the far-field helicopter based on the cyclostationarity of the main rotor noise. The Sparsity-Enhanced Spectral Coherence (SESC) is derived from the spectral coherence decomposition to improve the detection robustness at far distances and low SNR. Furthermore, a global detector is constructed to detect helicopters adaptively by fusing the detection of multiple orders of BPF, which can be performed on the calculated SESC. More accurate and robust detection of the far-field helicopter can be obtained by the proposed SESC detector. The effectiveness of the proposed detection approach is demonstrated and contrasted by using simulation and far-field flight test measurements of the ROBINSON R22 helicopter.

Well-tempered metadynamics applied to field-theoretic simulations of diblock copolymer melts.

Well-tempered metadynamics (WTMD) is applied to field-theoretic simulations (FTS) to locate the order-disorder transition (ODT) in incompressible melts of diblock copolymer with an invariant polymerization index of N̄=104. The polymers are modeled as discrete Gaussian chains with N = 90 monomers, and the incompressibility is treated by a partial saddle-point approximation. Our implementation of WTMD proves effective at locating the ODT of the lamellar and cylindrical regions, but it has difficulty with that of the spherical and gyroid regions. In the latter two cases, our choice of order parameter cannot sufficiently distinguish the ordered and disordered states because of the similarity in microstructures. The gyroid phase has the added complication that it competes with a number of other morphologies, and thus, it might be beneficial to extend the WTMD to multiple order parameters. Nevertheless, when the method works, the ODT can be located with impressive accuracy (e.g., ΔχN ∼ 0.01).

Historical Global Review of Acid-Volatile Sulfide Sediment Monitoring Data

Acid-volatile sulfides (AVS) are strongly associated with the bioavailability of some divalent metals such as cadmium, copper, lead, nickel and zinc. However, the global spatial variability of AVS for aquatic systems is unknown. The specific goals of this study were to: (1) summarize all available AVS monitoring data from all types of freshwater and saltwater waterbodies (streams/creeks, rivers, lakes/ponds/reservoirs and estuarine/marine areas) and (2) compare AVS concentrations from these various types of waterbodies considering both soil type classification and biomes. AVS measurements were reported from 21 different countries. A total of 17 different soil types were reported for all waterbody types and both podzols and luvisols were found in all waterbody types. Nine different biomes were sampled for all waterbody types. The temperate broadleaf and mixed forest biome was sampled for AVS in all waterbody types. Mean AVS concentrations ranged from 0.01 to 503 µmoles/g for 140 different waterbody types and the 90th centile for all these waterbodies was 49.4 µmoles/g. A ranking of waterbody type means from low to high AVS measurements showed the lowest mean value was reported for streams/creeks (5.12 µmoles/g; range from 0.1 to 39.8 µmoles/g) followed by lakes/ponds/reservoirs (11.3 µmoles/g; range from 0.79 to 127 µmoles/g); estuarine/marine areas (27.2 µmoles/g; range from 0.06 to 503 µmoles/g) and rivers (27.7 µmoles/g; range from 1.13 to 197 µmoles/g). The data provided in this study are compelling as it showed that the high variability of AVS measurements within each waterbody type as well as the variability of AVS within specific locations were often multiple orders of magnitude differences for concentration ranges. Therefore, a comprehensive spatial and temporal scale sampling of AVS in concert with divalent metals analysis is critical to avoid possible errors when evaluating the potential ecological risk of divalent metals in sediment.

A polynomial framework for design of drag reducing periodic two-dimensional textured surfaces

Periodic and symmetric two-dimensional textures with various cross-sectional profiles have been employed to improve and optimize the physical response of the surfaces such as drag force, superhydrophobicity, and adhesion. While the effect of the height and spacing of the textures have been extensively studied, the effect of the shape of the textures has only been considered qualitatively. Here, a polynomial framework is proposed to mathematically define the cross-sectional profiles of the textures and offer a quantitative measure for comparing the physical response of the textured surfaces with various shapes. As a case study, textured surfaces designed with this framework are tested for their hydrodynamic frictional response in the cylindrical Couette flow regime in Taylor–Couette flows. With the reduction in torque as the objective, experimental and numerical results confirm that textures with height-to-half-spacing ratios of lower and equal to unity with concave profiles offer a lower torque compared to both smooth surfaces and triangular textures. In addition, across multiple polynomial orders, textures defined by second-order polynomials offer a wide range of responses, eliminating the need for considering polynomials of higher orders and complexity. While the case study here is focused on the laminar flow regime and the frictional torque, the same type of analysis can be applied to other surface properties and physical responses as well.

Phase field modeling of coupling evolution of fracture and dielectric breakdown in ferroelectric materials

Ferroelectric materials have attracted increasing attention due to their distinguished electromechanical coupling properties. Fracture and dielectric breakdown are two main failure behaviours of ferroelectric materials, which may occur simultaneously under large mechanical and electrical loads. Understanding the coupling evolution of fracture and dielectric breakdown is crucial for the application of ferroelectric materials under complex electromechanical environments. In the present study, a phase field model with multiple order parameters is developed to investigate the coupling evolution behaviours of fracture and dielectric breakdown of ferroelectric materials subjected to the mechanical and electrical loadings. In the phase field model, coupled electromechanical degradation functions are proposed to model the interface boundary conditions of fracture and dielectric breakdown. Generalized configurational forces of fracture and dielectric breakdown are obtained from the phase field model to analyse the driving forces for the evolution of fracture and dielectric breakdown. The interaction between fracture and dielectric breakdown is predicted by using the phase field model with a set of interface boundary conditions. The simulation results show that dielectric breakdown occurs at impermeable crack tips due to the concentrated electric field, while the stress distribution around the crack tip and the driving force of crack propagation are influenced by dielectric breakdown. The attraction of crack to electrical treeing path and the deflection of electrical treeing to crack path are predicted during the coupling evolution of fracture and dielectric breakdown. The results of this work provide an in-depth insight into the electromechanical coupling failure of fracture and breakdown of ferroelectric materials. The developed model framework can be employed to investigate more complex electromechanical coupling failures of dielectric, piezoelectric and ferroelectric materials and structures.

Design and Analysis of Dual-Band High-Gain THz Antenna Array for THz Space Applications

In this paper, a high-gain THz antenna array is presented. The array uses a polyimide substrate with a thickness of 10 μm, a relative permittivity of 3.5, and an overall volume of 2920 μm × 1055 μm × 10 μm, which can be employed for THz band space communication and other interesting applications. The dual-band single-element antenna is designed in four steps, while operating at 0.714 and 0.7412 THz with −10 dB bandwidths of 4.71 and 3.13 GHz, providing gain of 5.14 and 5 dB, respectively. In order to achieve a high gain, multiple order antenna arrays are designed such as the 2 × 1 antenna array and the 4 × 1 antenna array, named type B and C, respectively. The gain and directivity of the proposed type C THz antenna array are 12.5 and 11.23 dB, and 12.532 and 11.625 dBi at 0.714 and 0.7412 THz, with 99.76 and 96.6% radiation efficiency, respectively. For justification purposes, the simulations of the type B antenna are carried out in two simulators such as the CST microwave studio (CSTMWS) and the advance design system (ADS), and the performance of the type B antenna is compared with an equivalent circuit model on the bases of return loss, resulting in strong agreement. Furthermore, the parametric analysis for the type C antenna is done on the basis of separation among the radiating elements in the range 513 to 553 μm. A 64 × 1 antenna array is used to achieve possible gains of 23.8 and 24.1 dB, and directivity of 24.2 and 24.5 dBi with good efficiencies of about 91.66 and 90.35% at 0.7085 and 0.75225 THz, respectively, while the 128 × 1 antenna array provides a gain of 26.8 and 27.2 dB, and directivity of 27.2 and 27.7 dBi with good efficiency of 91.66 and 90.35% at 0.7085 and 0.75225 THz, respectively. All the results achieved in this manuscript ensure the proposed design is a feasible candidate for high-speed and free space wireless communication systems.

An optimum solution for coal permeability estimation from mesoscopic scale calibrated stochastic and deterministic discrete fracture network models

The efficiency of a coal bed as a reservoir of methane and CO2 sink primarily depends on its fracture permeability. Conventional techniques of permeability estimation from drilled-core plugs in the laboratory and analytical computations are often different from in-situ measurements by multiple orders. Hence, pre-drill estimation of coal bed permeability remains one of the major challenges for the operators. Researchers usually try to solve this issue by building DFN models based on the direct use of µCT images of millimeters scale coal chips. However, such small-scale DFN models also overestimate permeability by many folds. As a solution, we propose the deployment of calibrated Discrete Fracture Network (DFN) modeling technique based on mesoscale coal images. Deterministic and stochastic DFN models are simulated by replicating thirteen coal specimens collected from the Gondwana coal seams in the Raniganj coalfield of India. Different model outputs such as directional permeabilities, permeability anisotropy, connectivity, etc. are used to distinguish and calibrate these models. The derived average permeabilities from these models closely match with available field measurements. Thus, the proposed technique helps in minimizing the huge gap between the laboratory and well-test measurements. In the process, we derive the threshold sample/model size that would sufficiently address the average cleat permeability at the well-drainage scale. Additionally, sensitivity analysis of key fracture attributes is carried out to quantify their effect on coal permeability.

Chemical evolution of secondary organic aerosol tracers during high-PM2.5 episodes at a suburban site in Hong Kong over 4 months of continuous measurement

Abstract. Secondary organic aerosol (SOA) makes a sizable contribution to fine-particulate-matter (PM2.5) pollution, especially during high-PM episodes. Past studies of SOA evolution at the episode scale mainly rely on measurements of bulk SOA mass, with few studies probing individual SOA molecular tracers. In this study, we continuously monitored (at a bi-hourly resolution) SOA tracers specific to a few common volatile organic compound (VOC) precursors at a suburban site in Hong Kong for a 4-month period from the end of August to December 2020. The SOA molecules include tracers for SOA derived from biomass burning (BB) emissions, monoaromatics, naphthalene/methylnaphthalenes, and three biogenic VOCs (isoprene, monoterpene, and sesquiterpene). Generally, the SOA tracers showed regional characteristics for both anthropogenic and biogenic SOA as well as for the BB-derived SOA. This work focused on the seasonal variation and evolution characteristics of SOA tracers during 11 city-wide PM2.5 episodes, which are defined as periods with PM2.5 concentrations exceeding 35 µg m−3 at 3 or more of the 15 general air quality monitoring stations cross the city. Mass increment ratios (MIR), calculated as the ratio of the mass concentration prior to an episode to that during an episode, were examined for individual species during each episode. During most episodes, the SOA tracer concentrations were enhanced (i.e. MIR >1), and the maximum MIR values were in the range of 5.5–11.0 for SOA tracers of different precursors. Episodes on summer and fall days showed notably larger MIR values than those falling on winter days, indicating the higher importance of SOA to the formation of summer/fall PM2.5 episodes. Simultaneous monitoring of six tracers for isoprene SOA revealed the dominance of the low-NOx pathway in forming isoprene SOA in our study region. The multiple monoterpene SOA products suggested fresher SOA in winter, evidenced by the increased presence of the early-generation products. Thus, the current study has shown by example the precursor-specific SOA chemical evolution characteristics during PM2.5 episodes in different seasons. This study also suggests the necessity to apply high-time-resolution organic marker measurement at multiple sites in order to fully capture the spatial heterogeneity of haze pollution at the city scale.

Prediction of changes in the vegetation cover of Ukraine due to climate warming

Purpose. To study ecological regularities of the formation of vegetation cover in Ukraine depending on the climatic conditions and analyze its possible changes due to global warming. Methodology. The research methodology involves the following: differentiation of the climatic conditions and evaluation of the significance of climatic indices at the level of territorial units of a geobotanical zoning based on variance analysis; multiple ordering of the geographic locations in terms of climatic indices basing on the analysis of main components (Principle Component Analysis); development of a typological scheme of the vegetation cover relying upon a discriminant analysis; statistic processing of the climatic parameters. Findings. Modelling of spatial differentiation of climatic indices depending on the location latitude and longitude as well as altitude above the sea level helped analyze the connection of vegetation cover and climate. It has been determined that peculiarities of the vegetation cover formation according to a scheme of geobotanical zoning is characterized in the most accurate way by the difference of such climatic indices as: air temperature in January, July, and August; monthly precipitation amounts during June-September; duration of a frost-free period; and hydrothermal coefficient of T.G. Selianinov. The main regularity of the formation of Ukrainian vegetation cover has been defined. The regularity means the following structure of interrelation between the climatic indices: along with the growing average monthly temperatures of June-September and decreasing precipitation amounts from April to September, the indices of climatic water availability decrease along with the increasing heat availability indices (duration of an active vegetation period and total of temperatures per that period, average annual temperature). This regularity represents a gradient of climatic indices from the Ukrainian Carpathians towards the southern Crimean coast. It has been determined that during some years of the early 21st century, the conditions peculiar for a steppe area were formed for the forest and forest steppe areas. Originality. The vegetation cover of Ukraine is characterized by the ecological range, which is evaluated basing on ordering of the geographical locations in terms of coverage of complex climatic environmental gradients. A typological scheme of the Ukrainian vegetation cover can be represented in a two-dimensional space in the form of square parabola, whose left branch shows a gradient of climatic factors and changes in vegetation cover from the west to the north-east and right branch indicates it from the north-east to the south. Graphic visualization of climatic information on the basis of ecograms and typological schemes of vegetation cover can be used to predict the vegetation cover dynamics due to certain climatic changes. Practical value. While understanding the climatic conditions of geographic locations during certain periods of time, one can identify their location in the ecological and coenotic range of Ukrainian vegetation and predict their stability and possible changes in the vegetation cover due to global warming.

Integrated Impacts of Non-Ideal Factors on the Vibration Characteristics of Permanent Magnet Synchronous Motors for Electric Vehicles

The nonlinear electromagnetic vibration of the motor is a major factor that deteriorates the noise, vibration, and hardness (NVH) performance of a vehicle’s electric drive system. Considering the nonlinear characteristics of the inverter, the nonsinusoidal distribution of the air-gap magnetic field, the cogging torque, and the current measurement error, a mathematical model of a permanent magnet synchronous motor of an electric vehicle was established, and its dynamic and electromagnetic vibration characteristics under different speed–load conditions were simulated and analyzed. The results show that the nonlinear characteristics of the inverter and nonsinusoidal distribution of the air-gap magnetic field cause the odd current harmonics, such as the 5th, 7th, 11th, and 13th, which lead to the 6th and its integer multiple order fluctuations of the electromagnetic torque. Moreover, the vibration amplitude is intensified under the coupling action of the nonlinear characteristics of the inverter and the nonsinusoidal distribution of the air-gap magnetic field. The current measurement error produces the 1st and 2nd harmonics of the d- and q-axes currents, which result in the 1st and 2nd order fluctuations of the electromagnetic torque. The cogging torque mainly leads to a 12th order torque ripple of the electromagnetic torque. In addition, the non-ideal factors cause a sharp deterioration in the system vibration state under high-speed and heavy-load conditions. This study provides a theoretical reference for the mathematical modeling and electromagnetic vibration research of permanent magnet synchronous motors, considering non-ideal factors comprehensively.

About r-primitive and k-normal elements in finite fields

In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of k-normal elements: an element \(\alpha \in {\mathbb {F}}_{q^n}\) is k-normal over \({\mathbb {F}}_q\) if the greatest common divisor of the polynomials \(g_{\alpha }(x)= \alpha x^{n-1}+\alpha ^qx^{n-2}+\cdots +\alpha ^{q^{n-2}}x+\alpha ^{q^{n-1}}\) and \(x^n-1\) in \({\mathbb {F}}_{q^n}[x]\) has degree k, generalizing the concept of normal elements (normal in the usual sense is 0-normal). In this paper we discuss the existence of r-primitive k-normal elements in \({\mathbb {F}}_{q^n}\) over \({\mathbb {F}}_{q}\), where an element \(\alpha \in {\mathbb {F}}_{q^n}^*\) is r-primitive if its multiplicative order is \(\frac{q^n-1}{r}\). We provide many general results about the existence of this class of elements and we work a numerical example over finite fields of characteristic 11.