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The integral inequalities have become a very popular area of research in recent years. The present paper deals with some important generalizations of convex stochastic processes. Several mean square integral inequalities are derived for this generalization. The involvement of the beta function in the results makes the inequalities more convenient for applied sciences.

Excessive nitrogen (N) application is widespread in Southern China. The effects of N fertilization on soil properties and crop physiology are poorly understood in tropical red loam soil. We conducted a field experiment to evaluate the effect of nitrogen fertilization rates on physiological attributes (chlorophyll, plant metabolic enzymes, soluble matters) on banana leaves, soil properties (soil enzymes, soil organic matter (SOM), soil available nutrients) as well as banana crop yield in a subtropical region of southern China. The N rates tested were 0 (N0), 145 (N145), 248 (N248), 352 (N352), 414 (NFT), and 455 (N455) g N per plant. The correlations among soil factors, leaf physiological factors and crop yield were evaluated. The results indiated that the high rates of N fertilization (NFT and N455) significantly decreased soil available potassium (K) content, available phosphorus (P) content, glutamine synthetase (GS) activity, and soluble protein and sugar contents compared with lower N rates. The N352 treatment had the highest crop yields compared with higher N rates treatments, followed by the N455 treatment. However, there were no significant differences in crop yields among N fertilization treatments. Factor analysis showed that the N352 treatment had the highest integrated score for soil and leaf physiological factors among all treatments. Moreover, the N352 treatment was the most effective in improving carbon and nitrogen metabolism in banana. Crop yield was significantly and positively linearly correlated with the integrated score (r = 0.823, p < 0.05). Path analysis revealed that invertase, SOM and sucrose synthase (SS) had a strong positive effect on banana yield. Canonical correspondence analysis (CCA) suggested that available K, invertase, acid phosphatase and available P were the most important factors impacting leaf physiological attributes. Cluster analysis demonstrated distinct differences in N application treatment related to variations in soil and leaf factors. This study suggested that excessive N fertilization had a negative effect on soil fertility, crop physiology and yield. The lower N rates were more effective in improving crop yield than higher rates of N fertilization. The N rate of 352 g N per plant (N352) was recommended to reduce excess N input while maintaining the higher yield for local farmers’ banana planting.

Abstract Cardiac‐resident macrophages (CRMs) play critical roles in maintaining cardiac homoeostasis and removing senescent and dying cells. Recent preclinical data have re‐energised the area of cardioimmunology and provided improved understanding of the modulation of compositional and functional phenotypes of CRMs. These data can aid in achieving improved cardiac regeneration, repair and functional remodelling following cardiac injury. In this review, we discuss the composition and renewal of various subsets of CRMs. Specific attention has been given to delineate the roles of various CRM subsets with respect to (1) facilitation of cardiac development and maintenance of physiological function such as electrical conduction and rhythm; (2) promotion of cardiac regeneration, inflammation resolution and functional remodelling following a cardiac injury; and (3) therapeutic potential. We have also highlighted the relationship between CRM replenishment and cardiomyocyte senescence as well as cardiovascular diseases development. Finally, we have addressed future perspectives and directions in basic research and potentially clinical applications of CRMs.

Abstract Substantia nigra (SN) is a complex and critical region of the brain wherein Parkinson's disease (PD) arises from the degeneration of dopaminergic neurons. Miniature SN‐like structures (mini‐SNLSs) constructed from novel combination of nanomaterials and cell technologies exhibit promise as potentially curative cell therapies for PD. In this work, a rapid self‐organization of mini‐SNLS, with an organizational structure and neuronal identities similar to those of the SN in vivo, is achieved by differentiating neural stem cells in vitro on biocompatible silica nanozigzags (NZs) sculptured by glancing angle deposition, without traditional chemical growth factors. The differentiated neurons exhibit electrophysiological activity in vitro. Diverse physical cues and signaling pathways that are determined by the nanomatrices and lead to the self‐organization of the mini‐SNLSs are clarified and elucidated. In vivo, transplantation of the neurons from a mini‐SNLS results in an early and progressive amelioration of PD in rats. The sculptured medical device reported here enables the rapid and specific self‐organization of region‐specific and functional brain‐like structures without an undesirable prognosis. This development provides promising and significant insights into the screening of potentially curative drugs and cell therapies for PD.

Compact neural networks are inclined to exploit “sparsely-connected”convolutions, such as depthwise convolution and group convolution for employment in mobile applications. Compared with standard “fully-connected”convolutions, these convolutions are more computationally economical. However, “sparsely-connected”convolutions block the inter-group information exchange, which induces severe performance degradation. To address this issue, we present two novel operations named merging and evolution to leverage the inter-group information. Our key idea is encoding the inter-group information with a narrow feature map, and then combining the generated features with the original network for better representation. Taking advantage of the proposed operations, we then introduce the Merging-and-Evolution (ME) module, an architectural unit specifically designed for compact networks. Finally, we propose a family of compact neural networks called MENet based on the ME modules. Extensive experiments on CIFAR, SVHN, ILSVRC 2012, and PASCAL VOC 2007 data sets demonstrate that MENet consistently outperforms other state-of-the-art compact networks under different computational budgets. For instance, under the computational budget of 140 MFLOPs, MENet surpasses ShuffleNet by 1% and MobileNet by 1.95% on ILSVRC 2012 top-1 accuracy, while by 2.9% and 4.1% on PASCAL VOC 2007 mAP, respectively.

Oleanolic acid (OA), a bioactive ingredient of Panax ginseng, exhibits neuroprotective pharmacological effects. However, the protective role of OA in cerebral ischemia and involved mechanisms remain unclear. This study attempted to explore the therapeutic effects of OA both in vitro and in vivo. OA attenuated cytotoxicity and overproduction of intracellular reactive oxygen species (ROS) by regulation of glycogen synthase kinase-3β (GSK-3β)/heme oxygenase-1 (HO-1) signal in oxygen-glucose deprivation/reoxygenation (OGD/R)-exposed SH-SY5Y cells. Additionally, OA administration significantly reduced the area of cerebral infarction and the neurological scores in the rat models of cerebral ischemia with middle cerebral artery occlusion (MCAO). The OA administration group showed a higher percentage of Nissl+ and NeuN+ cells, along with lower TUNEL+ ratios in the infarct area of MCAO rats. Moreover, OA administration reduced ROS production while it suppressed the GSK-3β activation and upregulated the HO-1 expression in infarcted tissue. Our results illustrated that OA significantly counteracted cerebral ischemia-mediated injury through antioxidant effects induced by the regulation of the GSK-3β/HO-1 signaling pathway, implicating OA as a promising neuroprotective drug for the therapy of ischemic stroke.

Marine actinomycetes are prolific chemical sources of complex and novel natural products, providing an excellent chance for new drug discovery. The chemical investigation of the marine-derived Streptomyces sp. ITBB-ZKa6, from Zhaoshu island, Hainan, led to the discovery of two unique antimycin-type depsipeptides, zhaoshumycins A (1) and B (2), along with the isolation of the four known neoantimycins A (3), F (4), D (5), and E (6). The structures of the new compounds 1 and 2 were elucidated on the basis of the analysis of diverse spectroscopic data and biogenetic consideration. Zhaoshumycins A (1) and B (2) represent a new class of depsipeptides, featuring two neoantimycin monomers (only neoantimycin D or neoantimycins D and E) linked to a 1,4-disubstituted benzene ring via an imino group. Initial toxicity tests of 1–6 in MCF7 human breast cancer cells revealed that compounds 5 and 6 possess weak cytotoxic activity. Further structure–activity relationship analysis suggested the importance of the NH2 group at C-34 in 5 and 6 for cytotoxicity in MCF7 cells.

Abstract Background Schistosomiasis is a water-borne disease caused by trematode worms belonging to genus Schistosoma, which is prevalent most of the developing world. Transmission of the disease is usually associated with multiple biological characteristics and social factors but also factors can play a role. Few studies have assessed the exact and interactive influence of each factor promoting schistosomiasis transmission. Methods We used a series of different detectors (i.e., specific detector, risk detector, ecological detector and interaction detector) to evaluate separate and interactive effects of the environmental factors on schistosomiasis prevalence. Specifically, (i) specific detector quantifies the impact of a risk factor on an observed spatial disease pattern, which were ranked statistically by a value of Power of Determinate (PD) calculation; (ii) risk detector detects high risk areas of a disease on the condition that the study area is stratified by a potential risk factor; (iii) ecological detector explores whether a risk factor is more significant than another in controlling the spatial pattern of a disease; (iv) interaction detector probes whether two risk factors when taken together weaken or enhance one another, or whether they are independent in developing a disease. Infection data of schistosomiasis based on conventional surveys were obtained at the county level from the health authorities in Anhui Province, China and used in combination with information from Chinese weather stations and internationally available environmental data. Results The specific detector identified various factors of potential importance as follows: Proximity to Yangtze River (0.322) > Land cover (0.285) > sunshine hours (0.256) > population density (0.109) > altitude (0.090) > the normalized different vegetation index (NDVI) (0.077) > land surface temperature at daytime (LSTday) (0.007). The risk detector indicated that areas of schistosomiasis high risk were located within a buffer distance of 50 km from Yangtze River. The ecological detector disclosed that the factors investigated have significantly different effects. The interaction detector revealed that interaction between the factors enhanced their main effects in most cases. Conclusion Proximity to Yangtze River had the strongest effect on schistosomiasis prevalence followed by land cover and sunshine hours, while the remaining factors had only weak influence. Interaction between factors played an even more important role in influencing schistosomiasis prevalence than each factor on its own. High risk regions influenced by strong interactions need to be targeted for disease control intervention.

Asparagus stem wilt, is a significant and devastating disease, typically leading to extensive economic losses in the asparagus industry. To obtain transgenic plants resistant to stem wilt, the hevein-like gene, providing broad spectrum bacterial resistance was inserted into the asparagus genome through Agrobacterium tumefaciens-mediated transformation. The optimal genetic transformation system for asparagus was as follows: pre-culture of embryos for 2 days, inoculation using a bacterial titre of OD600 = 0.6, infection time 10 min and co-culturing for 4 days using an Acetosyringone concentration of 200 μmol/L. Highest transformation frequencies reached 21% and ten transgenic asparagus seedlings carrying the hevein-like gene were identified by polymerase chain reaction. Moreover, integration of the hevein-like gene in the T1 generation of transgenic plants was confirmed by southern blot hybridization. Analysis showed that resistance to stem wilt was enhanced significantly in the transgenic plants, in comparison to non- transgenic plants. The results provide additional data for genetic improvement and are of importance for the development of new disease-resistant asparagus varieties.

We study the planar symmetric central configurations of the 1 + 4-body problem where the symmetry axis does not contain any infinitesimal masses. Under certain assumptions, we find analytically some central configurations for suitable positive masses and also get some numerical results of symmetric central configurations where infinitesimal masses are not necessarily equal.

Abstract In this note, we try to generalize the classical Cauchy-Lipschitz-Picard theorem on the global existence and uniqueness for the Cauchy initial value problem of the ordinary differential equation with global Lipschitz condition, and we try to weaken the global Lipschitz condition. We can also get the global existence and uniqueness.

In order to save people’s shopping time and reduce labor cost of supermarket operations, this paper proposes to design a supermarket service robot based on deep convolutional neural networks (DCNNs). Firstly, according to the shopping environment and needs of supermarket, the hardware and software structure of supermarket service robot is designed. The robot uses a robot operating system (ROS) middleware on Raspberry PI as a control kernel to implement wireless communication with customers and staff. So as to move flexibly, the omnidirectional wheels symmetrically installed under the robot chassis are adopted for tracking. The robot uses an infrared detection module to detect whether there are commodities in the warehouse or shelves or not, thereby grasping and placing commodities accurately. Secondly, the recently-developed single shot multibox detector (SSD), as a typical DCNN model, is employed to detect and identify objects. Finally, in order to verify robot performance, a supermarket environment is designed to simulate real-world scenario for experiments. Experimental results show that the designed supermarket service robot can automatically complete the procurement and replenishment of commodities well and present promising performance on commodity detection and recognition tasks.

In order to study the relationship between carbon emission and economic growth in China’s transportation industry and the driving factors of carbon emission, the Logarithmic Mean Divisia Index (LMDI) decomposition model of carbon emission elastic decoupling and carbon emission driving factors is established. The results show that the carbon emission decoupling value of China’s transportation industry has obvious spatial disequilibrium characteristics, and the decoupling state of “weak decoupling-expansion negative decoupling- growth linkage” appears from coastal areas to inland are as a whole. From 2009 to 2016, there are five types of carbon emission Kuznets curves of China’s transportation industry, namely linear type, U type, inverted-U type, inverted-N type and wave type. The carbon emission intensity of transportation industry in many provinces still tends to increase with the increase of economy. The energy structure effect and the energy intensity effect have a relatively weak negative impact on the carbon emission in the transportation industry. However, the industry scale effect has a significant positive impact, which is the main factor to promote the growth of carbon emission.

Agave species are an important family of crassulacean acid metabolism (CAM) plants with remarkable tolerance to heat and drought stresses (Agave deserti) in arid regions and multiple agricultural applications, such as spirit (Agave tequilana) and fiber (Agave sisalana) production. The agave genomes are commonly too large to sequence, which has significantly restricted our understanding to the molecular basis of stress tolerance and economic traits in agaves. In this study, we collected three transcriptome databases for comparison to reveal the phylogenic relationships and evolution patterns of the three agave species. The results indicated the close but distinctly domesticated relations between A. tequilana and A. sisalana. Natural abiotic and biotic selections are very important factors that have contributed to distinct economic traits in agave domestication together with artificial selection. Besides, a series of candidate unigenes regulating fructan, fiber, and stress response-related traits were identified in A. tequilana, A. sisalana, and A. deserti, respectively. This study represents the first transcriptome comparison within domesticated and wild agaves, which would serve as a guidance for further studies on agave evolution, environmental adaptation, and improvement of economically important traits.

We obtain the existence of homoclinic orbits at infinity for a class of second-order Hamiltonian systems with fixed energy. We use the limit for a sequence of approximate solutions which are obtained by variational methods.

DCAMKL1 (doublecortin and CaM kinase-like 1) has been found to be overexpressed and function as an oncogene in several types of cancer, but there are limited reports on the role of DCAMKL1 in bladder cancer. The messenger RNA and protein expression of DCAMKL1 in bladder cancer tissues and cell lines was measured by quantitative reverse transcription polymerase chain reaction, western blotting, or immunohistochemistry. The correlation between DCAMKL1 protein expression and clinicopathological characteristics was analyzed. Univariate and multivariate Cox regression models were adopted to evaluate prognostic significance of DCAMKL1 in bladder cancer patients. In our results, DCAMKL1 messenger RNA and protein were overexpressed in bladder cancer tissues compared with adjacent normal tissues. DCAMKL1 protein overexpression was positively associated with clinical stage, muscularis invasion, lymph node metastasis, and distant metastasis. The univariate and multivariate analyses suggested DCAMKL1 protein overexpression was an unfavorable prognostic factor in bladder cancer patients. In conclusion, DCAMKL1 is an independent poor prognostic factor for bladder cancer patients.

Sparse coding is an active research subject in signal processing, computer vision, and pattern recognition. A novel method of facial expression recognition via non-negative least squares (NNLS) sparse coding is presented in this paper. The NNLS sparse coding is used to form a facial expression classifier. To testify the performance of the presented method, local binary patterns (LBP) and the raw pixels are extracted for facial feature representation. Facial expression recognition experiments are conducted on the Japanese Female Facial Expression (JAFFE) database. Compared with other widely used methods such as linear support vector machines (SVM), sparse representation-based classifier (SRC), nearest subspace classifier (NSC), K-nearest neighbor (KNN) and radial basis function neural networks (RBFNN), the experiment results indicate that the presented NNLS method performs better than other used methods on facial expression recognition tasks.

Recently, compressive sensing (CS) has attracted increasing attention in the areas of signal processing, computer vision and pattern recognition. In this paper, a new method based on the CS theory is presented for robust facial expression recognition. The CS theory is used to construct a sparse representation classifier (SRC). The effectiveness and robustness of the SRC method is investigated on clean and occluded facial expression images. Three typical facial features, i.e., the raw pixels, Gabor wavelets representation and local binary patterns (LBP), are extracted to evaluate the performance of the SRC method. Compared with the nearest neighbor (NN), linear support vector machines (SVM) and the nearest subspace (NS), experimental results on the popular Cohn-Kanade facial expression database demonstrate that the SRC method obtains better performance and stronger robustness to corruption and occlusion on robust facial expression recognition tasks.

Facial expression recognition is an interesting and challenging subject. Considering the nonlinear manifold structure of facial images, a new kernel-based manifold learning method, called kernel discriminant isometric mapping (KDIsomap), is proposed. KDIsomap aims to nonlinearly extract the discriminant information by maximizing the interclass scatter while minimizing the intraclass scatter in a reproducing kernel Hilbert space. KDIsomap is used to perform nonlinear dimensionality reduction on the extracted local binary patterns (LBP) facial features, and produce low-dimensional discrimimant embedded data representations with striking performance improvement on facial expression recognition tasks. The nearest neighbor classifier with the Euclidean metric is used for facial expression classification. Facial expression recognition experiments are performed on two popular facial expression databases, i.e., the JAFFE database and the Cohn-Kanade database. Experimental results indicate that KDIsomap obtains the best accuracy of 81.59% on the JAFFE database, and 94.88% on the Cohn-Kanade database. KDIsomap outperforms the other used methods such as principal component analysis (PCA), linear discriminant analysis (LDA), kernel principal component analysis (KPCA), kernel linear discriminant analysis (KLDA) as well as kernel isometric mapping (KIsomap).

Schistosomiasis mapping using data obtained from parasitological surveys is frequently used in planning and evaluation of disease control strategies. The available geostatistical approaches are, however, subject to the assumption of stationarity, a stochastic process whose joint probability distribution does not change when shifted in time. As this is impractical for large areas, we introduce here the sandwich method, the basic idea of which is to divide the study area (with its attributes) into homogeneous subareas and estimate the values for the reporting units using spatial stratified sampling. The sandwich method was applied to map the county-level prevalence of schistosomiasis japonica in Anhui Province, China based on parasitological data collected from sample villages and land use data. We first mapped the county-level prevalence using the sandwich method, then compared our findings with block Kriging. The sandwich estimates ranged from 0.17 to 0.21% with a lower level of uncertainty, while the Kriging estimates varied from 0 to 0.97% with a higher level of uncertainty, indicating that the former is more smoothed and stable compared to latter. Aside from various forms of reporting units, the sandwich method has the particular merit of simple model assumption coupled with full utilization of sample data. It performs well when a disease presents stratified heterogeneity over space.

BACKGROUND:The Chinese national surveillance system showed that the risk of Schistosoma japonicum infection fluctuated temporally. This dynamical change might indicate periodicity of the disease, and its understanding could significantly improve targeted interventions to reduce the burden of schistosomiasis. The goal of this study was to investigate how the schistosomiasis risk varied temporally and spatially in recent years. METHODOLOGY/PRINCIPAL FINDINGS:Parasitological data were obtained through repeated cross-sectional surveys that were carried out during 1997-2010 in Anhui Province, East China. A multivariate autoregressive model, combined with principal oscillation pattern (POP) analysis, was used to evaluate the spatio-temporal variation of schistosomiasis risk. Results showed that the temporal changes of schistosomiasis risk in the study area could be decomposed into two sustained damped oscillatory modes with estimated period of approximately 2.5 years. The POPs associated with these oscillatory components showed that the pattern near the Yangtze River varied markedly and that the disease risk appeared to evolve in a Southwest/Northeast orientation. The POP coefficients showed decreasing tendency until 2001, then increasing during 2002-2005 and decaying afterwards. CONCLUSION:The POP analysis characterized the variations of schistosomiasis risk over space and time and demonstrated that the disease mainly varied temporally along the Yangtze River. The schistosomiasis risk declined periodically with a temporal fluctuation. Whether it resulted from previous national control strategies on schistosomiasis needs further investigations.

BACKGROUND:Schistosomiasis japonica still remains of public health and economic significance in China, especially in the lake and marshland areas along the Yangtze River Basin, where the control of transmission has proven difficult. In the study, we investigated spatio-temporal variations of S. japonicum infection risk in Anhui Province and assessed the associations of the disease with key environmental factors with the aim of understanding the mechanism of the disease and seeking clues to effective and sustainable schistosomiasis control. METHODOLOGY/PRINCIPAL FINDINGS:Infection data of schistosomiasis from annual conventional surveys were obtained at the village level in Anhui Province, China, from 2000 to 2010 and used in combination with environmental data. The spatio-temporal kriging model was used to assess how these environmental factors affected the spatio-temporal pattern of schistosomiasis risk. Our results suggested that seasonal variation of the normalized difference vegetation index (NDVI), seasonal variation of land surface temperature at daytime (LSTD), and distance to the Yangtze River were negatively significantly associated with risk of schistosomiasis. Predictive maps showed that schistosomiasis prevalence remained at a low level and schistosomiasis risk mainly evolved along the Yangtze River. Schistosomiasis risk also followed a focal spatial pattern, fluctuating temporally with a peak (the largest spatial extent) in 2005 and then contracting gradually but with a scattered distribution until 2010. CONCLUSION:The fitted spatio-temporal kriging model can capture variations of schistosomiasis risk over space and time. Combined with techniques of geographic information system (GIS) and remote sensing (RS), this approach facilitates and enriches risk modeling of schistosomiasis, which in turn helps to identify prior areas for effective and sustainable control of schistosomiasis in Anhui Province and perhaps elsewhere in China.

For $n$-body problems, a central configuration (CC) plays an important role. In this paper, we establish the relation between the spatial pyramidal central configuration (PCC) and the planar central configuration. We prove that the base of PCC is also a CC and we also prove that for some given conditions a planar CC can be extended to a PCC. In particular, if the pyramidal central configuration has a regular polygon base, then the masses of base are equal and the distance between the top vertex and the base is fixed and the mass of the top vertex is selective. Furthermore, the pyramidal central configuration gives rise to an example of a perverse solution in $mathbb{R}^3$.

Stylosanthes sp. is the most important forage legume in tropical areas worldwide. Stylosanthes anthracnose, which is mainly caused by Colletotrichum gloeosporioides, is a globally severe disease in stylo production. Little progress has been made in anthracnose molecular pathogenesis research. In this study, Agrobacterium tumefaciens-mediated transformation was used to transform Stylosanthes colletotrichum strain CH008. The major factors of the genetic transformation system of S. colletotrichum were optimized as follows: A. tumefaciens' AGL-1 concentration (OD(600)), 0.8; concentration of Colletotrichum conidium, 1 × 10(6) conidia/mL; acetosyringone concentration, 100 mmol/L; induction time, 6 h; co-culture temperature, 25 °C; and co-culture time, 3 d. Thus, the transformation efficiency was increased to 300-400 transformants per 106 conidia. Based on the optimized system, a mutant library containing 4616 mutants was constructed, from which some mutants were randomly selected for analysis. Results show that the mutants were single copies that could be stably inherited. The growth rate, spore amount, spore germination rate, and appressorium formation rate in some mutants were significantly different from those in the wild-type strain. We then selected the most appropriate method for the preliminary screening and re-screening of each mutant's pathogenic defects. We selected 1230 transformants, and obtained 23 strains with pathogenic defects, namely, 18 strains with reduced pathogenicity and five strains with lost pathogenicity. Thermal asymmetric interlaced PCR was used to identify the transfer DNA (T-DNA) integration site in the mutant that was coded 2430, and a sequence of 476 bp was obtained. The flanking sequence of T-DNA was compared with the Colletotrichum genome by BLAST, and a sequence of 401 bp was found in Contig464 of the Colletotrichum genome. By predicting the function of the flanking sequence, we discovered that T-DNA insertion in the promoter region of the putative gene had 79% homology with the aspartate aminotransferase gene in Magnaporthe oryzae (XP_003719674.1).

Speech emotion recognition is currently an active research subject and has attracted extensive interest in the science community due to its vital application to human-robot interaction. Most speech emotion recognition systems employ high-dimensional speech features, indicating human emotion expression, to improve emotion recognition performance. To effectively reduce the size of speech features, in this paper, a new nonlinear dimensionality reduction method, called ‘enhanced kernel isometric mapping’ (EKIsomap), is proposed and applied for speech emotion recognition in human-robot interaction. The proposed method is used to nonlinearly extract the low-dimensional discriminating embedded data representations from the original high-dimensional speech features with a striking improvement of performance on the speech emotion recognition tasks. Experimental results on the popular Berlin emotional speech corpus demonstrate the effectiveness of the proposed method.

We use Jacobi's necessary condition for the variational minimizer to study the periodic solution for spatial restricted N+1-body problems with a zero mass on the vertical axis of the plane for N equal masses. We prove that the minimizer of the Lagrangian action on the anti-T/2 or odd symmetric loop space must be a nonconstant periodic solution for any 2≤N≤472; hence the zero mass must oscillate, so that it cannot be always in the same plane with the other bodies. This result contradicts with our intuition that the small mass should always be at the origin.

We present a new extension of Serrin's lower semicontinuity theorem. We prove that the variational functional defined on is lower semicontinuous with respect to the strong convergence in , under the assumptions that the integrand has the locally absolute continuity about the variable .