Volumen 5 (2020): Heft 2 (July 2020)

This work is a review of current trends in the stray flux signal processing techniques applied to the diagnosis of electrical machines. Initially, a review of the most commonly used standard methods is performed in the diagnosis of failures in induction machines and using stray flux; and then specifically it is treated and performed the algorithms based on statistical analysis using cumulants and polyspectra. In addition, the theoretical foundations of the analyzed algorithms and examples applications are shown from the practical point of view where the benefits that processing can have using HOSA and its relationship with stray flux signal analysis, are illustrated.

Aim of the present paper is to establish fractional integral formulas by using fractional calculus operators involving the generalized (p, q)-Mathieu type series. Then, their composition formulas by using the integral transforms are introduced. Further, a new generalized form of the fractional kinetic equation involving the series is also developed. The solutions of fractional kinetic equations are presented in terms of the Mittag-Leffler function. The results established here are quite general in nature and capable of yielding both known and new results.

In this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods constructed for solving partial differential equations with CF and AB fractional derivatives.

In this paper, we study on normal complex contact metric manifold admitting a semi symmetric metric connection. We obtain curvature properties of a normal complex contact metric manifold admitting a semi symmetric metric connection. We also prove that this type of manifold is not conformal flat, concircular flat, and conharmonic flat. Finally, we examine complex Heisenberg group with the semi symmetric metric connection.

In this paper, the single center vortex method (SCVM) is extended to find some vortex solutions of finite core size for dissipative 2D Boussinesq equations. Solutions are expanded in to series of Hermite eigenfunctions. After confirmation the convergence of series of the solution, we show that, by considering the effect of temperature on the evolution of the vortex for the same initial condition as in [19] the symmetry of the vortex destroyed rapidly.

Three main tools to study graphs mathematically are to make use of the vertex degrees, distances and matrices. The classical graph energy was defined by means of the adjacency matrix in 1978 by Gutman and has a large number of applications in chemistry, physics and related areas. As a result of its importance and numerous applications, several modifications of the notion of energy have been introduced since then. Most of them are defined by means of graph matrices constructed by vertex degrees. In this paper we define another type of energy called q-distance energy by means of distances and matrices. We study some fundamental properties and also establish some upper and lower bounds for this new energy type.

Topological indices play a very important role in the mathematical chemistry. The topological indices are numerical parameters of a graph. The degree sequence is obtained by considering the set of vertex degree of a graph. Graph operators are the ones which are used to obtain another broader graphs. This paper attempts to find degree sequence of vertex–F join operation of graphs for some standard graphs.

Many chemically important graphs can be obtained from simpler graphs by applying different graph operations. Graph operations such as union, sum, Cartesian product, composition and tensor product of graphs are among the important ones. In this paper, we introduce a new invariant which is named as the first general Zagreb coindex and defined as M¯1α(G)=∑uv∈E(G¯)[dG(u)α+dG(v)α]\overline{M}^\alpha_1(G)=\Sigma_{uv\in E(\overline{G})}[d_G(u)^\alpha+d_G(v)^\alpha] , where α ∈ ℝ, α ≠ 0. Here, we study the basic properties of the newly introduced invariant and its behavior under some graph operations such as union, sum, Cartesian product, composition and tensor product of graphs and hence apply the results to find the first general Zagreb coindex of different important nano-structures and molecular graphs.

This paper uses the data of China’s venture capital industry in the CVSource and Zero2IPO databases from 1999 to 2018 as a research sample. By using UCINET software to build the entire network of venture capital industry for social network analysis, I calculate the group tie density based on network interaction, status heterogeneity and faults based on network centrality, to analyse whether those group network characteristics affect the complementarity effect. Based on this, I use probit model regression test to carry on an examination.

The kinetic mechanism of chemical reaction was used to calculate the coupling of the fluid kinetics with the urea decomposition reaction and the SCR reaction kinetics. Combined with the engine test and simulation, the distribution uniformity of the urea injection and ammonia gas was studied. Through numerical simulation on urea spray and exhaust flow in Urea-SCR system, the flow field characteristics in whole after-treatment system are gotten. By using numerical calculation in different urea injection angle and orifice sizes, the urea-crystallization and ammonia distribution have been studied and the optimal urea spray angle and nozzle size are given.

In this paper, we have established certain theorems involving Bailey pairs and WP-Bailey pairs. Further, making use of some known WP-Bailey pairs and theorems for constructing new WP-Bailey pairs, we have also established transformation formulas for q-hypergeometric series.

In this paper, we present a comprehensive analytical study on the perturbed restricted three bodies problem. We formulate the equations of motion of this problem, in the event of the asteroids belt perturbation. We find the locations of equilibrium points (collinear and triangular points) and analysis their linear stability. Furthermore the periodic orbits around both collinear and triangular points are found.

In the present manuscript, Crank Nicolson finite difference method is going to be applied to get the approximate solutions for the fractional Burgers equation. The fractional derivative used in this equation is going to be taken into consideration in the Caputo sense. The L1 type discretization formula is going to be applied to this equation. For checking the efficiency of proposed methods, the error norms L₂ and L_∞ have at the same time been calculated. Those newly got solutions using the presented method illustrate the easy usage and efficiency of the approach presented in this manuscript.

We present a mathematical analysis of a mixed ODE-PDE model describing the spatial distribution and temporal evolution of tumor and normal cells within a tissue subject to the effects of a chemotherapeutic drug. The model assumes that the influx of chemotherapy is restricted to a limited region of the tissue, mimicking a blood vessel passing transversely. We provide results on the existence and uniqueness of the model solution and numerical simulations illustrating different model behaviors.

In this paper we prove the existence of a solution for mean-field reflected backward doubly stochastic differential equations (MF-RBDSDEs) with one continuous barrier and discontinuous generator (left-continuous). By a comparison theorem establish here for MF-RBDSDEs, we provide a minimal or a maximal solution to MF-RBDSDEs.

In this paper, computational algorithm with the aid of Mathematica software is specifically designed for the gravitational two–body problem. Mathematical module is established to find the position and velocity vectors. Application of this module for different kind of orbits (elliptic, parabolic and hyperbolic) leads to accurate results, which proved module efficiency and to be skillful. The classical power series method is to be utilized as the methodology.

The electronically controlled fuel injection system of the diesel engine has multi-disciplinary characteristics and strong coupling. The programming method based on a mathematical model cannot be applied to the fuel system with a diversified structure arrangement. With the rapid development of hydrodynamics, more and more simulation software is applied in the actual research. Hydsim simulation platform is one of the best software for the simulation of the electronic control injection system. In this paper, the Hydsim simulation platform is used to simulate the electronic control injection system, and a test-bed for verification is built. Based on the experimental results, the injection characteristics of high-pressure common rail are studied. The validity and accuracy of the simulation model are verified by comparing the simulation value with the test value and the standard value. Finally, the influence of the structural parameters such as the inlet and outlet orifices and control pistons of the injection system on the injection characteristics is analysed, and the selection principle of each parameter in the system design is proposed, which provides a reliable basis for the structural optimisation.

Packet sets and inverse packet sets are two kinds of novel mathematical tools to analyze dynamic information systems. With advances in inverse packet sets, random inverse packet information is proposed by introducing random characteristics into inverse packet sets. Hence, random inverse packet information has dynamic and random characteristics, and is an extended form of inverse packet sets. Furthermore, random feature, dynamic feature, and identification relation about the random inverse packet information are discussed. Finally, based on the above theory, an instance is used to illustrate the applications of intelligent acquisition of investment information.

The rapid growth of fossil fuels and the aggravation of emission pollution can be improved by the green and diversified energy structure of road transport vehicles. In this article, Cui–Lawson model is first introduced to analyse the change trend of pure electric vehicle, hybrid electric vehicle, gas–fuel vehicle, fuel cell vehicle, biofuel vehicle and traditional vehicle in the next 50 years. Considering the climate, environment and economic development of each region, the future ownership of the above six models was studied by region. Moreover, the climate and topography affect the developing trend strongly. The ownership of traditional automobile is more than other kinds of vehicles until 35 and 40 years in Xibei and Dongbei regions for the cold temperature.

This paper sets up an individual fixed-effect model. Taking environmental information disclosure index and independent report as an alternative to environmental information disclosure, this paper studies the impact of environmental information disclosure on Certified Public Accountants’ audit fees and audit opinions. The results show that the environmental information disclosure level of the Chinese listed companies in the energy industry is positively correlated with the audit fees of CPA, and negatively correlated with the issuance of non-standard audit opinions, but whether the disclosure of independent reports has no significant impact on audit fees.

The burgeoning trend of globalization gives rise to the formation of the manufacturing ecosystem. This study aims to identify the critical factors of sustainable manufacturing for countries and regions across the globe finding their unique ecological niches. From the perspective of the ecological niche, we develop an evaluation system of the manufacturing niche. By using the fuzzy Decision-Making Trial and Evaluation Laboratory (DEMATEL) method, the critical factors, and its causal relationships of the manufacturing niche can be quantified and visualized. The results indicate that: (1) the evaluation system of the manufacturing niche is characterized by complexity and interactivity; (2) technical factors have the strongest impact on the evaluation system, among which R&D investment intensity and the input-output ratio of new products are key indicators; and (3) technical and policy factors are decisive for the system and actively influence economic and ecological factors. Theoretically, it is beneficial to augment the niche theory and industrial economics. Practically, it helps to create a win-win situation to facilitate governments to enact suitable industrial strategies and assist the manufacturing toward a more sustainable trajectory.

Workplace spirituality, job engagement, and perceived organisational support have become the themes over which heated discussion is held in the fields of management and public research. With the development of science and technology and the arrival and upgrading of burgeoning technologies, the competition among enterprises has been aggravated, while the employee's job engagement has become one of the key aspects to keep the competitive edge. By taking workplace spirituality as the premise variable of job engagement and perceived organisational support, the influences of workplace spirituality, perceived organisational support, and job engagement are discussed in this paper. In addition, by taking perceived organisational support as the regulated variable, a theoretical model was built and the regulating effect of perceived organisational support on workplace spirituality and job engagement was discussed. The empirical data of the model was obtained through 187 questionnaires. Data analysis and research hypothesis was implemented through SPSS23.0, theory method, and computational simulation. In sum, workplace spirituality and job involvement are remarkable and are positively associated with each other. Perceived organisational support plays a role in regulating the job engagement. These results provide new perspectives for exploring unemployment problems existing due to lack of job engagement. The organisation can intervene in employees’ work by cultivating workplace spirituality.

In this article, based on the collaborative deep learning (CDL) and convolutional matrix factorisation (ConvMF), the language model BERT is used to replace the traditional word vector construction method, and the bidirectional long–short time memory network Bi-LSTM is used to construct an improved collaborative filtering model BMF, which not only solves the phenomenon of ‘polysemy’, but also alleviates the problem of sparse scoring matrix data. Experiments show that the proposed model is effective and superior to CDL and ConvMF. The trained MSE value is 1.031, which is 9.7% lower than ConvMF.

Traditional mathematics learning always paid excessive attention to the memorisation and repeated practice of theoretical knowledge and ignored guidance on learning method, resulting in bad learning outcome and insufficient problem-solving ability. Based on the theory of ‘chunks’ and ‘objective’, this paper analyses the mathematics learning process from the perspective of experimental simulation, designs a mathematics learning process based on ‘chunk-objective’, and simulates it. The data show that the learning mode is scientific and effective, which can help students learn better according to cognitive rules, thus forming their knowledge structure and improving their ability to solve problems.

This paper mainly proposed and researched based on wavelet transform, and then used the X-map denoising technique of value filter. In other words, the value image was filtered in the spatial domain, and the value filtering was used as the standard pulse (salt) noise, also used as in the wavelet domain. After the filtered image was decomposed by biorthogonal double wavelet transform, a wavelet coefficient matrix was generated, and a soft threshold quantisation process was performed on the wavelet coefficients to produce a new wavelet coefficient matrix. In the end, they used a new wavelet coefficient matrix for image reconstruction. The processing resulted that the denoising method proposed in this paper showed that the X image can be denoised, which not only reduced the X-picture-like noise but also preserved the X-picture-like details as much as possible. It also helped to enhance diagnostic accuracy and reduced the difference in reading.

The scale and complexity of health information service system has increased dramatically, and its development activities and management are difficult to control. In the field of, Traditional methods and simple mathematical statistics methods are difficult to solve the problems caused by the explosive growth of data and information, which will adversely affect health information service system management finally. So, it is particularly important to find valuable information from the source code, design documents and collected software datasets and to guide the development and maintenance of software engineering. Therefore, some experts and scholars want to use mature data mining technologies to study the large amount of data generated in software engineering projects (commonly referred to as software knowledge base), and further explore the potential and valuable information inherently hidden behind the software data. This article initially gives a brief overview of the relevant knowledge of data mining technology and computer software technology, using decision tree graph mining algorithm to mine the function adjustment graph of the software system definition class, and then source code annotations are added to the relevant calling relationships. Data mining technology and computer software technology are deeply integrated, and the decision tree algorithm in data mining is used to mine the knowledge base of computer software. Potential defect changes are listed as key maintenance objects. The historical versions of source code change files with defects are found dynamically and corrected in time, to avoid the increase of maintenance cost in the future.

This paper extends the use of network theory to the study of region network modeling between China and five central Asian countries (Kazakhstan, Kirghiz Tanzania, Tajikistan, Turkmenistan, Uzbekistan) using the DMSP/OLS nighttime light data. In this paper, first, regional scale index and region interaction index are designed and calculated. Then based on these indices, the 0–1 region incidence matrix is obtained and the region network is constructed. Furthermore, the regional economic activity is identified by using the region network. This study deepens the application of complex network in the field of regional cooperation.

In this paper, the CAViaR model was used to measure the HS300, HSI and N225 risk levels and result showed that CAViaR model was able to measure the sample risk value at the 5% confidence level. However, at the 1% confidence level, the CAViaR model severely underestimated the sample risk value. In order to solve this question, this paper introduces the POT model and CAViaR model to construct the extreme risk measure model based on POT-CAViaR model and the test shows that the improved model can effectively improve the accuracy of extreme risk prediction and the model validity is strengthened.

This paper described the volatility characteristic of the rate of return of financial asset by using QR-GARCH model, through introducing EVT model and constructing the extreme risk measure model based on QR-GARCH-EVT. In this paper, HS300 index data test was applied to show that under 5% significance level, and QR-GARCH-EVT model can effectively measure the risk value of the sample, but under 1% significance level. QR-GARCH-EVT model will underestimate the risk value of the sample to a certain degree, but generally speaking, compared with other models, the risk value measured by QR-GARCH-EVT model has a higher accuracy to enhance effectiveness.

This paper uses the interval quadratic preference loss function to establish the threshold model of exchange rate volatility by Taylor series expansion, and GMM method to study the intervention behavior of exchange rate. It found that the central bank has asymmetric intervention preference for exchange rate, discontinuous and nonlinear intervention for exchange rate, and there is an intervention threshold, and the central bank has a certain degree of “fear of appreciation” trend. To some extent, asymmetric interval intervention preference has resulted in the rapid growth of China's foreign exchange reserves.

With the rapid development of online shopping, how to explore the value of online reviews, so as to give full play to their role in potential users’ purchasing decisions. Based on text mining and quantitative analysis, this paper studies the sentiment analysis of online reviews on B2C shopping website. The main attributes of commodity or service are extracted based on the order of word frequency in the online reviews. Text analysis method is used to judge the relationship between attributes of commodity or service and its emotional words. The fine-grained sentimental polarity and intensity of attributes are identified to analyze users’ concerns and preferences. The research shows that users pay more attention to the configuration and after-sales service of mobile, and have a positive sentimental orientation to most of attributes, especially unlocking function, hand feeling attribute and logistics service; and have a neutral sentimental orientation towards the attributes of battery and memory, and a negative sentimental orientation towards the membrane of mobile phone. The results can provide a reference for consumers to make purchasing decisions, for enterprises to improve product quality, and for shopping platform to optimize service.

The mathematical algorithm of zero order optimization is a general function approach optimization method. Based on this method, many sets of analytical procedures are developed using ANSYS to study the design of 600m scale concrete arch bridge and the optimal arch axis coefficient, change mode and cross section sizes are obtained and verified. Results show: the mathematical algorithm of zero order optimization can effectively study the design of 600m arch bridge. The rich conclusions about design parameters can be a significant reference for design and further research of 600 m scale concrete arch bridge.

With the continuous development of China's Internet economy, the model of grasping Internet economy benefit is diversified. Since 2016, the online celebrity economy has formed a trend of Internet marketing, and the online celebrity economic industry chain further has established along with the further development of MCN institutions. The relationship among the overall industry upstream and downstream sectors become closer and closer and even appear the trend of vertical cooperation. This paper is aimed at expressing the pricing mechanism of the online celebrity industrial chain and applies game theory and mathematical analysis to propose a reasonable pricing model

Photovoltaic power system is taking a significant percentage of power system and the demands for accurate forecasting of the power outputs is surging. In prior works, the forecasting problem was formulated as a regression problem, however, which most cannot guarantee that the forecasted outputs is nonnegative. To solve this problem, we proposed a novel probabilistic model by using nonlinear regression and Bayesian learning method. In the paper, we present the detailed theoretical derivations and interpretations. The simulation results show the validity and feasibility of the proposed algorithm by comparing with the traditional SVM algorithm.

Local officials play active and important roles in the high-quality agricultural development process in China. In this study, the DEA-SBM model was used to measure the level of agricultural quality development in China, and Tobit regression was conducted to analyze the impacts of officials on agricultural quality development in China. The results showed that between 1997 and 2016, the agricultural GTFP exhibited a downward trend before following an upward trend in China. The personal characteristics of officials had important effects on the development of agricultural quality. In different development stages, officials had diverse effects on agricultural quality development.

This paper studies Inventory location routing problem in supply chain distribution network planning under vendor inventory management while considering customer inventory holding cost. In order to minimize the total cost of supply chain, an optimization model is established and an improved tabu search algorithm is used to solve the problem. From the analysis, it shows that the total cost decreases as the total vehicles capacity increases, and the maximum utilization of alternative vehicles and the minimum cost of the system don’t occur at the same time in some cases.

How to reduce the hardware cost and high power consumption of RF link of communication device is the key problem to be solved for multi-transmitting antenna and multi-receiving antenna system (MIMO). Always choose the best antennas connection a limited number of RF circuits, which is called antenna selection technology (AS), are a perfect solution to the problem, Assuming that the spatial range of the antenna meets the requirements of signal multiplexing and based on the maximum capacity criterion of the selected MIMO system, the manuscript proposes a low computational complexity (CC) and high performance joint transmitting and receiving antenna selection technique (JTRAS). Starting from the traditional capacity formula and the full matrix of MIMO channel, we utilize a simplified channel capacity expression through repeatedly iterating to delete a row and a column of the equivalent decrement channel matrix, which is to remove a pair of transmitting and receiving antennas. Based on the decreasing JTRAS (DJTRAS) algorithm, the capacity results of simulating calculation indicate that its median capacity overtakes other ones, such as optimum selection (OS), AS based on Frobenius 2 norm (NBS), and concise joint AS criterion (CJAS) etc., and the novel DJTRAS scheme can significantly reduce computational complexity (CC) compared to the exhaustive search method with maximum capacity, which defined as optimal algorithm in the curve graphs. This new technology of the AS is particularly suited to large number of selected antennas, such as L_t ≥ N_T/2,L_r ≥ N_R/2.

This paper uses the law of one price (LOP) and the DCC-GARCH method based on ten-day price sequences. The findings indicate that, compared with the international refined oil markets with mature market-oriented pricing mechanisms, only in the Chinese market do gasoline and diesel prices meet the LOP. This finding shows that, in the context of the gradual integration of the global refined oil market, the international level of China's refined oil price is still quite low. The price reforms pertaining to China's refined oil products still need to be pushed forward in the direction of marketization and internationalization.

In the present paper, we establish some compositions formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with k-Struve function Sν,ckS_{\nu ,c}^k as of the kernel. The results are presented in terms of generalized k-Wright function pΨqk_p\Psi _q^k .

The behavior of a viscous incompressible fluid on a rotating sphere is described by the nonlinear barotropic vorticity equation (BVE). Conditions for the existence of a bounded set that attracts all BVE solutions are given. In addition, sufficient conditions are obtained for a BVE solution to be a global attractor. It is shown that, in contrast to the stationary forcing, the dimension of the global BVE attractor under quasiperiodic forcing is not limited from above by the generalized Grashof number.

This article highlights particular mixed-mode oscillations (MMO) based on canard explosion observed in a fractional-order Fitzhugh-Nagumo (FFHN) model. In order to rigorously analyze the dynamics of the FFHN model, a recently introduced mathematical notion, the Hopf-like bifurcation (HLB), which provides a precise definition for the change between a fixed point and an S−asymptotically T−periodic solution, is used. The existence of HLB in this FFHN model is proved and the appearance of MMO based on canard explosion in the neighborhoods of such HLB points are numerically investigated using a new algorithm: the global-local canard explosion search algorithm. This MMO is constituted of various patterns of solutions with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied simultaneously. On the basis of such numerical experiment, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional-order close to one. Therefore, this very simple two-dimensional FFHN model, presents an incredible ability to mimic the complex dynamics of neurons.

For a collection of infinite loaded graphs, random perturbations of special type are considered. It is shown that some known classes of these graphs are stable with respect to small random perturbations of this type, while the rest are not.

In the present paper we construct an example of 4-dimensional flows on 𝕊³ × 𝕊¹ whose saddle periodic orbit has a wildly embedded 2-dimensional unstable manifold. We prove that such a property has every suspension under a non-trivial Pixton's diffeomorphism. Moreover we give a complete topological classification of these suspensions.

The article continues a series of works studying cylindrical transformations having discrete orbits (Besicovitch cascades). For any γ ∈ (0,1) and any ɛ > 0 we construct a Besicovitch cascade over some rotation with bounded partial quotients, and with a γ–Hölder function, such that the Hausdorff dimension of the set of points in the circle having discrete orbits is greater than 1 − γ− ɛ.

The purpose of this article is to review the author's results on the existence and structure of minimal sets and attractors of conformal foliations. Results on strong transversal equivalence of conformal foliations are also presented. Connections with works of other authors are indicated. Examples of conformal foliations with exceptional, exotic and regular minimal sets which are attractors are constructed.

We study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps T_c,ɛ(x) = (−1 + c|x|^1−ɛ) · sgn(x). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to the birth of Lorenz attractor (when the saddle value becomes positive). We indicate those regions in the bifurcation plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.

We prove that given any closed 3-manifold M³, there is an A-flow f^t on M³ such that the non-wandering set NW (f^t) consists of 2-dimensional non-orientable expanding attractor and trivial basic sets.

Let X be a dendrite, f : X → X be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω-limit set ω(x, f ) of any point x ∈ X has the next properties: (1)

ω(x,f)⊆Per(f)¯\omega (x,f) \subseteq \overline {Per(f)} , where Per( f ) is the set of periodic points of f ;

In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that (3)

Ω(f)=Per(f)¯\Omega (f) = \overline {Per(f)} , where Ω( f ) is the set of non-wandering points of f.

The aim of this note is to show that the above results (1) – (3) do not hold for monotone maps on dendroids.

In this paper we investigate stability of the integrability property of skew products of interval maps under small C¹-smooth perturbations satisfying some conditions. We obtain here (sufficient) conditions of the partial integrability for maps under considerations. These conditions are formulated in the terms of properties of the unperturbed skew product. We give also the example of the partially integrable map.

We study the class of finite additive shift invariant measures on the real separable Hilbert space E. For any choice of such a measure we consider the Hilbert space ℋ of complex-valued functions which are square-integrable with respect to this measure. Some analogs of Sobolev spaces of functions on the space E are introduced. The analogue of Gauss theorem is obtained for the simplest domains such as the rectangle in the space E. The correctness of the problem for Poisson equation in the rectangle with homogeneous Dirichlet condition is obtained and the variational approach of the solving of this problem is constructed.