Group Classification Problem Research Articles

Group Classification of the Unsteady Axisymmetric Boundary Layer Equation

Unsteady equations of flat and axisymmetric boundary layers are considered. For the unsteady axisymmetric boundary layer equation, the problem of group classification is solved. It is shown that the kernel of symmetry operators can be extended by no more than four-dimensional Lie algebra. The kernel of symmetry operators of the unsteady flat boundary layer equation is found and it is shown that it can be extended by no more than a five-dimensional Lie algebra. The non-existence of the unsteady analogue of the Stepanov–Mangler transformation is proved.

Lie symmetries for the cosmological field equations in brane-world gravitywith bulk scalar field

We address the group classification problem for gravitational field equations within the context of brane-world cosmology, considering the presence of a bulk scalar field. Our investigation revolves around a five-dimensional spacetime, with the four-dimensional Friedmann–Lemaître–Robertson–Walker geometry embedded within it. Additionally, we assume that the scalar field exists in this five-dimensional geometry (bulk) and possesses a nonzero mass. The resulting field equations constitute a system of nonlinear partial differential equations. We apply the Lie symmetry condition to identify all functional forms of the scalar field potential, ensuring that the field equations remain invariant under one-parameter point transformations. Consequently, we find that only the exponential potential exhibits Lie symmetries. Finally, the Lie invariants are used to construct similarity transformations, which enable us to derive exact solutions for the system.

FROM GEOMETRY TO ARITHMETIC OF COMPACT HYPERBOLIC COXETER POLYTOPES

We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.

Mapping method of group classification

We revisit the entire framework of group classification of differential equations. After introducing the notion of weakly similar classes of differential equations, we develop the mapping method of group classification for such classes, which generalizes all the versions of this method that have been presented in the literature. The mapping method is applied to group classification of various classes of Kolmogorov equations and of Fokker–Planck equations in the case of space dimension one. The equivalence groupoids and the equivalence groups of these classes are computed. The group classification problems for these classes with respect to the corresponding equivalence groups are reduced to finding all inequivalent solutions of heat equations with inequivalent potentials admitting Lie-symmetry extensions. This reduction allows us to exhaustively solve the group classification problems for the classes of Kolmogorov and Fokker–Planck equations with time-independent coefficients.

Precise Diagnosis of Unknown Fault of High-Speed Train Bogie Using Novel FBM-Net

As the preferred mode of travel, the high-speed train (HST) and its healthy operation have received extensive attention. In the long-term service of HST, the track irregularity and wheel-rail wear may cause all kinds of faults towards the components of bogie, which is the only connection between the train body and the track. Taking into account the unknownness of bogie fault during actual operation, it is inappropriate to simply convert the issue of bogie fault diagnosis to the problem of group classification of known faults, as conducted in almost all reference works. In this paper, aiming at resolving the defect that supervised learning cannot identify unknown categories, the fractional Brownian motion (FBM) is integrated into an one-dimensional convolutional neural network (1D-CNN) to distinguish unknown bogie faults from known ones. The method takes the advantages of both 1D-CNN and FBM. On the one hand, the limitation in extracting feature artificially is broken by using convolution algorithm, and the deep features of original signals are extracted through stacking convolution kernels. On the other hand, different from one-class theory, e.g., one-class support vector machine (OCSVM), FBM brings randomness into the network to make the model sensitive to unknown faults. At last, four diagnosis strategies, i.e., SDE-Net, PSO-SVDD, CNN-LSTM-FCM, and OCSVM-ELM, are introduced to compare with the method proposed to verify the effectiveness and superiority. Experimental results reveal that the Fractional-Brownian-Motion-based Network (FBM-Net) can not only classify known faults efficiently, but also distinguish unknown faults with an accuracy of more than 93%.

RFMonitor: Monitoring smoking behavior of minors using COTS RFID devices

The protection laws of minors in various countries have clearly banned minors from smoking. However, due to various social factors and the psychological conditions of minors, smoking among minors cannot be effectively controlled. A typical example is that many countries ban smoking in public places, and many places have special smoking areas. Previous work designed wearable sensor devices or used wireless signals (e.g., WIFI and Bluetooth etc.) to identify smoking activities. System deployment and implementation are rarely considered in existing system application scenarios. Here, we show that we can use the backscatter signal of the passive RFID tag, combined with the improved radio frequency chromatographic imaging technology, to analyze the RSSI signal of the human smoking action. Therefore, the system is specially designed for application scenarios and can realize the height detection of minors and the recognition of smoking behavior. In the end, related comparative experiments were also carried out to verify the effectiveness and real-time performance of the system. Aiming at the classification problem of different age groups, this paper proposed to distinguish population categories and identify smoking behaviors by analyzing RSSI signals extracted from commercial readers. Finally, the validity and real-time performance of the system are verified by comparative experiments.

Group Classification Invariant Solutions of Burgers' Equation

The aims of the present paper is to solve the problem of the group classification of the general Burgers’ equation u_t=f(x,u) u_x^2+g(x,u)u_xx, where f and g are arbitrary smooth functions of the variables x and u, by using Lie method. The paper is one of the few applications of an algebraic approach to the problem of group classification: We followed the analysis mathematical method using the method of preliminary group classification. A number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained. The result of the work is a wide class of equations summarized in table form.

Symmetry analysis for the 2 + 1 generalized quantum Zakharov-Kuznetsov equation

We solve the group classification problem for the 2 + 1 generalized quantum Zakharov-Kuznetsov equation. Particularly we consider the generalized equation , and the time-dependent Zakharov-Kuznetsov equation . Function and , are determine in order the equations to admit additional Lie symmetries. Finally, we apply the Lie invariants to find similarity solutions for the generalized quantum Zakharov-Kuznetsov equation.

Cohomological induction and uniform measure equivalence

We construct a general cohomological induction isomorphism from a uniform measure equivalence of locally compact, second countable, unimodular groups which, as a special case, yields that the graded cohomology rings of quasi-isometric, connected, simply connected, nilpotent Lie groups are isomorphic. This unifies results of Shalom and Sauer and also provides new insight into the quasi-isometry classification problem for low dimensional nilpotent Lie groups.

Técnicas de visão computacional para classificação de peças

Computer vision is an area of artificial intelligence that has a great impact on various sectors of society. The ability of the machine to see objects of interest in an image and produce a classification or detection response is very important in the context of automation. The computer vision techniques allow, through image processing steps and the use of classifiers, to provide answers to several problems that arise. The objective of this work is to analyze the responsiveness of a specific classifier, the K-nearest neighbors (KNN), for a group classification problem and analyze its performance through hit rate and accuracy parameters. First, four groups of parts were created in a three-dimension (3D) design software and cut on a laser cutting machine, then pictures were taken of each of the parts individually. Later, using the OpenCV libraries, we could carry out the image processing, such as changing the red-green-blue (RGB) pattern to grayscale, binarization, noise correction, and obtaining invariant moments; and in the Scikit-Learn library, the classifier was trained and tested. It was possible to conclude that the classifier, in tests 1 and 2, was able to classify properly the parts groups having recall and precision above 65 and 80%, while, in test 3, it had a recall of 42% and a precision of 55%, showing that, with more analyzed classes, the classifier decreases its efficiency.

LogSum\u2009+\u2009L2 penalized logistic regression model for biomarker selection and cancer classification

Biomarker selection and cancer classification play an important role in knowledge discovery using genomic data. Successful identification of gene biomarkers and biological pathways can significantly improve the accuracy of diagnosis and help machine learning models have better performance on classification of different types of cancer. In this paper, we proposed a LogSum + L2 penalized logistic regression model, and furthermore used a coordinate decent algorithm to solve it. The results of simulations and real experiments indicate that the proposed method is highly competitive among several state-of-the-art methods. Our proposed model achieves the excellent performance in group feature selection and classification problems.

Three Group Classification Problem Approach Based on Fuzzy Goal Programming

In this study, a new fuzzy logic and mathematical programming based model was proposed to solve three-group classification problem. Determination of cut-off value, which corresponds to discrimination axis in classification problems, has importance. Status of the cut-off value such as asymmetric triangle fuzzy number, trapezoid fuzzy number and gauss fuzzy number was examined. The proposed approach displayed better performance when compared to Fisher's Linear Discriminant Function and some mathematical programming-based models by using three group data sets used frequently in the literature.

Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions

The famous Toms effect (1948) consists of a substantial increase of the critical Reynolds number when a small amount of soluble polymer is introduced into water. The most noticeable influence of polymer additives is manifested in the boundary layer near solid surfaces. The task includes the ratio of two characteristic length scales, one of which is the Prandtl scale, and the other is defined as the square root of the normalized coefficient of relaxation viscosity (Frolovskaya and Pukhnachev, 2018) and does not depend on the characteristics of the motion. In the limit case, when the ratio of these two scales tends to zero, the equations of the boundary layer are exactly integrated. One of the goals of the present paper is group analysis of the boundary layer equations in two mathematical models of the flow of aqueous polymer solutions: the second grade fluid (Rivlin and Ericksen, 1955) and the Pavlovskii model (1971). The equations of the plane non-stationary boundary layer in the Pavlovskii model are studied in more details. The equations contain an arbitrary function depending on the longitudinal coordinate and time. This function sets the pressure gradient of the external flow. The problem of group classification with respect to this function is analyzed. All functions for which there is an extension of the kernels of admitted Lie groups are found. Among the invariant solutions of the new model of the boundary layer, a special place is taken by the solution of the stationary problem of flow around a rectilinear plate.

Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations

Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the equivalence generated by its equivalence group. This gives an exhaustive description of its equivalence groupoid. After extending the algebraic method of group classification to non-normalized classes of differential equations, we solve the complete group classification problem for the class under study up to both usual and general point equivalences. The solution includes the complete preliminary group classification of the class and the construction of singular Lie-symmetry extensions, which are not related to subalgebras of the equivalence algebra. The complete preliminary group classification is based on classifying appropriate subalgebras of the entire infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel invariance algebra. The results obtained can be used to construct exact solutions of nonlinear wave and elliptic equations.

Equivalence groupoids and group classification of multidimensional nonlinear Schrödinger equations

We study admissible and equivalence point transformations between generalized multidimensional nonlinear Schrödinger equations and classify Lie symmetries of such equations. We begin with a wide superclass of Schrödinger-type equations, which includes all the other classes considered in the paper. Showing that this superclass is not normalized, we partition it into two disjoint normalized subclasses, which are not related by point transformations. Further constraining the arbitrary elements of the superclass, we construct a hierarchy of normalized classes of Schrödinger-type equations. This gives us an appropriate normalized superclass for the non-normalized class of multidimensional nonlinear Schrödinger equations with potentials and modular nonlinearities and allows us to partition the latter class into three families of normalized subclasses. After a preliminary study of Lie symmetries of nonlinear Schrödinger equations with potentials and modular nonlinearities for an arbitrary space dimension, we exhaustively solve the group classification problem for such equations in space dimension two.

Групповая классификация, инвариантные решения и законы сохранения нелинейного двумерного ортотропного уравнения фильтрации с дробной производной Римана-Лиувилля по времени

Рассматривается нелинейное двумерное ортотропное уравнение фильтрации с дробной производной Римана-Лиувилля по времени. Доказывается, что такое уравнение может допускать группы точечных преобразований только линейно-автономного типа. Решается задача групповой классификации рассматриваемого уравнения по его точечным симметриям относительно коэффициентов пьезопроводности, являющихся функциями квадрата модуля градиента давления. Доказывается, что если порядок дробного дифференцирования меньше единицы, основная допускаемая уравнением группа точечных преобразований является четырехпараметрической и расширяется до пятипараметрической в изотропном случае. Для степенных зависимостей коэффициентов пьезопроводности допускаемая группа становится пятипараметрической в ортотропном случае и шестипараметрической в изотропном случае. Также выделяется специальный вид степенной зависимости коэффициентов, не имеющий аналога в случае классического уравнения фильтрации, при котором происходит дополнительное расширение группы оператором проективного преобразования. Для уравнения с порядком дробного дифференцирования $\alpha \in (1,2)$ размерности всех допускаемых групп оказываются на единицу больше за счет допускаемого оператора, соответствующего сдвигу решения на дополнительное частное решение этого уравнения. На основе проведенной групповой классификации для соответствующих алгебр Ли инфинитезимальных операторов групп точечных преобразований, допускаемых различными видами рассматриваемого нелинейного ортотропного дробно-дифференциального уравнения, выписываются представления инвариантно-групповых решений, соответствующие оптимальным системам двумерных подалгебр. Приводятся примеры уравнений, получающихся в результате симметрийной редукции исходного дробно-дифференциального уравнения, а также некоторые их решения. Доказывается, что рассматриваемое дробно-дифференциальное уравнение фильтрации является нелинейно самосопряженным, что дает возможность строить его законы сохранения. Соответствующие компоненты всех найденных сохраняющихся векторов приводятся в явном виде.

Conservation laws of the equation of one-dimensional shallow water over uneven bottom in Lagrange’s variables

The systems of equations of one-dimensional shallow water over uneven bottom in Euler’s and Lagrange’s variables are considered. The intermediate system of equations is introduced. Hydrodynamic conservation laws of intermediate system of equations are used to find all first order conservation laws of shallow water equations in Lagrange’s variables for all bottom profiles. The obtained conservation laws are compared with the hydrodynamic conservation laws of the system of equations of one-dimensional shallow water over uneven bottom in Euler’s variables. Bottom profiles, providing additional conservation laws, are given. The problem of group classification of contact transformations of the shallow water equation in Lagrange’s variables is solved. First order conservation laws of the shallow water equation in Lagrange’s variables are obtained using Noether’s theorem. In the considered cases, the correspondence between non-divergence symmetries and the original Lagrangian is shown. A similar correspondence is valid for an arbitrary ordinary differential equation of the second order. It is shown that the application of Lagrange’s identity did not find all first order conservation laws of the shallow water equation in Lagrange’s variables.

Contact symmetries and conservation laws of the first order of the equation of one-dimensional shallow water over a rough bottom in Lagrange’s variables

The systems of equations of one-dimensional shallow water over a rough bottom in Euler’s and Lagrange’s variables are considered. The problem of group classification of contact symmetries of the shallow water equation in Lagrange’s variables is solved. First order conservation laws of the shallow water equation in Lagrange’s variables obtained using the Noether’s theorem. Bottom profiles, providing additional conservation laws, are given.

A multi Ability CP-ABE access control scheme for public cloud storage

The main personality of the cryptography structure based on communication with writings of stable figures and private keys [1-4]. Our development is a key encapsulation mechanism (KEM), so long messages can be coded under a short symmetric key. In our response, the writings of the figures and the private keys have stable dimensions and the people in general are directly in the maximum estimate of s. Furthermore, in our plan, the private key generator (PKG) can potentially include new people without modifying the already widespread data (as in the EIB’s plans). We also note that there is no chain of importance between the characters, despite HIBE. The income of the general population is directed at the maximum size of S, and not in the amount of decoding keys that can be transmitted, which is the amount of conceivable characters. In this case, use a simple situation to find out about the group classification and main administration test problems. Think about a source that sends information to a provision of beneficiaries in a multicast session. Session security is supervised by two principles of useful substances: a Group Controller (GC) responsible for confirmation, approval and control and a Key Server (KS). To ensure classification in the middle of the multicast session, the sender (source) shares a mysterious symmetric key with all the individuals in the legitimate collection, called the Traffic Encryption Key (TEK). To multicast a mysterious message, the source encodes the message with the TEK using a symmetric encryption calculation. From previous documents, we look at how to share protected information in the cloud without losing keys. In this article, we present a new digital brand, an SSH key, hash functionality, and major escrow calculations.

Group classification of dynamics equations of self-gravitating gas

In the paper, a group classification problem is solved for a system of equations which describes motion of self-gravitating gas. A parameter in group classification problem is a function which is determined by an equation of state. A kernel of Lie algebras admitted by the system and an algebra of equivalence transformations group are derived. All specifications of the parameter that lead to extensions of the kernel are found.