Let ΡΥ（X） be the semigroup of all partial transformations on X, Υ（X） and Ι（X） be the subsemigroups of ΡΥ（X） of all full transformations on X and of all injective partial transformations on X, respectivel...Let ΡΥ（X） be the semigroup of all partial transformations on X, Υ（X） and Ι（X） be the subsemigroups of ΡΥ（X） of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let ΡΥ（X, Y） = {α∈ ΡΥ（X） ： Xα Y}, Υ（X, Y） = ΡΥ（X, Y） ∩Υ（X） and Ι（X, Y） = ΡΥ（X, Y） ∩ Ι（X）. In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green＇s relations of Υ（X,Y）. In this paper, we present analogous results for bothΡ Υ（X, Y） and Ι（X, Y）. For a finite set X with ｜x｜≥ 3, the ranks of ΡΥ（X） = ΡΥ（X, X）, Υ（X） = Υ（X, X） and.Ι（X） = Ι（X, X） are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of ΡΥ（X,Y）, Υ（X, Y） and Ι（X, Y） for any proper non-empty subset Y of X.展开更多
A new loop algebra and a new Lax pair are constructed, respectively. It follows that the integrable coupling of the TC hierarchy of equations, which is also an expanding integrable model, is obtained. Specially, the i...A new loop algebra and a new Lax pair are constructed, respectively. It follows that the integrable coupling of the TC hierarchy of equations, which is also an expanding integrable model, is obtained. Specially, the integrable coupling of the famous KdV equation is presented.展开更多
The paper describes some implementation aspects of an algorithm for approximate solution of the traveling salesman problem based on the construction of convex closed contours on the initial set of points (“cities”) ...The paper describes some implementation aspects of an algorithm for approximate solution of the traveling salesman problem based on the construction of convex closed contours on the initial set of points (“cities”) and their subsequent combination into a closed path (the so-called contour algorithm or “onion husk” algorithm). A number of heuristics related to the different stages of the algorithm are considered, and various variants of the algorithm based on these heuristics are analyzed. Sets of randomly generated points of different sizes (from 4 to 90 and from 500 to 10,000) were used to test the algorithms. The numerical results obtained are compared with the results of two well-known combinatorial optimization algorithms, namely the algorithm based on the branch and bound method and the simulated annealing algorithm. .展开更多
We investigate decomposition of codes and finite languages. A prime decomposition is a decomposition of a code or languages into a concatenation of nontrivial prime codes or languages. A code is prime if it cannot be ...We investigate decomposition of codes and finite languages. A prime decomposition is a decomposition of a code or languages into a concatenation of nontrivial prime codes or languages. A code is prime if it cannot be decomposed into at least two nontrivial codes as the same for the languages. In the paper, a linear time algorithm is designed, which finds the prime decomposition. If codes or finite languages are presented as given by its minimal deterministic automaton, then from the point of view of abstract algebra and graph theory, this automaton has special properties. The study was conducted using system for computational Discrete Algebra GAP. .展开更多
Considering the bundle of 2-jets as a realization of the holomorphic manifold over 3-dimensional nilpotent algebra,the authors introduce a new class of lifts of connections in the bundle of 2-jets which is a generaliz...Considering the bundle of 2-jets as a realization of the holomorphic manifold over 3-dimensional nilpotent algebra,the authors introduce a new class of lifts of connections in the bundle of 2-jets which is a generalization of the complete lifts.展开更多
In this work,we consider a stochastic epidemic model with vaccination,healing and relapse.We prove the existence and the uniqueness of the positive solution.We establish sufficient conditions for the extinction and th...In this work,we consider a stochastic epidemic model with vaccination,healing and relapse.We prove the existence and the uniqueness of the positive solution.We establish sufficient conditions for the extinction and the persistence in mean of the stochastic system.Moreover,we also establish sufficient conditions for the existence of ergodic stationary distribution to the model,which reveals that the infectious disease will persist.The graphical illustrations of the approximate solutions of the stochastic epidemic model have been performed.展开更多
Existing pre-trained models like Distil HuBERT excel at uncovering hidden patterns and facilitating accurate recognition across diverse data types, such as audio and visual information. We harnessed this capability to...Existing pre-trained models like Distil HuBERT excel at uncovering hidden patterns and facilitating accurate recognition across diverse data types, such as audio and visual information. We harnessed this capability to develop a deep learning model that utilizes Distil HuBERT for jointly learning these combined features in speech emotion recognition (SER). Our experiments highlight its distinct advantages: it significantly outperforms Wav2vec 2.0 in both offline and real-time accuracy on RAVDESS and BAVED datasets. Although slightly trailing HuBERT’s offline accuracy, Distil HuBERT shines with comparable performance at a fraction of the model size, making it an ideal choice for resource-constrained environments like mobile devices. This smaller size does come with a slight trade-off: Distil HuBERT achieved notable accuracy in offline evaluation, with 96.33% on the BAVED database and 87.01% on the RAVDESS database. In real-time evaluation, the accuracy decreased to 79.3% on the BAVED database and 77.87% on the RAVDESS database. This decrease is likely a result of the challenges associated with real-time processing, including latency and noise, but still demonstrates strong performance in practical scenarios. Therefore, Distil HuBERT emerges as a compelling choice for SER, especially when prioritizing accuracy over real-time processing. Its compact size further enhances its potential for resource-limited settings, making it a versatile tool for a wide range of applications.展开更多
With the development of edge devices and cloud computing,the question of how to accomplish machine learning and optimization tasks in a privacy-preserving and secure way has attracted increased attention over the past...With the development of edge devices and cloud computing,the question of how to accomplish machine learning and optimization tasks in a privacy-preserving and secure way has attracted increased attention over the past decade.As a privacy-preserving distributed machine learning method,federated learning(FL)has become popular in the last few years.However,the data privacy issue also occurs when solving optimization problems,which has received little attention so far.This survey paper is concerned with privacy-preserving optimization,with a focus on privacy-preserving data-driven evolutionary optimization.It aims to provide a roadmap from secure privacy-preserving learning to secure privacy-preserving optimization by summarizing security mechanisms and privacy-preserving approaches that can be employed in machine learning and optimization.We provide a formal definition of security and privacy in learning,followed by a comprehensive review of FL schemes and cryptographic privacy-preserving techniques.Then,we present ideas on the emerging area of privacy-preserving optimization,ranging from privacy-preserving distributed optimization to privacy-preserving evolutionary optimization and privacy-preserving Bayesian optimization(BO).We further provide a thorough security analysis of BO and evolutionary optimization methods from the perspective of inferring attacks and active attacks.On the basis of the above,an in-depth discussion is given to analyze what FL and distributed optimization strategies can be used for the design of federated optimization and what additional requirements are needed for achieving these strategies.Finally,we conclude the survey by outlining open questions and remaining challenges in federated data-driven optimization.We hope this survey can provide insights into the relationship between FL and federated optimization and will promote research interest in secure federated optimization.展开更多
For decades, Lychrel numbers have been studied on many bases. Their existence has been proven in base 2, 11 or 17. This paper presents a probabilistic proof of the existence of Lychrel number in base 10 and provides s...For decades, Lychrel numbers have been studied on many bases. Their existence has been proven in base 2, 11 or 17. This paper presents a probabilistic proof of the existence of Lychrel number in base 10 and provides some properties which enable a mathematical extraction of new Lychrel numbers from existing ones. This probabilistic approach has the advantage of being extendable to other bases. The results show that palindromes can also be Lychrel numbers.展开更多
A semigroup S with a sub-band B is called a good B-quasi-Ehresmann semigroup if it is a Bsemiabundant semigroup satisfying the congruence condition such that B((ab)+)abB((ab)*)aB(a*)B(b+)b for all a,b ∈S.We show that...A semigroup S with a sub-band B is called a good B-quasi-Ehresmann semigroup if it is a Bsemiabundant semigroup satisfying the congruence condition such that B((ab)+)abB((ab)*)aB(a*)B(b+)b for all a,b ∈S.We show that every good B-quasi-Ehresmann semigroup has a global representation and a standard representation.As a special case,the structure of good quasi-adequate semigroups is described.展开更多
An embedding from a group algebra to a matrix algebra is given in this paper. By using it, a criterion for an invertible element in a group algebra is proven.
Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in com...Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the same time, it is still a subject of ongoing research.This paper presents a selection of open problems in this context. The authors would be curious to hear about any substantial progress on any of these problems.展开更多
This paper investigates a solution technique for solving a two-dimensional Kuramoto-Sivashinsky equation discretized using a finite difference method. It consists of an order reduction method into a coupled system of ...This paper investigates a solution technique for solving a two-dimensional Kuramoto-Sivashinsky equation discretized using a finite difference method. It consists of an order reduction method into a coupled system of second-order equations, and to formulate the fully discretized, implicit time-marched system as a Lyapunov- Sylvester matrix equation. Convergence and stability is examined using Lyapunov criterion and manipulating generalized Lyapunov-Sylvester operators. Some numerical implementations are provided at the end to validate the theoretical results.展开更多
Taking into account the increasing volume of text documents,automatic summarization is one of the important tools for quick and optimal utilization of such sources.Automatic summarization is a text compression process...Taking into account the increasing volume of text documents,automatic summarization is one of the important tools for quick and optimal utilization of such sources.Automatic summarization is a text compression process for producing a shorter document in order to quickly access the important goals and main features of the input document.In this study,a novel method is introduced for selective text summarization using the genetic algorithm and generation of repetitive patterns.One of the important features of the proposed summarization is to identify and extract the relationship between the main features of the input text and the creation of repetitive patterns in order to produce and optimize the vector of the main document features in the production of the summary document compared to other previous methods.In this study,attempts were made to encompass all the main parameters of the summary text including unambiguous summary with the highest precision,continuity and consistency.To investigate the efficiency of the proposed algorithm,the results of the study were evaluated with respect to the precision and recall criteria.The results of the study evaluation showed the optimization the dimensions of the features and generation of a sequence of summary document sentences having the most consistency with the main goals and features of the input document.展开更多
This paper aims to introduce the novel concept of the bipolar picture fuzzy set(BPFS)as a hybrid structure of bipolar fuzzy set(BFS)and picture fuzzy set(PFS).BPFS is a new kind of fuzzy sets to deal with bipolarity(b...This paper aims to introduce the novel concept of the bipolar picture fuzzy set(BPFS)as a hybrid structure of bipolar fuzzy set(BFS)and picture fuzzy set(PFS).BPFS is a new kind of fuzzy sets to deal with bipolarity(both positive and negative aspects)to each membership degree(belonging-ness),neutral membership(not decided),and non-membership degree(refusal).In this article,some basic properties of bipolar picture fuzzy sets(BPFSs)and their fundamental operations are introduced.The score function,accuracy function and certainty function are suggested to discuss the comparability of bipolar picture fuzzy numbers(BPFNs).Additionally,the concept of new distance measures of BPFSs is presented to discuss geometrical properties of BPFSs.In the context of BPFSs,certain aggregation operators(AOs)named as“bipolar picture fuzzy weighted geometric(BPFWG)operator,bipolar picture fuzzy ordered weighted geometric(BPFOWG)operator and bipolar picture fuzzy hybrid geometric(BPFHG)operator”are defined for information aggregation of BPFNs.Based on the proposed AOs,a new multicriteria decision-making(MCDM)approach is proposed to address uncertain real-life situations.Finally,a practical application of proposed methodology is also illustrated to discuss its feasibility and applicability.展开更多
We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-tr...We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-triple as the minimum reduced modulus of the mapping Q（a）. It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW^＊-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C^＊-algebras and von Neumann algebras are also studied.展开更多
Suppose that A is an n×n complex matrix. Denote by G(A)the contragradient Lie algebra corresponding to the matrix A. If A is a generalized Cartan matrix, we call G(A) a Kac-Moody Lie algebra.
Let H be a Hopf algebra over a field with bijective antipode, and A a commutative cleft right H-comodule algebra. In this paper, we investigate the ho-mological dimensions and the semisimplicity of the category of rel...Let H be a Hopf algebra over a field with bijective antipode, and A a commutative cleft right H-comodule algebra. In this paper, we investigate the ho-mological dimensions and the semisimplicity of the category of relative Hopf modulesAMH.展开更多
文摘Let ΡΥ（X） be the semigroup of all partial transformations on X, Υ（X） and Ι（X） be the subsemigroups of ΡΥ（X） of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let ΡΥ（X, Y） = {α∈ ΡΥ（X） ： Xα Y}, Υ（X, Y） = ΡΥ（X, Y） ∩Υ（X） and Ι（X, Y） = ΡΥ（X, Y） ∩ Ι（X）. In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green＇s relations of Υ（X,Y）. In this paper, we present analogous results for bothΡ Υ（X, Y） and Ι（X, Y）. For a finite set X with ｜x｜≥ 3, the ranks of ΡΥ（X） = ΡΥ（X, X）, Υ（X） = Υ（X, X） and.Ι（X） = Ι（X, X） are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of ΡΥ（X,Y）, Υ（X, Y） and Ι（X, Y） for any proper non-empty subset Y of X.
文摘A new loop algebra and a new Lax pair are constructed, respectively. It follows that the integrable coupling of the TC hierarchy of equations, which is also an expanding integrable model, is obtained. Specially, the integrable coupling of the famous KdV equation is presented.
文摘The paper describes some implementation aspects of an algorithm for approximate solution of the traveling salesman problem based on the construction of convex closed contours on the initial set of points (“cities”) and their subsequent combination into a closed path (the so-called contour algorithm or “onion husk” algorithm). A number of heuristics related to the different stages of the algorithm are considered, and various variants of the algorithm based on these heuristics are analyzed. Sets of randomly generated points of different sizes (from 4 to 90 and from 500 to 10,000) were used to test the algorithms. The numerical results obtained are compared with the results of two well-known combinatorial optimization algorithms, namely the algorithm based on the branch and bound method and the simulated annealing algorithm. .
文摘We investigate decomposition of codes and finite languages. A prime decomposition is a decomposition of a code or languages into a concatenation of nontrivial prime codes or languages. A code is prime if it cannot be decomposed into at least two nontrivial codes as the same for the languages. In the paper, a linear time algorithm is designed, which finds the prime decomposition. If codes or finite languages are presented as given by its minimal deterministic automaton, then from the point of view of abstract algebra and graph theory, this automaton has special properties. The study was conducted using system for computational Discrete Algebra GAP. .
文摘Considering the bundle of 2-jets as a realization of the holomorphic manifold over 3-dimensional nilpotent algebra,the authors introduce a new class of lifts of connections in the bundle of 2-jets which is a generalization of the complete lifts.
文摘In this work,we consider a stochastic epidemic model with vaccination,healing and relapse.We prove the existence and the uniqueness of the positive solution.We establish sufficient conditions for the extinction and the persistence in mean of the stochastic system.Moreover,we also establish sufficient conditions for the existence of ergodic stationary distribution to the model,which reveals that the infectious disease will persist.The graphical illustrations of the approximate solutions of the stochastic epidemic model have been performed.
文摘Existing pre-trained models like Distil HuBERT excel at uncovering hidden patterns and facilitating accurate recognition across diverse data types, such as audio and visual information. We harnessed this capability to develop a deep learning model that utilizes Distil HuBERT for jointly learning these combined features in speech emotion recognition (SER). Our experiments highlight its distinct advantages: it significantly outperforms Wav2vec 2.0 in both offline and real-time accuracy on RAVDESS and BAVED datasets. Although slightly trailing HuBERT’s offline accuracy, Distil HuBERT shines with comparable performance at a fraction of the model size, making it an ideal choice for resource-constrained environments like mobile devices. This smaller size does come with a slight trade-off: Distil HuBERT achieved notable accuracy in offline evaluation, with 96.33% on the BAVED database and 87.01% on the RAVDESS database. In real-time evaluation, the accuracy decreased to 79.3% on the BAVED database and 77.87% on the RAVDESS database. This decrease is likely a result of the challenges associated with real-time processing, including latency and noise, but still demonstrates strong performance in practical scenarios. Therefore, Distil HuBERT emerges as a compelling choice for SER, especially when prioritizing accuracy over real-time processing. Its compact size further enhances its potential for resource-limited settings, making it a versatile tool for a wide range of applications.
基金supported in part by the National Natural Science Foundation of China (62136003,62302147,62103150,62006053,and 62306097)in part by the China Postdoctoral Science Foundation (2021M691012)+1 种基金in part by the Natural Science Foundation of Guangdong Province (2022A1515010443)in part by the National Research,Development and Innovation Fund of Hungary under the Establishment of Competence Centers,Development of Research Infrastructure Programme funding scheme (2019-1.3.1-KK-2019-00011).
文摘With the development of edge devices and cloud computing,the question of how to accomplish machine learning and optimization tasks in a privacy-preserving and secure way has attracted increased attention over the past decade.As a privacy-preserving distributed machine learning method,federated learning(FL)has become popular in the last few years.However,the data privacy issue also occurs when solving optimization problems,which has received little attention so far.This survey paper is concerned with privacy-preserving optimization,with a focus on privacy-preserving data-driven evolutionary optimization.It aims to provide a roadmap from secure privacy-preserving learning to secure privacy-preserving optimization by summarizing security mechanisms and privacy-preserving approaches that can be employed in machine learning and optimization.We provide a formal definition of security and privacy in learning,followed by a comprehensive review of FL schemes and cryptographic privacy-preserving techniques.Then,we present ideas on the emerging area of privacy-preserving optimization,ranging from privacy-preserving distributed optimization to privacy-preserving evolutionary optimization and privacy-preserving Bayesian optimization(BO).We further provide a thorough security analysis of BO and evolutionary optimization methods from the perspective of inferring attacks and active attacks.On the basis of the above,an in-depth discussion is given to analyze what FL and distributed optimization strategies can be used for the design of federated optimization and what additional requirements are needed for achieving these strategies.Finally,we conclude the survey by outlining open questions and remaining challenges in federated data-driven optimization.We hope this survey can provide insights into the relationship between FL and federated optimization and will promote research interest in secure federated optimization.
文摘For decades, Lychrel numbers have been studied on many bases. Their existence has been proven in base 2, 11 or 17. This paper presents a probabilistic proof of the existence of Lychrel number in base 10 and provides some properties which enable a mathematical extraction of new Lychrel numbers from existing ones. This probabilistic approach has the advantage of being extendable to other bases. The results show that palindromes can also be Lychrel numbers.
基金supported by National Basic Research Program of China (Grant No.2003CB317001)Natural Science Foundation of Hunan (Grant No.06JJ20025)+1 种基金Scientific Research Fund of HunanProvincial Education Department (Grant No.09K084)UGC (HK) (Grant No.2160210 (03/04))
文摘A semigroup S with a sub-band B is called a good B-quasi-Ehresmann semigroup if it is a Bsemiabundant semigroup satisfying the congruence condition such that B((ab)+)abB((ab)*)aB(a*)B(b+)b for all a,b ∈S.We show that every good B-quasi-Ehresmann semigroup has a global representation and a standard representation.As a special case,the structure of good quasi-adequate semigroups is described.
文摘An embedding from a group algebra to a matrix algebra is given in this paper. By using it, a criterion for an invertible element in a group algebra is proven.
基金supported by the National Natural Science Foundation of China under Grant No.11501552the President Fund of the Academy of Mathematics and Systems Science,CAS(2014-cjrwlzx-chshsh)+1 种基金a Starting Grant from the Ministry of Education of Chinasupported by the Austrian FWF under Grant Nos.F5004,Y464-N18,and W1214
文摘Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the same time, it is still a subject of ongoing research.This paper presents a selection of open problems in this context. The authors would be curious to hear about any substantial progress on any of these problems.
文摘This paper investigates a solution technique for solving a two-dimensional Kuramoto-Sivashinsky equation discretized using a finite difference method. It consists of an order reduction method into a coupled system of second-order equations, and to formulate the fully discretized, implicit time-marched system as a Lyapunov- Sylvester matrix equation. Convergence and stability is examined using Lyapunov criterion and manipulating generalized Lyapunov-Sylvester operators. Some numerical implementations are provided at the end to validate the theoretical results.
基金I thank a referee for his very useful remarks.Research is partially supported by Grant HERMES CODE 26872,Departamento de Matemticas,Universidad Nacional de Colombia,Bogotá.
文摘The paper contains a classification of derivations of skew PBW-extensions of rings.
文摘Taking into account the increasing volume of text documents,automatic summarization is one of the important tools for quick and optimal utilization of such sources.Automatic summarization is a text compression process for producing a shorter document in order to quickly access the important goals and main features of the input document.In this study,a novel method is introduced for selective text summarization using the genetic algorithm and generation of repetitive patterns.One of the important features of the proposed summarization is to identify and extract the relationship between the main features of the input text and the creation of repetitive patterns in order to produce and optimize the vector of the main document features in the production of the summary document compared to other previous methods.In this study,attempts were made to encompass all the main parameters of the summary text including unambiguous summary with the highest precision,continuity and consistency.To investigate the efficiency of the proposed algorithm,the results of the study were evaluated with respect to the precision and recall criteria.The results of the study evaluation showed the optimization the dimensions of the features and generation of a sequence of summary document sentences having the most consistency with the main goals and features of the input document.
文摘This paper aims to introduce the novel concept of the bipolar picture fuzzy set(BPFS)as a hybrid structure of bipolar fuzzy set(BFS)and picture fuzzy set(PFS).BPFS is a new kind of fuzzy sets to deal with bipolarity(both positive and negative aspects)to each membership degree(belonging-ness),neutral membership(not decided),and non-membership degree(refusal).In this article,some basic properties of bipolar picture fuzzy sets(BPFSs)and their fundamental operations are introduced.The score function,accuracy function and certainty function are suggested to discuss the comparability of bipolar picture fuzzy numbers(BPFNs).Additionally,the concept of new distance measures of BPFSs is presented to discuss geometrical properties of BPFSs.In the context of BPFSs,certain aggregation operators(AOs)named as“bipolar picture fuzzy weighted geometric(BPFWG)operator,bipolar picture fuzzy ordered weighted geometric(BPFOWG)operator and bipolar picture fuzzy hybrid geometric(BPFHG)operator”are defined for information aggregation of BPFNs.Based on the proposed AOs,a new multicriteria decision-making(MCDM)approach is proposed to address uncertain real-life situations.Finally,a practical application of proposed methodology is also illustrated to discuss its feasibility and applicability.
基金I+D MEC Projects No.MTM 2005-02541,MTM 2004-03882Junta de Andalucfa Grants FQM 0199,FQM 0194,FQM 1215the PCI Project No.A/4044/05 of the Spanish AECI
文摘We revise the notion of von Neumann regularity in JB^＊-triples by finding a new characterisation in terms of the range of the quadratic operator Q（a）. We introduce the quadratic conorm of an element a in a JB^＊-triple as the minimum reduced modulus of the mapping Q（a）. It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW^＊-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C^＊-algebras and von Neumann algebras are also studied.
文摘Suppose that A is an n×n complex matrix. Denote by G(A)the contragradient Lie algebra corresponding to the matrix A. If A is a generalized Cartan matrix, we call G(A) a Kac-Moody Lie algebra.
基金The Fundation of Key Research Program (02021029) and the NSF (2004kj352) of Anhui Province, China.
文摘Let H be a Hopf algebra over a field with bijective antipode, and A a commutative cleft right H-comodule algebra. In this paper, we investigate the ho-mological dimensions and the semisimplicity of the category of relative Hopf modulesAMH.
