In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations an...In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic.In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive(implicative)derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97-113] and [Information Sciences, 178(2008),307-316].展开更多
Lattice implication algebras is an algebraic structure which is established by combining lattice and implication algebras. In this paper,the relationship between lattice implication algebras and MV algebra was discuss...Lattice implication algebras is an algebraic structure which is established by combining lattice and implication algebras. In this paper,the relationship between lattice implication algebras and MV algebra was discussed,and then proved that both of the categorys of the two algebras are categorical equivalence. Finally,the infinitely distributivity in lattice implication algebras were proved.展开更多
In this paper,X is a locally compact Hausdorff space and A is a Banach algebra.First,we study some basic features of C0(X,A)related to BSE concept,which are gotten from A.In particular,we prove that if C0(X,A)has the ...In this paper,X is a locally compact Hausdorff space and A is a Banach algebra.First,we study some basic features of C0(X,A)related to BSE concept,which are gotten from A.In particular,we prove that if C0(X,A)has the BSE property then A has so.We also establish the converse of this result,whenever X is discrete and A has the BSE-norm property.Furthermore,we prove the same result for the BSE property of type I.Finally,we prove that C0(X,A)has the BSE-norm property if and only if A has so.展开更多
Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bound...Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L<sup>∞</sup>(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L<sup>2</sup>(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .展开更多
The most crucial requirement in radiation therapy treatment planning is a fast and accurate treatment planning system that minimizes damage to healthy tissues surrounding cancer cells. The use of Monte Carlo toolkits ...The most crucial requirement in radiation therapy treatment planning is a fast and accurate treatment planning system that minimizes damage to healthy tissues surrounding cancer cells. The use of Monte Carlo toolkits has become indispensable for research aimed at precisely determining the dose in radiotherapy. Among the numerous algorithms developed in recent years, the GAMOS code, which utilizes the Geant4 toolkit for Monte Carlo simula-tions, incorporates various electromagnetic physics models and multiple scattering models for simulating particle interactions with matter. This makes it a valuable tool for dose calculations in medical applications and throughout the patient’s volume. The aim of this present work aims to vali-date the GAMOS code for the simulation of a 6 MV photon-beam output from the Elekta Synergy Agility linear accelerator. The simulation involves mod-eling the major components of the accelerator head and the interactions of the radiation beam with a homogeneous water phantom and particle information was collected following the modeling of the phase space. This space was po-sitioned under the X and Y jaws, utilizing three electromagnetic physics mod-els of the GAMOS code: Standard, Penelope, and Low-Energy, along with three multiple scattering models: Goudsmit-Saunderson, Urban, and Wentzel-VI. The obtained phase space file was used as a particle source to simulate dose distributions (depth-dose and dose profile) for field sizes of 5 × 5 cm<sup>2</sup> and 10 × 10 cm<sup>2</sup> at depths of 10 cm and 20 cm in a water phantom, with a source-surface distance (SSD) of 90 cm from the target. We compared the three electromagnetic physics models and the three multiple scattering mod-els of the GAMOS code to experimental results. Validation of our results was performed using the gamma index, with an acceptability criterion of 3% for the dose difference (DD) and 3 mm for the distance-to-agreement (DTA). We achieved agreements of 94% and 96%, respectively, between simulation and experimentation for the three electromagnetic physics models and three mul-tiple scattering models, for field sizes of 5 × 5 cm<sup>2</sup> and 10 × 10 cm<sup>2</sup> for depth-dose curves. For dose profile curves, a good agreement of 100% was found between simulation and experimentation for the three electromagnetic physics models, as well as for the three multiple scattering models for a field size of 5 × 5 cm<sup>2</sup> at 10 cm and 20 cm depths. For a field size of 10 × 10 cm<sup>2</sup>, the Penelope model dominated with 98% for 10 cm, along with the three multiple scattering models. The Penelope model and the Standard model, along with the three multiple scattering models, dominated with 100% for 20 cm. Our study, which compared these different GAMOS code models, can be crucial for enhancing the accuracy and quality of radiotherapy, contributing to more effective patient treatment. Our research compares various electro-magnetic physics models and multiple scattering models with experimental measurements, enabling us to choose the models that produce the most reli-able results, thereby directly impacting the quality of simulations. This en-hances confidence in using these models for treatment planning. Our re-search consistently contributes to the progress of Monte Carlo simulation techniques in radiation therapy, enriching the scientific literature.展开更多
Some relationships between the representation of Hom-Jacobi-Jordan algebra and that of Jacobi-Jordan algebra are studied.Moreover,by using the notion ofαk-anti-derivation,a property theorem of multiplicative Hom-Jaco...Some relationships between the representation of Hom-Jacobi-Jordan algebra and that of Jacobi-Jordan algebra are studied.Moreover,by using the notion ofαk-anti-derivation,a property theorem of multiplicative Hom-Jacobi-Jordan algebras is also given.展开更多
In this paper,we consider the algebraic structure of derivative Hardy Spaces.By using the method of[6,12,15],we get the Duhamel product forming Banach algebra in derivative Hardy Spaces,and invertibility criterion,and...In this paper,we consider the algebraic structure of derivative Hardy Spaces.By using the method of[6,12,15],we get the Duhamel product forming Banach algebra in derivative Hardy Spaces,and invertibility criterion,and describe the extended eigenvalue of the integral operator V.We generalize the results in[1,2,6,11,16].展开更多
In this paper, from the spacetime algebra associated with the Minkowski space ℝ3,1by means of a change of signature, we describe a quaternionic representation of the split-tetraquaternion algebra which incorporates th...In this paper, from the spacetime algebra associated with the Minkowski space ℝ3,1by means of a change of signature, we describe a quaternionic representation of the split-tetraquaternion algebra which incorporates the Pauli algebra, the split-biquaternion algebra and the split-quaternion algebra, we relate these algebras to Clifford algebras and we show the emergence of the stabilized Poincaré-Heisenberg algebra from the split-tetraquaternion algebra. We list without going into details some of their applications in Physics and in Born geometry.展开更多
Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent m...Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence E0,E1,⋯,Es⋯, which leads to the chain complex Es0→Δs0Ess→Δs1⋯→ΔsiEs(i+1)s→Δsi+1⋯of Esby discussing the chain complex E10→Δ10E11→Δ11⋯→Δ1r−1E1r→Δ1r⋯of E1and proves that Es+1i≅Hi(Es)=KerΔsi+1/ImΔsiand therefore Es+1≅H(Es)by the chain complex of Es(see Theorem 2).展开更多
Using the theory of derivations on finitely generated and graded Lie algebras, we determine that derivations of the BMS-Weyl algebra are all inner. On this basis, it is proved that every 2-local derivation of the BMS-...Using the theory of derivations on finitely generated and graded Lie algebras, we determine that derivations of the BMS-Weyl algebra are all inner. On this basis, it is proved that every 2-local derivation of the BMS-Weyl algebra is a derivation.展开更多
Some properties of BZMV dM -algebra are proved, and a new operator is introduced. It is shown that the substructure of BZMV dM -algebra can produce a quasi-lattice implication algebra. The relations between BZMV...Some properties of BZMV dM -algebra are proved, and a new operator is introduced. It is shown that the substructure of BZMV dM -algebra can produce a quasi-lattice implication algebra. The relations between BZMV dM -algebra and other algebras are discussed in detail. A pseudo-distance function is defined in linear BZMV dM -algebra, and its properties are derived.展开更多
The current article intends to introduce the reader to the concept of injective and projective modules and to describe the CFT. We present a clear view to show the homological algebra and injective and projective modu...The current article intends to introduce the reader to the concept of injective and projective modules and to describe the CFT. We present a clear view to show the homological algebra and injective and projective modules.展开更多
The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers ...The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers would only provide dramatic speedups for a few specific problems, for example, factoring integers and breaking cryptographic codes in the conventional quantum computing approach. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In the conventional approach, it is implemented through the tensor product of qubits. In the suggested geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on a three-dimensional sphere, which is very different from the usual Hilbert space scheme.展开更多
In this paper,we first give the general forms of skew commuting maps and skew anti-commuting maps by the Peirce decomposition on a unital ring with a nontrivial idempotent,respectively,and then,as applications,we obta...In this paper,we first give the general forms of skew commuting maps and skew anti-commuting maps by the Peirce decomposition on a unital ring with a nontrivial idempotent,respectively,and then,as applications,we obtain the concrete characterizations of all nonadditive skew(anti-)commuting maps on some operator algebras.展开更多
基金Supported by a grant of National Natural Science Foundation of China(12001243,61976244,12171294,11961016)the Natural Science Basic Research Plan in Shaanxi Province of China(2020JQ-762,2021JQ-580)。
文摘In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic.In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive(implicative)derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97-113] and [Information Sciences, 178(2008),307-316].
文摘Lattice implication algebras is an algebraic structure which is established by combining lattice and implication algebras. In this paper,the relationship between lattice implication algebras and MV algebra was discussed,and then proved that both of the categorys of the two algebras are categorical equivalence. Finally,the infinitely distributivity in lattice implication algebras were proved.
文摘In this paper,X is a locally compact Hausdorff space and A is a Banach algebra.First,we study some basic features of C0(X,A)related to BSE concept,which are gotten from A.In particular,we prove that if C0(X,A)has the BSE property then A has so.We also establish the converse of this result,whenever X is discrete and A has the BSE-norm property.Furthermore,we prove the same result for the BSE property of type I.Finally,we prove that C0(X,A)has the BSE-norm property if and only if A has so.
文摘Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L<sup>∞</sup>(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L<sup>2</sup>(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .
文摘The most crucial requirement in radiation therapy treatment planning is a fast and accurate treatment planning system that minimizes damage to healthy tissues surrounding cancer cells. The use of Monte Carlo toolkits has become indispensable for research aimed at precisely determining the dose in radiotherapy. Among the numerous algorithms developed in recent years, the GAMOS code, which utilizes the Geant4 toolkit for Monte Carlo simula-tions, incorporates various electromagnetic physics models and multiple scattering models for simulating particle interactions with matter. This makes it a valuable tool for dose calculations in medical applications and throughout the patient’s volume. The aim of this present work aims to vali-date the GAMOS code for the simulation of a 6 MV photon-beam output from the Elekta Synergy Agility linear accelerator. The simulation involves mod-eling the major components of the accelerator head and the interactions of the radiation beam with a homogeneous water phantom and particle information was collected following the modeling of the phase space. This space was po-sitioned under the X and Y jaws, utilizing three electromagnetic physics mod-els of the GAMOS code: Standard, Penelope, and Low-Energy, along with three multiple scattering models: Goudsmit-Saunderson, Urban, and Wentzel-VI. The obtained phase space file was used as a particle source to simulate dose distributions (depth-dose and dose profile) for field sizes of 5 × 5 cm<sup>2</sup> and 10 × 10 cm<sup>2</sup> at depths of 10 cm and 20 cm in a water phantom, with a source-surface distance (SSD) of 90 cm from the target. We compared the three electromagnetic physics models and the three multiple scattering mod-els of the GAMOS code to experimental results. Validation of our results was performed using the gamma index, with an acceptability criterion of 3% for the dose difference (DD) and 3 mm for the distance-to-agreement (DTA). We achieved agreements of 94% and 96%, respectively, between simulation and experimentation for the three electromagnetic physics models and three mul-tiple scattering models, for field sizes of 5 × 5 cm<sup>2</sup> and 10 × 10 cm<sup>2</sup> for depth-dose curves. For dose profile curves, a good agreement of 100% was found between simulation and experimentation for the three electromagnetic physics models, as well as for the three multiple scattering models for a field size of 5 × 5 cm<sup>2</sup> at 10 cm and 20 cm depths. For a field size of 10 × 10 cm<sup>2</sup>, the Penelope model dominated with 98% for 10 cm, along with the three multiple scattering models. The Penelope model and the Standard model, along with the three multiple scattering models, dominated with 100% for 20 cm. Our study, which compared these different GAMOS code models, can be crucial for enhancing the accuracy and quality of radiotherapy, contributing to more effective patient treatment. Our research compares various electro-magnetic physics models and multiple scattering models with experimental measurements, enabling us to choose the models that produce the most reli-able results, thereby directly impacting the quality of simulations. This en-hances confidence in using these models for treatment planning. Our re-search consistently contributes to the progress of Monte Carlo simulation techniques in radiation therapy, enriching the scientific literature.
基金National Natural Science Foundation of China(12071405,11571145)。
文摘Some relationships between the representation of Hom-Jacobi-Jordan algebra and that of Jacobi-Jordan algebra are studied.Moreover,by using the notion ofαk-anti-derivation,a property theorem of multiplicative Hom-Jacobi-Jordan algebras is also given.
基金Supported by National Natural Science Foundation of China(11801094).
文摘In this paper,we consider the algebraic structure of derivative Hardy Spaces.By using the method of[6,12,15],we get the Duhamel product forming Banach algebra in derivative Hardy Spaces,and invertibility criterion,and describe the extended eigenvalue of the integral operator V.We generalize the results in[1,2,6,11,16].
文摘In this paper, from the spacetime algebra associated with the Minkowski space ℝ3,1by means of a change of signature, we describe a quaternionic representation of the split-tetraquaternion algebra which incorporates the Pauli algebra, the split-biquaternion algebra and the split-quaternion algebra, we relate these algebras to Clifford algebras and we show the emergence of the stabilized Poincaré-Heisenberg algebra from the split-tetraquaternion algebra. We list without going into details some of their applications in Physics and in Born geometry.
文摘Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence E0,E1,⋯,Es⋯, which leads to the chain complex Es0→Δs0Ess→Δs1⋯→ΔsiEs(i+1)s→Δsi+1⋯of Esby discussing the chain complex E10→Δ10E11→Δ11⋯→Δ1r−1E1r→Δ1r⋯of E1and proves that Es+1i≅Hi(Es)=KerΔsi+1/ImΔsiand therefore Es+1≅H(Es)by the chain complex of Es(see Theorem 2).
基金National Natural Science Foundation of China(11971315)。
文摘Using the theory of derivations on finitely generated and graded Lie algebras, we determine that derivations of the BMS-Weyl algebra are all inner. On this basis, it is proved that every 2-local derivation of the BMS-Weyl algebra is a derivation.
基金SupportedbytheNationalNaturalScienceFoundationofChina (No .69972 0 36)
文摘Some properties of BZMV dM -algebra are proved, and a new operator is introduced. It is shown that the substructure of BZMV dM -algebra can produce a quasi-lattice implication algebra. The relations between BZMV dM -algebra and other algebras are discussed in detail. A pseudo-distance function is defined in linear BZMV dM -algebra, and its properties are derived.
文摘The current article intends to introduce the reader to the concept of injective and projective modules and to describe the CFT. We present a clear view to show the homological algebra and injective and projective modules.
文摘The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers would only provide dramatic speedups for a few specific problems, for example, factoring integers and breaking cryptographic codes in the conventional quantum computing approach. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In the conventional approach, it is implemented through the tensor product of qubits. In the suggested geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on a three-dimensional sphere, which is very different from the usual Hilbert space scheme.
基金supported by the National Natural Science Foundation of China (Nos.12171290,12301152)the Natural Science Foundation of Shanxi Province (No.202203021222018)。
文摘In this paper,we first give the general forms of skew commuting maps and skew anti-commuting maps by the Peirce decomposition on a unital ring with a nontrivial idempotent,respectively,and then,as applications,we obtain the concrete characterizations of all nonadditive skew(anti-)commuting maps on some operator algebras.