A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power...A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.展开更多
Let R be a commutative ring and (S, ≤) a strictly totally ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. In this paper we show that if RM is a PS-module, then the module [[MS≤]] of genera...Let R be a commutative ring and (S, ≤) a strictly totally ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. In this paper we show that if RM is a PS-module, then the module [[MS≤]] of generalized power series over M is a PS [[RS,≤]]-module.展开更多
Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized p...Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.展开更多
The mathematical modeling of a rotating tapered Timoshenko beam with preset and pre-twist angles is constructed. The partial differential equations governing the six degrees, i.e., three displacements in the axial, fl...The mathematical modeling of a rotating tapered Timoshenko beam with preset and pre-twist angles is constructed. The partial differential equations governing the six degrees, i.e., three displacements in the axial, flapwise, and edgewise directions and three cross-sectional angles of torsion, flapwise bending, and edgewise bending, are obtained by the Euler angle descriptions. The power series method is then used to inves- tigate the natural frequencies and the corresponding complex mode functions. It is found that all the natural frequencies are increased by the centrifugal stiffening except the twist frequency, which is slightly decreased. The tapering ratio increases the first transverse, torsional, and axial frequencies, while decreases the second transverse frequency. Because of the pre-twist, all the directions are gyroscopically coupled with the phase differences among the six degrees.展开更多
In this article, the authors study the vector-valued random power series on the unit ball of C^n and get vector-valued Salem-Zygmund theorem for them by using martingale technique. Further, the relationships between v...In this article, the authors study the vector-valued random power series on the unit ball of C^n and get vector-valued Salem-Zygmund theorem for them by using martingale technique. Further, the relationships between vector-valued random power series and several function spaces are also studied.展开更多
In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power s...In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.展开更多
Suppose that {X(n)(omega)} are independent random complex variable sequence, E(X(n)) = 0 and [GRAPHICS] (V(X(n) = sigma(n)2). If reversed capital E-epsilon > 0 such that for all P (H) > 1-epsilon, we have [GRAPH...Suppose that {X(n)(omega)} are independent random complex variable sequence, E(X(n)) = 0 and [GRAPHICS] (V(X(n) = sigma(n)2). If reversed capital E-epsilon > 0 such that for all P (H) > 1-epsilon, we have [GRAPHICS] Then the circle {\Z\ = rho} is almost surely a natural boundary of the random series [GRAPHICS]展开更多
Let R be a ring such that all left semicentral idempotents are central and (S, ≤) a strictly totally ordered monoid satisfying that 0 ≤s for all s ∈S. It is shown that [[R^S≤]], the ring of generalized power ser...Let R be a ring such that all left semicentral idempotents are central and (S, ≤) a strictly totally ordered monoid satisfying that 0 ≤s for all s ∈S. It is shown that [[R^S≤]], the ring of generalized power series with coefficients in R and exponents in S, is right p.q.Baer if and only if R is right p.q.Baer and any S-indexed subset of I(R) has a generalized join in I(R), where I(R) is the set of all idempotents of R.展开更多
To the serf-similar analytical solution of the Boussinesq equation of groundwater flow in a semi-infinite porous medium, when the hydraulic head at the boundary behaved like a power of time, Barenblatt obtained a powe...To the serf-similar analytical solution of the Boussinesq equation of groundwater flow in a semi-infinite porous medium, when the hydraulic head at the boundary behaved like a power of time, Barenblatt obtained a power series solution. However, he listed only the first three coefficients and did not give the recurrent formula among the coefficients. A formal proof of convergence of the series did not appear in his works. In this paper, the recurrent formula for the coefficients is obtained by using the method of power series expansion, and the convergence of the series is proven. The results can be easily understood and used by engineers in the catchment hydrology and baseflow studies as well as to solve agricultural drainage problems.展开更多
Let R be a ring and (S, 〈) be a strictly totally ordered monoid satisfying that 0 〈 s for all s C S. It is shown that if A is a weakly rigid homomorphism, then the skew generalized power series ring [[RS,-〈, λ]]...Let R be a ring and (S, 〈) be a strictly totally ordered monoid satisfying that 0 〈 s for all s C S. It is shown that if A is a weakly rigid homomorphism, then the skew generalized power series ring [[RS,-〈, λ]] is right p.q.-Baer if and only if R is right p.q.-Baer and any S-indexed subset of S,(R) has a generalized join in S,(R). Several known results follow as consequences of our results.展开更多
Exponential integral for real arguments is evaluated by employing a fast-converging power series originally developed for the resolution of Grandi’s paradox. Laguerre’s historic solution is first recapitulated and t...Exponential integral for real arguments is evaluated by employing a fast-converging power series originally developed for the resolution of Grandi’s paradox. Laguerre’s historic solution is first recapitulated and then the new solution method is described in detail. Numerical results obtained from the present series solution are compared with the tabulated values correct to nine decimal places. Finally, comments are made for the further use of the present approach for integrals involving definite functions in denominator.展开更多
In this paper, a new approach called Power Series Approximation Method (PSAM) is developed for the numerical solution of a generalized linear and non-linear higher order Boundary Value Problems (BVPs). The proposed me...In this paper, a new approach called Power Series Approximation Method (PSAM) is developed for the numerical solution of a generalized linear and non-linear higher order Boundary Value Problems (BVPs). The proposed method is efficient and effective on the experimentation on some selected thirteen-order, twelve-order and ten-order boundary value problems as compared with the analytic solutions and other existing methods such as the Homotopy Perturbation Method (HPM) and Variational Iteration Method (VIM) available in the literature. A convergence analysis of PSAM is also provided.展开更多
This paper provides a power series solution to the Duffing-harmonic oscillator and compares the frequencies with those obtained by the harmonic balance method.To capture the periodic motion of the oscillator,the power...This paper provides a power series solution to the Duffing-harmonic oscillator and compares the frequencies with those obtained by the harmonic balance method.To capture the periodic motion of the oscillator,the power series expansion is used upon transforming the time variable into an“oscillating time”which reduces the governing equation to a form well-conditioned for a power series solution.A recurrence equation for the series coefficients is established in terms of the“oscillating time”frequency which is then determined by employing Rayleigh’s energy principle.The response of the oscillator is compared with a numerical solution and good agreement is demonstrated.展开更多
An analytical approach based on the power series method is used to analyze the free vibration of a cantilever beam with geometric and inertia nonlinearities.The time variable is transformed into a“harmonically oscill...An analytical approach based on the power series method is used to analyze the free vibration of a cantilever beam with geometric and inertia nonlinearities.The time variable is transformed into a“harmonically oscillating time”variable which transforms the governing equation into a form well-conditioned for a power series analysis.Rayleigh’s energy principle is also used to determine the vibration frequency.Convergence of the power series solution is demonstrated and excellent agreement is seen for the vibration response with a numerical solution.展开更多
Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either (S,≤) satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generali...Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either (S,≤) satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].展开更多
Let A,B be associative rings with identity,and(S.≤)a strictly totally ordered monoid which is also artinian and finitely generated.For any bimodule AaMB. we show that the bimodule [[A^(S.≤)]][M^(S.≤)][[B^(S.≤)]]de...Let A,B be associative rings with identity,and(S.≤)a strictly totally ordered monoid which is also artinian and finitely generated.For any bimodule AaMB. we show that the bimodule [[A^(S.≤)]][M^(S.≤)][[B^(S.≤)]]defines a Morita duality if and only if _AM_B defines a Morita duality and A is left noetherian.B is right noetherian.As a corollary,it.is shown that the ring[[A^(S.≤)]]of generalized power series over A has a Morita duality if and only if A is a left noetherian ring with a Morita duality induced by a bimodule _AM_B such that B is right noetherian.展开更多
As a generalization of power series rings, Ribenboim introduced the notion of the rings of generalized power series. Let R be a commutative ring, and (S.≤) a strictly totally ordered monoid. We prove that (1) the...As a generalization of power series rings, Ribenboim introduced the notion of the rings of generalized power series. Let R be a commutative ring, and (S.≤) a strictly totally ordered monoid. We prove that (1) the ring [[R<sup>(</sup>S.≤]] of generalized power series is a PP-ring if and only if R is a PP-ring and every S-indexed subset C of B(R) (the set of all idempotents of R) has a least upper bound in B(R). and (2) if (S. ≤) also satisfies the condition that 0≤s for any s∈S, then the ring [[R<sup>(</sup>S.≤]] is weakly PP if and only if R is weakly PP.展开更多
Let A be a ring.In this paper we generalize some results introduced by Aliabad and Mohamadian.We give a relation bet ween the z-ideals of A and t hose of the formal power series rings in an infinite set of indetermiii...Let A be a ring.In this paper we generalize some results introduced by Aliabad and Mohamadian.We give a relation bet ween the z-ideals of A and t hose of the formal power series rings in an infinite set of indetermiiiates over A.Consider A[[Xa]]3 and its subrings A[[X_(A)]]_(1),A[[X_(A)]]_(2),and A[[X_(A)]]_(α),where a is an infinite cardinal number.In fact,a z-ideal of the rings defined above is of the form I+(X_(A))i,where i=1,2,3 or an infinite cardinal number and I is a z-ideal of A.In addition,we prove that the same condition given by Aliabad and Mohamadian can be used to get a relation between the minimal prime ideals of the ring of the formal power series in an infinite set of indeterminates and those of the ring of coefficients.As a natural result,we get a relation between the z°-ideals of the formal power series ring in an infinite set of indeterminates and those of the ring of coefficients.展开更多
文摘A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.
基金The NNSF (10171082) of China and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.C.
文摘Let R be a commutative ring and (S, ≤) a strictly totally ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. In this paper we show that if RM is a PS-module, then the module [[MS≤]] of generalized power series over M is a PS [[RS,≤]]-module.
文摘Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.
基金Project supported by the National Natural Science Foundation of China(Nos.11672007,11402028,11322214,and 11290152)the Beijing Natural Science Foundation(No.3172003)the Key Laboratory of Vibration and Control of Aero-Propulsion System Ministry of Education,Northeastern University(No.VCAME201601)
文摘The mathematical modeling of a rotating tapered Timoshenko beam with preset and pre-twist angles is constructed. The partial differential equations governing the six degrees, i.e., three displacements in the axial, flapwise, and edgewise directions and three cross-sectional angles of torsion, flapwise bending, and edgewise bending, are obtained by the Euler angle descriptions. The power series method is then used to inves- tigate the natural frequencies and the corresponding complex mode functions. It is found that all the natural frequencies are increased by the centrifugal stiffening except the twist frequency, which is slightly decreased. The tapering ratio increases the first transverse, torsional, and axial frequencies, while decreases the second transverse frequency. Because of the pre-twist, all the directions are gyroscopically coupled with the phase differences among the six degrees.
文摘In this article, the authors study the vector-valued random power series on the unit ball of C^n and get vector-valued Salem-Zygmund theorem for them by using martingale technique. Further, the relationships between vector-valued random power series and several function spaces are also studied.
文摘In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.
文摘Suppose that {X(n)(omega)} are independent random complex variable sequence, E(X(n)) = 0 and [GRAPHICS] (V(X(n) = sigma(n)2). If reversed capital E-epsilon > 0 such that for all P (H) > 1-epsilon, we have [GRAPHICS] Then the circle {\Z\ = rho} is almost surely a natural boundary of the random series [GRAPHICS]
基金TRAPOYT(200280)the Cultivation Fund(704004)of the Key Scientific and Technical Innovation Project,Ministry of Education of China
文摘Let R be a ring such that all left semicentral idempotents are central and (S, ≤) a strictly totally ordered monoid satisfying that 0 ≤s for all s ∈S. It is shown that [[R^S≤]], the ring of generalized power series with coefficients in R and exponents in S, is right p.q.Baer if and only if R is right p.q.Baer and any S-indexed subset of I(R) has a generalized join in I(R), where I(R) is the set of all idempotents of R.
基金Project supported by the National Natural Science Foundation of China (No.50425926)
文摘To the serf-similar analytical solution of the Boussinesq equation of groundwater flow in a semi-infinite porous medium, when the hydraulic head at the boundary behaved like a power of time, Barenblatt obtained a power series solution. However, he listed only the first three coefficients and did not give the recurrent formula among the coefficients. A formal proof of convergence of the series did not appear in his works. In this paper, the recurrent formula for the coefficients is obtained by using the method of power series expansion, and the convergence of the series is proven. The results can be easily understood and used by engineers in the catchment hydrology and baseflow studies as well as to solve agricultural drainage problems.
基金The Youth Foundation(QN2012-14)of Hexi University
文摘Let R be a ring and (S, 〈) be a strictly totally ordered monoid satisfying that 0 〈 s for all s C S. It is shown that if A is a weakly rigid homomorphism, then the skew generalized power series ring [[RS,-〈, λ]] is right p.q.-Baer if and only if R is right p.q.-Baer and any S-indexed subset of S,(R) has a generalized join in S,(R). Several known results follow as consequences of our results.
文摘Exponential integral for real arguments is evaluated by employing a fast-converging power series originally developed for the resolution of Grandi’s paradox. Laguerre’s historic solution is first recapitulated and then the new solution method is described in detail. Numerical results obtained from the present series solution are compared with the tabulated values correct to nine decimal places. Finally, comments are made for the further use of the present approach for integrals involving definite functions in denominator.
文摘In this paper, a new approach called Power Series Approximation Method (PSAM) is developed for the numerical solution of a generalized linear and non-linear higher order Boundary Value Problems (BVPs). The proposed method is efficient and effective on the experimentation on some selected thirteen-order, twelve-order and ten-order boundary value problems as compared with the analytic solutions and other existing methods such as the Homotopy Perturbation Method (HPM) and Variational Iteration Method (VIM) available in the literature. A convergence analysis of PSAM is also provided.
文摘This paper provides a power series solution to the Duffing-harmonic oscillator and compares the frequencies with those obtained by the harmonic balance method.To capture the periodic motion of the oscillator,the power series expansion is used upon transforming the time variable into an“oscillating time”which reduces the governing equation to a form well-conditioned for a power series solution.A recurrence equation for the series coefficients is established in terms of the“oscillating time”frequency which is then determined by employing Rayleigh’s energy principle.The response of the oscillator is compared with a numerical solution and good agreement is demonstrated.
文摘An analytical approach based on the power series method is used to analyze the free vibration of a cantilever beam with geometric and inertia nonlinearities.The time variable is transformed into a“harmonically oscillating time”variable which transforms the governing equation into a form well-conditioned for a power series analysis.Rayleigh’s energy principle is also used to determine the vibration frequency.Convergence of the power series solution is demonstrated and excellent agreement is seen for the vibration response with a numerical solution.
基金National Natural science Foundation of China(10171082)the Cultivation Fund of the Key Scientific Technical Innovation Project,Ministry of Education of ChinaTRAPOYT
文摘Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either (S,≤) satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].
基金supported by National Natural Science Foundation of China(10171082)Foundation for University Key Teacherthe Ministry of Education(GG-110-10736-1001)
文摘Let A,B be associative rings with identity,and(S.≤)a strictly totally ordered monoid which is also artinian and finitely generated.For any bimodule AaMB. we show that the bimodule [[A^(S.≤)]][M^(S.≤)][[B^(S.≤)]]defines a Morita duality if and only if _AM_B defines a Morita duality and A is left noetherian.B is right noetherian.As a corollary,it.is shown that the ring[[A^(S.≤)]]of generalized power series over A has a Morita duality if and only if A is a left noetherian ring with a Morita duality induced by a bimodule _AM_B such that B is right noetherian.
基金Research supported by National Natural Science Foundation of China. 19501007Natural Science Foundation of Gansu. ZQ-96-01
文摘As a generalization of power series rings, Ribenboim introduced the notion of the rings of generalized power series. Let R be a commutative ring, and (S.≤) a strictly totally ordered monoid. We prove that (1) the ring [[R<sup>(</sup>S.≤]] of generalized power series is a PP-ring if and only if R is a PP-ring and every S-indexed subset C of B(R) (the set of all idempotents of R) has a least upper bound in B(R). and (2) if (S. ≤) also satisfies the condition that 0≤s for any s∈S, then the ring [[R<sup>(</sup>S.≤]] is weakly PP if and only if R is weakly PP.
文摘Let A be a ring.In this paper we generalize some results introduced by Aliabad and Mohamadian.We give a relation bet ween the z-ideals of A and t hose of the formal power series rings in an infinite set of indetermiiiates over A.Consider A[[Xa]]3 and its subrings A[[X_(A)]]_(1),A[[X_(A)]]_(2),and A[[X_(A)]]_(α),where a is an infinite cardinal number.In fact,a z-ideal of the rings defined above is of the form I+(X_(A))i,where i=1,2,3 or an infinite cardinal number and I is a z-ideal of A.In addition,we prove that the same condition given by Aliabad and Mohamadian can be used to get a relation between the minimal prime ideals of the ring of the formal power series in an infinite set of indeterminates and those of the ring of coefficients.As a natural result,we get a relation between the z°-ideals of the formal power series ring in an infinite set of indeterminates and those of the ring of coefficients.