中国科学院数学与系统科学研究院期刊网

15 January 1979, Volume 22 Issue 1
    

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  • Acta Mathematica Sinica, Chinese Series. 1979, 22(1): 1-13. https://doi.org/10.12386/A1979sxxb0001
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    In this abstract we denote the non-associative and non-distributive ring briefly by NAD-ring.Let {R_i} be a coUection of NAD-rings and denote L as the collection of all elements a = (a_1,a_2,…,a_i,…) where a_i ∈ R_i. As usual we can introduce into L the following additive operation and scalar multiplication: (a_1,…, a_i,…) + (a_1′,…, a_i′,…) = (a_1 + a_i′,…, a_i + a_i',…) aa = (aa_1,…,aa_i,…),aa= (a_1a,…, a_ia,…) where a∈. Let a be an additive subgroup of L and called a subgroup if and only if aa ∈a, aa ∈a for every a ∈and a ∈a. A-subgroup a is called a subsystem of L if and only if a can be represented by the usual complete direct sum of a collection {a_i}of systems a_i of R_i where a_i are the images of a into R_i under the natural homomorphism (a_1,…, a_i,…)→ (0,…, a_i, 0…) In this case we denote a=(a_1,…, a_i,…).Now we can introduce the multiplicative operation []into L.Let b=(b_1,…, b_i,…)∈L and a= (a_1,…, a_i,…) be a subsystem of L. We define the multiplication of b and a as follows [b, a] = ([b_1, a_1],…,[b_i,a_i],…) [a, b] = ([a_1, b_1],…,[a_i, b_i],…)It is easy to prove that L is an NAD-ring.Definition 1 : The above NAD-ring L is called the complete direct sum of the collection of NAD-rings R_i.Definition 2:A subsystem L is called a subring of L if and only if[L, L]L.A subring L is called an ideal if and only if [L, L]L, [L,L]L. A subring L ofL is called a subdirect sum of NAD-rings R_i if and only if the images of L under thenatural homomorphism (a_1,…,a_i,…) → ( 0,…,a_i, 0,…)are either zero or R_i forevery i.Definition 3: We say that an NAD-ring R can be imbedded into the complete direct sum L if and only if there exists a mapping which is a group isomorphism of the additive group R into the additive group L and the following conditions are satisfied: (i) ([b, a])[(b), (a)] for every b ∈ R and every subsystem a of R. (ii) The natural homomorphism (a_1,…, a_i,…)→ (0,…, a_i, 0,…) of(R) into R_i is an onto mapping for every i.It is clear that the above conceptions are not all coincided with the usual conceptions.Theorem 1: An NAD-ring R can be imbedded into the complete direct sum of a collection of NAD-rings R_i if and only if R contains a collection of ideals I_i such that ∩ I_i=0 and R/I_i R_i.Theorem 2: Let R be a semi-simple NAD-ring satisfying the W-maximal condition on ideals. Suppose that every prime ideal of R is maximal, then R can be imbedded into the complete direct sum of a collection of simple NAD-rings Ri. On the contrary, if R is isomorphic to a subdirect sum of a collection of simple rings R_i, then every prime ideal of R must be maximal.Theorem 3: Let R be a semi-simple NAD-ring satisfying the W-maximal condition on ideals of R and be isomorphic to a subdirect sum S of simple NAD-rings R_i, then every ideal a of R must be isomorphic to a subring (D_1,…, D_i,…)_s of S, where (D_1,…, D_i,…)_s denote the collection of the elements of (D_1,…, D_i,…) in S and D_i are either R_i or zero. Furthermore, every prime ideal of R must be isomorphic to (R_1, R_2,…, 0, R_a,…)_s in which only one zero appears, and every minimal ideal of R must be isomorphic to a R_i.Theorem 4: Let R be a semi-simple NAD-ring satisfying the maxiimal condition on ideals. Suppose that either R contains a proper ideal or every non-zero ideal a is a zero divisor, namely, there exists a non-zero ideal b such that [a, b] = 0, [b, a]=0. Then R is adirect sum of principal ideals.Definition 4: An NAD-ring R is called split if and only if for every pair ideals a and b such that there exists an ideal such that.Theorem 5: Let R be an NAD-ring satisfying the minimal condition on ideals, then R is split if and only if the intersection of maximal ideals of R is zero.Theorem 6: Let R be a split NAD-ring satisfying the minimal condition on ideals, and be the intersection of prime ideals of R. Suppose that R contains non-zero proper ideals. Then R = ⊕(a_1)⊕…⊕(a_n),= 0 and (a_i)~2= (a_i), where (a_i) are minimal ideals. i= 1,2,…,n. Let a be an ideal and then a=⊕(a_(i1))⊕…⊕(a_(is)) a
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(1): 14-27. https://doi.org/10.12386/A1979sxxb0002
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    In this paper we shall study differential and integral calculus on the field of generalized power series with real coefficients and real exponents. This field was investigated long ago by T. Levi-Civita, and A. Ostrowski, etc. Several years ago, D. Langwitz considered a theory of functions on it, and raised a question whether the functions considered by him satisfy the intermediate value theorem and the mean value theorem of the differential calculus.With the powerful tool of nonstandard analysis, A. Robinson answered partially this question and gave some sufficient conditions for the above two theorems to be true. But these sufficient conditions are not necessary conditions. In. the present paper we sbalt give necessary and sufficient conditions for the above two theorems to be true by using nonstandard analysis. Moreover, we shall also give the necessary and sufficient conditions which make almost all the other theorems fundamental in differential and integral calculus to be true, such as the theorem of assuming the maximum on a closed interval, Rolle's theorem, the Lagrange's form of remainder of Taylor's expansion, Cauchy's theorem, and the mean value theorem of integral calculus.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(1): 28-44. https://doi.org/10.12386/A1979sxxb0003
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    In this paper we discuss the observability and controllability of the elastic vibration system with the angular velocity, angular and acceleration inputted to the controller as feedback signals. Necessary and sufficient conditions for the controllability and observability are obtained. These conditions are intimately associated with the problem of whether the operator of open-loop system and the operator of closed-loop system have common spectrums or not.For the elastic beam, if the feedback signal is the angular velocity, the system is observable. Then the measurable point is not zero point of the first derivative of the modes of elastic vibration. If the system is controllable, the control point is not zero point of the modes of elastic vibration.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(1): 45-53. https://doi.org/10.12386/A1979sxxb0004
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    It is proved by Wu Fang, in 1974, that we can certainly find a solution for the optimal block search which satisfies the conditions:Ⅰ.k_1 is an even number.Ⅱ. |k_i-k_j|≤2(1≤i, j≤n);Ⅲ. All even numbers in k_2, k_3,…, k_n are equal.All optimal solutions satisfying these conditions are found by recent works [2]-[6]. Now we investigate the cases where these conditions are not satisfied. The main results are as follows. Theorem 3. If n = 2, k_1 + k_2 = N, then(1) When N = 4u, (2u, 2u) is optimal for any δ, and (2u + 1, 2u - 1) is optimal for δ = 0.(2) When N = 4u + 1, (2u, 2u + 1) is optimal for any δ, and (2u +2, 2u-1) is optimal for δ = 0.(3) When N = 4u + 2, (2u + 1 ± 1, 2u + 1 1) is optimal for any δ, and (2u + 3, 2u-1) and (2u+1, 2u+1) is optimal for δ=0.(4) When N = 4u + 3, (2u + 2, 2u + 1) is optimal for any δ.Theorem 6. If n > 2, conditions (1) are necessary for the solution (k_1, k_2,…, k_n) to be optimal, except that when δ = 0,(1) (2u-2, 2u, 2u-3), are optimal.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(1): 54-68. https://doi.org/10.12386/A1979sxxb0005
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    Let f(x) be a 2π-Periodic-Continuous function where we investigate-two-operators: whereWe obtain the degree of approximation of periodic continuous functions by these operators and an asymptotic representation of these operators.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(1): 69-99. https://doi.org/10.12386/A1979sxxb0006
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    The main results of the present paper are:1.If eanonical affine connections of the first type and the second type are identical on a homogeneous Kobayashi manifold, then the transitive analytic automorphie group of the manifold is semisimple.2.In the third and fourth chapter we give the necessary and sufficient conditions for the existence of an invariant complex structure on complex C-spaces of the compact type and the noncompact type.3. On a C-space of the noncompact type, Koszul form is non-degenerate if and only if Jt = tJ, where J.is an invariant complex structure and t is an involutorial automorphism.4. When B is a regular C-subalgebra of G, the C-spaces of the compact type are Hemitian, and Kahlerian, too. At this time Koszul form is negative. So we obtain the results of H. C. Wang [11] and J. L. Koszul for a bounded domain.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(1): 100-117. https://doi.org/10.12386/A1979sxxb0007
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    In this paper, we shall give a new non-Riemannian geometry - biscalar (ψ)tensor (g) geometry, and within the framework of this non-Riemannian geometry, we shall construct the scalar-tensor theory for which gravitation (including both the scalar and tensor fields) is geometrized in the spirit of general relativity. Then we shall discuss the variational principles of scalar-tensor theories of gravitation. Our basic idea is to set up a metric-preserving connection by means of torsion and to make it dispensable to introduce the conception of gauge, which will endow the abovemontioned geometry with a more simple and natural structure than that of Weyl's geometry or Lyra's geometry. Moreover, this geometry is a natural extension of Riemannian geometry.In §2 we shall give a structural theorem of this geometry and discuss the influence of the Bianchi's Identities on this geometry. And therby we shall arrive at a theorem concerning the existence and uniqueness of the general integral of a kind of systems of first-order nonlinear partial differential equations.In §3 we shall construct a geometrized model for gravitation and point out that Einstein's General Relativity and Dicke's scalar-tensor theory of gravitation are special cases of our theory, and D. K. Sen and K. A. Dunn's scalar-tensor theory of gravitation based on Lyra's geometry is very similar to another special form of our theory. In this section, we shall give torsion a new physical interpretation, i.e. torsion is caused by the uneven distribuation of material, which is a macrophenomenon. Our point of view is quite contrary to E. Cartan's, Sciama's and Kibble's usual View-points that torsion is caused by the internal spin of material, which is a microphenomenon.In §4 we shall extend variational principles of general scalar-tensor theories of gravitation. Finally, we shall give a geometrical background of the generalized Einstein's variational principle.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(1): 118-122. https://doi.org/10.12386/A1979sxxb0008
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    In this paper a criterion is given for convergence of P_j(k, a) as a→∞, where P_j(k, a) is obtained by replacing Cov(X_o, X_o) with aI in the estimation error eovariance matrix of the linear minimum covarianee estimate for x_j based on y_o,…, y_k. When j =0(k), this convergence is the definition of the stochastic observability (detectability) of the system. The necessary and sufficient conditions for the unbiasedness of the estimates without the knowledge of initial values are obtained, and if these conditions are satisfied it can be proved that the estimation error covariance matrix is just the limit of P_j(k, a).