15 July 1978, Volume 21 Issue 3

    

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  ON SPECTRUM DISTRIBUTION OF COMPLETELY CONTROLLABLE LINEAR SYSTEMS

  Acta Mathematica Sinica, Chinese Series. 1978, 21(3): 193-205. https://doi.org/10.12386/A1978sxxb0022

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  Let be a Hilbert space. A be a linear operator satisfying the condition (F) given as follows:(F_1) A is an unbounded spectral operator with spectral resolution where dim E(λ_k)= 1; in general, we assume that E(λ_k) (k) is self-adjoint, and φ_k(k ≥ 1) is the normal eigenveetor in E(λ_k);In this note, we proved the followingTheorem 1.1. Let A be a linear operator satisfying condition (F). Then, for a given b ∈and any given sequence A {ν_1,ν_2,…, ν_n,…} of complex numbers, there exists at least an element g ∈such that σ(A + <., g>b)= σ_p(A + <., g>b) = A, if and only ifTheorem 1.2. Let A boa linear operator satisfying condition (F), and J a set of indexes such that Then the necessary and sufficient condition for the stabilization of linear system dx/dt=Ax + bu(t) by the feedback law u(t) = 〈x(t), k〉 with some pair of g, b (∈), is
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  THE PERTURBATION THEORY FOR A CLASS OF LINEAR OPERATORS AND ITS APPLICATIONS

  Acta Mathematica Sinica, Chinese Series. 1978, 21(3): 206-222. https://doi.org/10.12386/A1978sxxb0023

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  This paper discusses the perturbation theory of linear operators of discrete type by using the theory of unconditional bases. With this perturbation theory, we have given a rigorous foundation for schSdinger's perturbation theory and the recurrent formulas of revisory, terms for perturbant eigenvalues and eigenvectors. Finally, an application for the problem of elastic oscillation is provided.
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  SOME NOTES ON THE CHROMATIC POLYNOMIALS OF GRAPHS

  Acta Mathematica Sinica, Chinese Series. 1978, 21(3): 223-230. https://doi.org/10.12386/A1978sxxb0024

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  The bounds for the coeffients of ehromatic polynomials are given in terms of the number of cycles with every length. In section 3, we give an explicit expression for the chromatic polynomials of outerplane graphs.
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  THE STRONG LAW OF LARGE NUMBERS FOR TRANSITION PROBABILITIES OF DENUMERABLE HOMOGENEOUS MARKOV CHAINS

  Acta Mathematica Sinica, Chinese Series. 1978, 21(3): 231-242. https://doi.org/10.12386/A1978sxxb0025

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  In this paper the following results have been proved:Theorem 1. Let x_n(n = 0, 1, 2,…) be a homogeneous Markov chain, S(k, n) be the number of occurrences of the state k and A (k, l, n) be the number of the state l occurring directly after the state k in the first n trials, and assume that P(D_k)>0, where D_k={ω:x_i=k for infinite i}, then holds almost everywhere in D_k, i.e.,Theorem 2. Let x_n(n = 0, 1, 2,…) be a homogeneous Markov chain, C be a irreducible class of the recurrent states, k ∈ C, S(k, n) be the number of occurrences of the state k and A(k, l, n) be the number of the state l occurring directly after the state k in the first n trials, if there exists j ∈C such that q_i = P(x_o = j) > 0, then holds almost everywhere in δ_j = {ω:x_o = j}, i.e.,In proof, the author has put forward a new purely analytical method for researches on strong limit theorems, a method of the function theory which is completely different from the usual methods in probability theory.
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  THE GENERALIZED CONTINUUM HYPOTHESES WITH PARAMETERS AND THE AXIOM OF CHOICE

  Acta Mathematica Sinica, Chinese Series. 1978, 21(3): 243-246. https://doi.org/10.12386/A1978sxxb0026

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  In this paper, we have solved :1. the problem of the relations between the Generalized Continuum Hypotheses with parameters as natural numbers or Alephs;2. the problem of the relations between the Generali, zed Continuum Hypotheses with parameters as natural numbers or Alephs and the Axiom of Choice.
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  AN INTEGRAL EQUALITY OF MEAN CURVATURE

  Acta Mathematica Sinica, Chinese Series. 1978, 21(3): 247-252. https://doi.org/10.12386/A1978sxxb0027

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  In this paper, we present an integral equality of mean curvature in the Sobolev space W_(2,o)~2 (Ω), where Ω is a bounded domain in R~n, with smooth boundary Ω. Its applications are also exhibited.
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  ON NEW HOMOTOPY INVARIANTS Ⅰ.A SHORT EXACT SEQUENCES

  Acta Mathematica Sinica, Chinese Series. 1978, 21(3): 253-262. https://doi.org/10.12386/A1978sxxb0028

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  Let K be an N-dimensional CW-complex, n = N- 1.We consider the groups H~n(K, Z) and Coker Sq~2, where Sq~2:H~(n-1)(K, Z) → H~(n+1)(K, Z_2) is the Steenrod square.Denote the p-primary component of G by G_((p)) and m_G = {g|mg = 0}. Assume Then On 2~lkH~n(k,Z),we have cohomology operations Each operation T~((1))(k)has the property: Definition.Define and call them the T~((1))(k) torsions of K.Obviously, the T~((1)) torsions are all homotopy numerical invariants.Using these invariants, we can list the generators and their order of cohomotopy groupπ~n(K)_((2)) as in theorem 4.The results can be generalized to obtain the generators and their order of cohomotopy group π~m(K)_((p)), for odd p and 2p- 3≤N-m<4p- 5, cf. theorem 8.
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  AN EXISTENCE AND UNIQUENESS THEOREM OF LIMIT CYCLES OF LINARD EQUATION

  Acta Mathematica Sinica, Chinese Series. 1978, 21(3): 263-269. https://doi.org/10.12386/A1978sxxb0029

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  In this paper, we mainly consider the Lienard equation x+f(x)x+x=0[Eq.(1)]. Let a F(x) =∫_o~x f(x)dx, F_1(x) = F(x) and F_2=(x) = F(-x) for x ≥ 0, then we haveTheorem 1. Let f(x) be continuous for all x. If (i) there exists a > 0 such that F_1(x) ≤0 ≤F_2(x) for 0 A respectively; (iii) F_1(+ ∞)> F_2(+∞); then Eq. (1) has a unique nonconstant periodic solution. Moreover, this solution is a stable limit cycle.A corollary of this theorem is stated under the following conditions" (i) f(x) is continuous for all x and there exists a > 0 such that xF(x) ≤ 0 for |x| < a, but F(x) is not identically zero for |x| << 1; (ii) F(x) is nondeceasing outside (-a, a); (iii) F(-∞)