15 April 1956, Volume 6 Issue 2

    

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  ON DIVISOR PROBLEMS(Ⅱ)

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 139-152. https://doi.org/10.12386/A1956sxxb0011

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  Let d_k(n) be the numbers of expressions of n as k factors, and let D_k(x).Let R_k(x) = (a_(k,o) + a_(k,1) In x + … + a_(k,k-1) ln~(k-1) x) x(x>0) be the residue of ξ~k(s)x~s/s at s=1.Define △_k(x)=D_k(x)-R_k(x).Theorem.This theorem is a generalization of Hardy's resultThe proof of the theorem is based on generalized identity, givtn by the authorin previous paper "On Divisor Problems(Ⅰ)".
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  О ПАРАХ СВЯЗНОСТЕЙ И ИНТЕГРАЛЬНЫХ МНОГООБРАЗИЯХ СИСТЕМЫ УРАВНЕНИЙ С ЧАСТНЫМИ ПРОИЗВОДНЫМИ 2-ГО ПОРЯДКА(Ⅰ)

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 153-162. https://doi.org/10.12386/A1956sxxb0012

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  О ПАРАХ СВЯЗНОСТЕЙ И ИНТЕГРАЛЬНЫХ МНОГООБРАЗИЯХ СИСТЕМЫ УРАВНЕНИЙ С ЧАСТНЫМИ ПРОИЗВОДНЫМИ 2-ГО ПОРЯДКА(Ⅱ).

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 163-169. https://doi.org/10.12386/A1956sxxb0013

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  THE ABSOLUTE SUMMABILITY A OF INFINITE SERIES

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 170-183. https://doi.org/10.12386/A1956sxxb0014

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  A series is said to be absolutely summable A, or simply summable |A|, when the function f(x) =∑c_nx~n is of bounded variation in the interval 0≤x≤1. The aim of this paper is to solve the following problem: when the series ∑c_n is summable |A|, under what condition or conditions the series ∑c_n converges? The answer is given by the followingTheorem. When ∑c_n is summable |A|, then the pair of conditions C_n=0(1) and C_0+C_1+…+C_n=O(1) is necessary and sufficient for the convergence of the series ∑c_n.The proof is based upon the followingRemark. Let p=p(σ) be an integer such that p→∞ and (σ—1) p~3 ≤ 1 as σ→1—0. ThenRemark. Lemma 4 in my English paper "Convergence of absolutely summable series", Scientia Sinica Ⅳ (1955), is invalid; it should be replaced by the Lemma stated above during the proof of Lemma 5.
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  THEORY OF REGIONAL ANALYSIS OF ORDINARY DIFFERENTIAL EQUATIONS(Ⅱ)

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 184-205. https://doi.org/10.12386/A1956sxxb0015

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  We take the following systemdx/dt=a_(oo)+a_(10)x+a_(01)y=X_1,dy/dt= b_(oo) + b_(10)x + b_(01)y = Y_1 to approach a system of differential equations dx/dt=X(x,y), dy/dt=Y(x,y) in a region S. The condition of expressibility of the previously paper is dropped since the system (Ⅰ) is integrable. The condition of approximation remains the same, i.e. the functional assumes the absolute minimum.In particular, we take any region symmetric with respect to both x- and y- axes as the region S, the coefficients of (Ⅰ) can be put in the following formulae:This paper contains two parts. In the first part we extend the classification of points of the previous paper. Shortly speaking, a point (x, y)is classified according to the characteristic equationAs application we consider the basic problem of small parameter from the point of view of regional analysis. Through concrete examples we compare our method with that of M. J. Lighthill and reveal the fundamental defect of the traditional method.In the second part we consider the approximation in the large. We take the location of limit cycle as an application. From formulae (Ⅳ) we roughly deduce the following formula This formula is applicable to equations of Van der Pol-Rayleigh type, equations of Pioncaretype, and equations satisfying symmetric principle,etc.
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  ON THE INTEGRAL CURVATURE OF A CLOSED SPACE CURVE

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 206-214. https://doi.org/10.12386/A1956sxxb0016

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  The purpose of this paper is to establish some theorems on differential geometry in the large. The main results are as follows:Theorem 1. If a closed space curve C has an angular point with its interior angle θ, then the integral curvature of C is not less than π+θ, i.e., if k denotes the curvature of C, then ∮_((c)) K ds ≥ π + θ. Corollary. The integral curvature of any closed space curve is not less than 2π.Theorem 2. If a closed space curve has a k multiple point then the integral curvature of this curve is not less than kπ.Theorem 3. The binormal indicatrix of any closed space curve has at least one multiple point.Theorem 4. The absolute value of the integral torsion of a closed space curve whose tangent indicatrix has no multiple point is less than 2π i.e., if σ denotes the torsion of C, then —2π<∮_((c)) σds<2π.Corollary. If a certain lines of curvature C of a surface is closed and its tangent indicatrix has no multiple point, then the integral torsion of C is zero.
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  ВЛОЖЕНИЕ ПРОСТРАНСТВА ФИНСЛЕРА В ПРОСТРАНСТВО МИНКОВСКОГО

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 215-232. https://doi.org/10.12386/A1956sxxb0017

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  ON THE BORSUK’S ABSOLUTE HOMOTOPY EXTENSION PROPERTY.

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 233-241. https://doi.org/10.12386/A1956sxxb0018

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  A topological space X is said to be an AHE (AHE~p) (read space possessing absolute homotopy extension property, (absolute P dimensional homotopy extension property)) if any mapping of B × (0, 1) + Y × (0) into X can be extended to a mapping of Y × (0, 1) into X for any normal, space Y and its non null closed subset B (For any separable metric space Y and its non null closed subset B and dim (Y-B)≤P.) The following theorems are proved. (1) Any product of AHE (AHE~p) (finite or infinite in number) is an AHE (AHE~p). (2) If an arcwise connected metric space X is an AHE~(p+1) and is P-Lc at one of its points, then X is P-Lc at all itspoints. (3) If an arcwise connected metric space X is an AHE and is locally contractible at one of its points, then X is locally contractible at all its points. (4) A necessary and sufficient condition for a P-dimensional arcwise connected separable metric space to be an ANR is that X is an AHE~(p+1) and is Lc~p at one of its points. (5) A necessary and sufficient condition for a sep arable metric space X to be an AR is that X is a contractible AHE. The following remarks are shown by examples. (1) A compact AHE of finite dimension which is locally contractible at one of its points is not necessarily an ANR. (2) An arcwise connected compact AHE of finite dimension is not necessarily an ANR. (3) An arcwise connected compact AHE of infinite dimension which is Lc~ω at one of its points is not necessarily an ANR. (4) A compactum of infinite dimension which is locally contractible at all its point is not necessarily an AHE. Whether an AHE of infinite dimension, locally contractible at one of its points, is an AHE or not, is an unsolved problem.
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  UNIQUENESS THEOREM FOR CHAPLYGIN’S PROBLEM(Ⅰ)

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 242-249. https://doi.org/10.12386/A1956sxxb0019

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  In this paper the uniqueness problem of the Chaplygin's equation K(y) u_(xx)+u_(yy)=0 (K(0)=0; dk/dy >0 for k≠0) is considered. The domain D is bounded by three curves showing in the figure, where Γ_1 is a characteristic define by the equation dy = —(—K)~(-1/2) dx, Γ_2 and Γ_3 are continuous and piecewise continuously differentiable curves, and moreover, Γ_2 satisfying the condition 0 ≤ dy ≤ (—K)~(-1/2) dx.We consider solution u whose first derivatives are continuous on the boundary except at O and A. When P ∈D, P→O and P→A, the conditions imposed on u_x, u_y are u_x= O (OP~α),u_y = O (OP~α) (—1<α≤0),u_x=O(AP~β),u_y=O(AP~β) (—1/2<β≤0).The result of this paper is as follows.Let two continuous and piecewise continuously differentiable functions B, C define in the upper half-plane and satisfying the following conditions: B = x, C = 0 (when 0≤x≤1, y=0),(1) C ≥ 0 (when y > 0),(2) B= O (OP), C = O (OP) (when P∈D,P→O),(3) B = O (1), C = O (AP) (when P ∈D, P→A),(4) (B_y,+K C_x)~2 ≤ (B_x—C_y) [(C K)_y—K B_x] .(5) be the polar coordinates of x,Y and It is easily seen from (2)and (5)that m≥0. otherwise let θ_o be the angle defined by the equation LetTheorem 1. The portion of Γ_3 lying in the region 0 < θ < θ_o, 0 < r < exp satisfying the condition B dy — C dx ≥ 0 and the solution u in D which vanishes on Γ_2 and Γ_3, then u = 0 in D.We consider two particular, cases of the theorem.Ⅰ. B=x, C=y satisfying (1)—(5).In this case m=0. Thus the uniqueness theorem holds if the portion of Γ_3 lying in the domain 0 < θ < θ_o, 0 < r < exp 1/n satisfying x dy — y dx ≥ 0 .This result is a little better than a theorem of Morawetz.Ⅱ. In the case of Tricomi's equation (K(y)=y), if l is a constant satisfying 1/2≤l≤1, then B=x, C=ly satisfying (1)—(5). In this case h(θ)=(cos~2 θ+9/4 l~2 sin~2 θ)~(-1/2),m=l/2-|3/2l-1|, θ_o is defined by the equation ln (1+m)~(1/m),n=min h(θ)≥min (1,2/3l).Thus the uniqueness theorem of Tricomi's equation holds if the portion of Γ_3 lying in the domain 0 < 0 <θ_o, 0 < r < satisfying x dy—ly dx≥0.Theorem 2. If the portion of Γ_3 lying in the region y>0, (x—1)~2+Y~2<1 satisfying the condition dy≥0 and the solution u in D which vanishes on Γ_2 and Γ_3, then u=0 in D.The method of proof is to consider the energy integral = 0 for suitable choice of a, b, c.
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  UNIQUENESS THEOREM FOR CHAPLYGIN’S PROBLEM(Ⅱ)

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 250-262. https://doi.org/10.12386/A1956sxxb0020

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  In this paper the uniqueness problem of the Chaplygin's equation K(y) u_(xx)+u_(yy)=0(K(0)=0; dK/dy > 0 for y ≠ 0) is considered. The domain D is bounded by three curves showing in the figure, where Γ_1 and Γ_2 are characteristics define by the equation dx~2+Kdy~2=0, Γ_3 is a continuous curve. Let the ordinate of P be yo and let 1+2 (K/K_y)_y=f(y). Let y_1=0 if f(y)≥0 for all yo≤y< 0, otherwise let y_1 be the upper bound of values y in the interval yo≤y<0 satisfying f(y)< 0.Theorem. If y_1<0 and the following relation holds: and if u is a quasi—regular solution which vanishes on Γ_2 +Γ_3 then u=0 in D.This result is a little better than a result of.The method of proof of the theorem is to the sum of the energy integral + cu_y) (Ku_(xx) + u_(yy) dx dy = 0 and the zero integral + ∮ [u~2 (q dx — p dy) + d(ru~2)] = 0 for suitable choice of a, b, c, p, q, r.
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  A SYSTEM OF FUNCTIONAL EQUATIONS

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 263-269. https://doi.org/10.12386/A1956sxxb0021

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  Let B_k~l denote B_k~l ≡ (l+k)!/(l-k)!k!z~k, is the kth coefficient of Bessel polynomial of degree l, yl(x) the study of Euler-Poisson-Darboux equation by Fourier transforms suggested us to consider the following system of functional equations and with the additional conditions where l=0, 1,…, n.In this note we prove that on the interval 0
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  ON PROPER ISOMORPHISMS OF (μ,△,γ) - SYSTEMS I.

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 270-301. https://doi.org/10.12386/A1956sxxb0022

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  According to [2] we define a (μ, △, γ,)-system, H[{(H~n}{H~n (m)}, μ, △, γ as a set of groups H~n(m) (m=0, 2; n=1, 2,...) and homomorphisms μ : H~n→H~n(2) , △: H~n(2)→_2H~n+1, γ_p~n: H~n(2)→H~(n+p)(2), where p is even and _2H~(n+1) a subgroup of H~(n+1) such that 2g= 0 if g ∈_2H~(n+1). A proper homomorphism, g, of a (μ, △, γ)-system, H, into another, namely, H [{H~n] }, {[H~n (2)}, μ, △, γ, is a set of homomorphisms, either of H~n into H~n or of H~n (2) into H~n (2), satisfying the following conditions μg = gμ, g△ = △g, gγ_p~n = γ_p~ng. Geometrically H~n and H~n (2) cohomology groups of a topological space, while μ, △, and γ_p~n are natural maps, 1/2δ and steenrod squares respectively. Let f: K→L be a homotopy equivalence between spaces K and L, then f induces a proper isomorphism f: H(L)→H(k), where H(L) and H(K) denote (μ,△, γ,)-systems associated with the spaces L and K respectively. Under proper isomorphisms there are invariants of (μ, △, γ)-systems. No doubt this contributes a new method to find new numercial invariants of a topological space.Let H and H denote two (μ, △, γ)-systems. If there is a system of isomorphisms h~n: H~n→H~n (n=1, 2…), then according to J. H. C. Whitehead [3] {h~n}may be extended to a (μ, △)-isomorphism h: H→H which induces, according to [1], a new (μ, △, γ)-system H = H [{H~n}, {H~n (2)}, μ, △, h~(-1) γ_p~n, h].Lemma 1. Let f: H→H be a (μ, △)-isomorphism. Then f is a proper isomorphism, if, and only if, h~(-1) f: H→H is a proper automorphism.This is a trivial extension of an argument in [1].Let E be a set of pairs of integers (n,p), where p is even. If we know a set, A, of invariants of a proper isomorphic class of (μ,△,γ)-systems, so that these invariants determine the normal forms, N_p~n of γ_p~n in case (n, p) ∈ E, there must be a proper automorphism g carrying the (μ, △, γ)-system H into another, namely g H, in which γ_p~n, becomes N_p~n if (n, p) ∈ E. In other words, we have gγ_p~n g~(-1)=N_p~n, if (n,p) ∈E. If the proper isomorphic class containing H has the same set of invariants A as the class containing H, then there is a proper automorphisms g of H so that g γ_p~n g~(-1)= N_p~n, if (n, p)∈ E. It is not certain whether H and H are proper automorphic. But we have.Lemma 2. The (μ, △,γ)-systems H and H are proper automorphic, if, and only if,g H and g H are properly automorphic.Hence our algebraic problem is to find the complete set of numerical invariants of {γ_p~n} ((n,p)E) under all the (μ, △)-automorphisms h which satisfies the following conditions h N_p~n=N_p~n h, (n,p)∈E; h γ_P~n= γ_p~n h, (n, p)E.Of course we must investigate the necessary and sufficient conditions, so that a system of automorphisms of {H~n(m)}(m = 0, 2; n = 1, 2,…)constitute a (μ,△)-automorphism.By the use of the above argument we solve a particular problem as follows: By we mean the set of (μ, △, γ)-systems, each of which consists of no more than two non-trivial γ_s~′, nomely, γ_p~n and γ_p~m with an additional condition that between the sets of groups(H~n, H~(n+1), H~(n+p), H~(n+p+1)) and (H~m, H~(m+1), H~(m+k), H~(m+k+1) there is a unique common group. We determine the complete set of numerical invariants of proper isomorphic class of (μ, △, γ)-systems of and reach the normal forms of classes of them. Some invariants are essentially new. With respect to a given topological space we may, in general choose a pair of Steenrod squares satisfying our condition so that these two squares together with the cohomology groups H~n(m) (m=0, 2; n=1, 2, ... ) of the topological space constitute a system of Consequently we contribute new numerical invariants to the topological space according to each choice which we have made.If (H~n, H~(n+1), H~(n+p), H~(n+p+1)) ∩ (H~m H~(m+1), H~(m+k), H~(m+k+1) is one group, then there are four cases as follows: (a) n=m+1, k>2, p≠k, k-2; or m=n+1, p>2, k≠p, p-2. (b) n+p=m+k+1, p>2, p≠k, k+2; or m+k=n+p+1, k>2, k≠p, p+2. (c) n=m+k+1 or m=n+p+1. (d) n+1=m+k,k>2, p>2; or m+1=n+p, p>2, k>2.Let
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  SOME DEFINITE INTEGRALS

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 302-312. https://doi.org/10.12386/A1956sxxb0023

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  It was proved indirectlyi that for n(≥3), m, n being integers, then where z is a vector with n complex components and zz' P_m~(ν)(t) is the ultraspherical polynomial of the m-th degree. The aim of the paper is to prove (1) directly.Firstly, we reduce (1) into a simple integral which is a special case of the following result: Let s be a complex number with real part > m + 2λ, λ, be a real number >-1/2. Then, we haveThe proof depends essentially on the following facts: Substituting P_m~(λ)(x) by its expansion and integrating term by term, we obtain a generalized hypergeometric series. Fortunately the series is of Saalschutz's type, it can be expressed in terms of gamma function.
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  THE MINIMUM MODULUS AND MINIMUM DISTORTION FOR THE CLASS T_r TYPICALLY REAL FUNCTIONS.

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 313-319. https://doi.org/10.12386/A1956sxxb0024

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  Golusin proved that any function of the class T_r of typically real functions can be put into the form of the integral with an increasing a(θ)such that where S(z,a) denotes the expression By means of this theorem, Golusin gave the upper bounds of |f(z)| and |f'(z)| on the circle |z|<1. In the present note we give the precise lowerbounds of |f(z)| as well as of |f′(z)|in the divisions (Ⅰ), (Ⅱ), (Ⅲ), (Ⅳ), (Ⅴ), (Ⅵ), (Ⅶ) of the unit circle |z|<1. The inequalities R(1+z~2/z)≥2,-2≤R(1+z~2/z)≤2,R(1+z~2/z≤-2 holds respectively on (Ⅰ), (Ⅱ), (Ⅲ), these three divisions cover the whole circle |z| < 1. And the inequalities |1+z~2/z|<2, R(1+6z~2+z~4/z+z~3)≥4,R(1+6z~2+z~4/z+z~3)≤-4 define (Ⅵ), (Ⅴ), and (Ⅶ) respectively. The division (Ⅵ) is represented by the three inequalities |1+z/z|<2,|R[1+6z~2+z~4/z+z~3]|<4,|z|<1 which contains two domains as shown in the figure Ⅱ. The four divisions (Ⅳ), (Ⅴ), (Ⅵ) and (Ⅶ) also cover the whole circle.We prove the following two theorems:Theorem 1. If f(z) ∈ T_r, then |f(z)|≥|z/1+z~2|,|(z/1-z~2)~2I[1+z~2/z]|,|z/(1-z)~2| hold predsely in (Ⅰ), (Ⅱ), (Ⅲ). The extremal functions are respectively f(z) =s(z,-1),z~2/(1-z~2)~2 1/S(z,a),S(z,1), where a being a suitable number in the interval [-1, 1 ].Theorem 2. If f(z) ∈ T_r, then |f′(z)|≥0,|1-z/(1-z)~3,|z+z~3/(1-z~2)~3 I(1+6z~2+z~4/z+z~3)|,|1+z/(1-z)~3| hold precisely in (Ⅳ), (Ⅴ), (Ⅵ), (Ⅶ). The corresponding extremal functions for (Ⅴ),(Ⅵ), and (Ⅶ) are respectively f(z)=S(z,-1),z~2/(1-z~2)~2 1/S(z,a) (-1≤a≤1),S(z,1). Besides the case (Ⅳ), the respective extremal functions for (Ⅰ), (Ⅱ), (Ⅲ), (Ⅴ), (Ⅵ) and (Ⅶ) given in Theorem 1 and Theorem 2 are unique.
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  ON THE DEFORMATION OF A RIEMANNIAN METRIC V_m IN A SPACE OF CONSTANT CURVATURE S_(m+1)

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 320-332. https://doi.org/10.12386/A1956sxxb0025

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  With the aid of the method of exterior differential forms N. N. Yanenko has recently given a complete classification of the deformation of m dimensional Riemannian metric ds~2 = gii du~i du~i (i,j=1,…,m) in Euclidean space E_(m+1). Here we propose to investigate the same problem in a space S_(m+1) of constant curvature k_(om+1). Introducing the definition of the k_o-rank of a metric, we obtain the following results:1. In general, a V_m S_(m+1) is indeformable and the only possible deformable metric must be of k_o-rank ≤ 2 (an extension of Beez's Theorem).2. When k_o-rank ≥4, the Peterson-Codazzi equations are consequences of Gauss equations (an extension of T. Y. Thomas' Theorem).A complete classification of deformable hypersur faces V_m S_(m+1) is given.
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  THE COEFFICIENTS AND SECTIONS OF THE SCHLICHT FUNCTIONS

  Acta Mathematica Sinica, Chinese Series. 1956, 6(2): 333-345. https://doi.org/10.12386/A1956sxxb0026

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  Let the k-symmetric function f_k(z) =be regular and schlicht in the circlez= |z|< 1.Write It is proved (see [3], [6], [7], [8]) that when k = 1, 2, 3 for all n the polynomials σ_n~((k))(z) and teh latter number can never replaced by any are sehlicht in the circle |z| 3, a problem, which remains now unsolved.In this paper, containing two parts, the author first makes some estimations of some beginning coefficients of the functions f3(z), f4(z) and f_1(z) with a_1~((1))=0 and then, in the latter part, using these results and based on the method of Kung-Sun, the author proves the following theorems concerning the sections of the 4-symmetric functions:Theorem Ⅰ. All polynomials σ_n~((4))(z), except n=4, are schlicht in the circle |z| < 4 andcan never be replaced by any larger one.Theorem. When n > 8, the polynomial σ_n~((4))(z)=z+a_1z~5+…+a_(n-1)z~4~((-1)+1) is schlicht in the circle |z| <(1-4 1n n/n)~1/4.