15 January 1957, Volume 7 Issue 1

    

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  MODAL SYSTEMS WITH A FINITE NUMBER OF MODALITIES

  Acta Mathematica Sinica, Chinese Series. 1957, 7(1): 1-27. https://doi.org/10.12386/A1957sxxb0001

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  If we add the modal concepts“◇p”(it is possible that p),“~◇p”(itis impossible that p),and“□p”(it is necessary that p)etc.,to the truth-valued logical system,we get a modal logical system.A proposition whichis formed by means of propositional variables and the two connectors “~”“□”alone is called a modality.As to the five modal systems proposed by C.I.Lewis S1-S5,it has beenproved that the systems S1 and S2 contain infinitely many modalities andthat the systems S3,S4 and S5 contain only a finite number of modalities,the number of which are 42,14 and 6 respectively.In the present paper we start with a basic modal system B,far weakerthan the system S2,and show that if we add to it the following proposi-tions(where“□”means n symbols“□”written successively)(?)then the resulting system B_1 will contain no more than 126 modalities;andif we add still the proposition(?)then the resulting system will contain exactly the same modalities as thatcontained in S3(see§2).We then generalize this result and show that ifwe add to the system B the following propositions(?)then the resulting system B_n will likewise contain only a finite number ofmodalities,no more than(see§3)(?)We show that the first proposition is neccessary for a system to containonly a finite number of modalities,in a next paper we shall discuss theneccessary and sufficient conditions and the corresponding modal systems.What we have assumed in the deduction are very weak,hence the wholediscussion and the results may be written in a pure mathmatical language.
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  ON DEMOULIN TRANSFORMS OF PROJEGTIVE MINIMAL SURFACES(Ⅰ)

  Acta Mathematica Sinica, Chinese Series. 1957, 7(1): 28-50. https://doi.org/10.12386/A1957sxxb0002

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  The main purpose of the present paper is to investigate the relationbetween the Godeaux sequences of a projective minimal surface S and oneof its Demoulin transform S.We demonstrate the followingTheorem.If the corresponding points of the Godeaux sequences of Sand(?)be arranged in three rows,(?)then in the line space R_5 the join of any two consecutive points of the se-cond row must intersect the join of the two consecutiue points(if any.)standing in the same columns of the first or third row.
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  GENERALIZED ITERATED LAPLACE OPERATORS

  Acta Mathematica Sinica, Chinese Series. 1957, 7(1): 51-62. https://doi.org/10.12386/A1957sxxb0003

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  Suppose that U(P) is a summable function defined in a domain D inthe m-dimensional Euclidean space.Let(?)be the hyperspherical mean of U on the hyperspherical surface C_m(P;r),lying entirely in D,with center P and radius r,where C(r)is the volume of C_m(P;r)and dS_Q is its volume element.Write(?)We introduce the generalized n-th iterated Laplace operator(?)related to the function U(P) as follows:(?)where V[μ_k;n,α],V[U;n,α] and D_n denote the following three deter-minates respectively:(?)If U(P)∈C~(2n),i.e.if U(P) is continuous together with its partial de-rivatives of the first 2n orders,then V_K~n,a~U(P)(K=0,1,2,…)exist andare all equal to the ordinary n-th iterated Laplace operator Δ~nU(P).The main theorem of this paper is as follows:If U(P)∈C~(2n-2)(D),and if ▽_K~(n,a) U(P)=0 for some non-negative integerK and α≧2,then U(P) is harmonic of order n in the domain D.We also obtain a theorem concerning subharmonic functions of ordern which runs as follows:If U(P)∈C~(2n-2)(D),and if (-1)~(n-1)▽_K_k~(n,a) U(P)≤0 for some non-negativeinteger K and α≥2, hen U(P) is subharmonic of order n in the domainD.
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  COMPARISON OF GRADIENT METHODS WITH APPLICATIONS

  Acta Mathematica Sinica, Chinese Series. 1957, 7(1): 63-78. https://doi.org/10.12386/A1957sxxb0004

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  In this paper,the problem of solving a system of n linear algebraicequationsAx+b=0,(1)is used to compare different gradient methods with regard to amount ofwork inyolved per iteration and effectiveness of the method.Kantoro-vich's process for solying(1)in case A is symmetric and positive definiteis interpreted as a process minimizing the residual |F| of an equivalentsystem of(1)F=Bx+c=0where B'B=A and B'c=b,along the direction of quickest descent of thefunction |F|.Following Kantorovich,the effectiveness of gradient methods is impro-ved by combining two consecutive steps into one.It is proved that if theprocess does not hit the solution at one stroke,then the improved processactually gives better results than that of two separate consecutive steps.Finally,my gradient method is used to find the latent vectors of asymmetric matrix.It is proved that if the starting vector is sufficientlyclose to a latent vector,then the process converges monotonically to thelatent vector.An example is given to show that the gradient method maygive better results than the ordinary iteration method.
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  ON THE REALIZATION OF COMPLEXES IN EUCLIDEAN SPACES Ⅱ

  Acta Mathematica Sinica, Chinese Series. 1957, 7(1): 79-101. https://doi.org/10.12386/A1957sxxb0005

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  For the realization problem of complexes or more general spaces in eu-clidean spaces we may cite the following results:1°In 1932 Van Kampen~[1]proved the existence of n-dimensional comp-lexes K not realizable in euclidean space of dimension 2n.His proof depends on certain invariant deduced from the reduced 2-foldsymmetric product K of K.The author~[2] has pointed out that Van Kampen'sinvariant is merely the extreme one of a system of invariants Φ~m∈H~m(K,I(m)).(I(m)=I or I_2,depending on m being even or odd),namely Φ~(2n),andΦ~m=0 is a necessary condition for K to be realizable in R~m.We provedalso in[2] the topological invariance of Φ~m,i.e.,the independence of Φ~m fromthe chosen subdivision K of the space of K,while Van Kampen,in the ex-treme case m=2n considered by him,has not even proved its combina-torial invariance.2°In the case of differentiable manifold,Whitney(cf.e.g.[3])hasintroduced a system of invariants which I have called the dual Whitneyclasses(?),and proved that(?)=O,k≥m-n(1)are necessary conditions for an n-dimensional closed differentiable manifoldto be realizable in R~m.3°For compact Hausdorff spaces X,Thom has proved([4]Th.Ⅲ.25)the following theorem:Let Q~i be certain operations deduced from Steenrodsquares,thenQ~iH~r(x,l_2)=0,2i+r≥m(2)are necessary conditions for X to be realizable in R~m The operations Q~iwas previously introduced by the author by applying Smith's theory of pe-riodic transformations and will henceforth be denoted by Sm~i,cf.[5]and [6].Besides,Flores~([7])get also same results as Van Kampen concerning theexistence of n-dimensional complexes K not realizable in R~(2n).However,whathe used to prove his results is get by imbedding first K in R~(2n+1),and isactually not an invariant but depends in general on the way of realization of K in R~(2n+1).This will be explained further in the sequel.In the various theories listed above,not only the methods used are quitedifferent from each other,but also the objects and the realization conceptinvolved are not the same.For example,the theory of Van Kampen studiesabout the semi-linear realization of finite complexes,the theory of Whitneyis applicable only for the differentiable realization of differentiable mani-folds,and the theory of Thom is concerned about topological realization ofmore general spaces.The present-paper gives a general theory including the various theoriesabove cited as its particular cases.This theory may be formulated in termsof the following easily proved fundamental theorem:For any Hausdorff space X let X be the space of all ordered pairs(x_1,x_2),where x_1,x_2 ∈X and x_1≠x_2.Let t:(?)≡(?)be the transformationt(x_1,x_2))=(x_2,x_1),X the modular space (?)/t.With respect to the pair((?),t),we may define according to P.A.Smith~([8])a system of cohomologyclassesΦ~m(X)∈H~m(X,I(m)),where H represents singular homology system.Then Φ~m(X)=0 is a nece-ssary condition for X to be topologically realizable in R~m.If X be a finite polyhedron,then according to results of[2],the classesΦ~m defined above are the same as the classes Φ~m defined in that paper ina quite different manner.It follows that Φ~m=0 is not only a necessarycondition for X to possess a subdivision K such that K is semilinearly rea-lizable in R~m,as shown in[2],but also one for X itself to be topologicallyrealizable in R~m.If X is a closed differentiable manifold of dimension n,then we provethat from ρ_2Φ~m(X)=0 we get(1),but not vice versa.Our fundamentaltheorem has therefore Whitney's theorem as its consequence.It followsalso that Whitney's condition(1)is not only necessary for the differentiablerealization,but also necessary for topological realization of X in R~m.We prove also that if X be a finite polyhedron,then from ρ_2Φ~m(X)=0we get(2).Hence our fundamental theorem has also,at least in the caseof finite polyhedron,Thom's theorem as its consequence.In the case of differentiable manifold,Whitney~([9])has introduced theconcept of“immersion”in a euclidean space.He proved that any differen-tiable manifold of dimension n may be immersed in R~(
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  ON THE SOLUTIONS OF THE SYSTEMS OF LINEAR EQUATIONS WITH PRIME VARIABLES

  Acta Mathematica Sinica, Chinese Series. 1957, 7(1): 102-122. https://doi.org/10.12386/A1957sxxb0006

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  Let α_(11),…,α_(1,2n+1),…,α_(n1),…,α_(n,2n+1),b_1,…,b_n,be 2n(n+1)integers,and P be a sufficiently large integer.My purpose is trying to find anasymptotic formula for the number of prime number solutions of the systemof equations(?)(μ=1,…,n),within the region 2≤ρ_v≤ρ(v=1,…,2n+1)under some conditions.Thisproblem was proposed by Prof.Hua in his book“Theory of Additive Primenumbers”,and is the natural extension of the famous“Goldbach-Vinogradov's”Theorem.Putting L=logP,e(x)=e~(2πix),(Φ(p))~(2n+1)s(p)/p~n be the number of solu-tions of(?)(μ=1,…,n)within the range 1≤l_v≤ρ-1,thenⅠproved in this paper the Theorem 1.If all the n-th minors of the matrix(?)is not equal to zero,and have no common divisor other thanⅠ,then thenumber of systems of prime number solutions of the equations(?)(μ=1,…,n),within the region 2≤ρ_v≤ρ(1≤v≤2n+1)is equal to(?)where the constant in the symbol O is independent of b's,R=(logL)~n or1 when n is greater than or equal to 1,and(?)(?)where(?)Without any difficulty,by the same method as in the proof of Theorem1, we can proveTheorem 1'.Let m≥2n+1,if all the n-th minors of the matrixis not equal to zero,and have no common divisor other than 1,then(?)where I_1(b;P),B_1(b;P),s_1(p) have the same meaning as in Theorem 1,andthe constant involved in the symbol O is also independent of b's. When n=1,Ⅰprove the followingTheorem 2.Let(α_1,…,α_m)=1,m≥3,b≡sum from v=1 to m α_v(mod 2).If(α_v_1,…,α_v_(m-1),b)=1.is always true for any m—1 different α_v,then the number of prime num-ber solutions of the equationα_1p_1+…+α_mp_m=bwithin the range 2≤P_v≤P,(1≤v≤m),is equal to(?)where(?)(b)is an infinite product and is greater than an absolute constant.Moreover,if α_v>0 for all v,then(?)where A_m denotes the product α_1…α_m.
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  ON DEMOULIN TRANSFORMS OF PROJECTIVE MINIMAL SURFACES(Ⅲ)

  Acta Mathematica Sinica, Chinese Series. 1957, 7(1): 123-127. https://doi.org/10.12386/A1957sxxb0007

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  As a generalization of the main theorem given in the first paper wegive here an almost geometrical demonstration of the followingTheorem.If the corresponding points of the Godeaux sequences of aprojective minimal surface S and one of its Demoulin transforms(?)in theline space S_5 be arranged in three rows:…U(n+1)U_n…U_3 U_2 U_1 U…V_m V_(m+1)…,(?)…U_(n-3) U_(n-4) …V V_1 V_2 V_3 … V_(m+4) V_(m+5)…,then the join of any two consecutive points in the middle row must intersectthe join of the two consecutive points standing in the same columns of thefirst or third row,and therefore the corresponding points of the Godeauxsequences constitute two broeken lines,generally extended in both directions,which intersect each other at intervals of three sides.
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  THE BASIC CALCULUS

  Acta Mathematica Sinica, Chinese Series. 1957, 7(1): 132-143. https://doi.org/10.12386/A1957sxxb0009

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  The Basic Calculus may be conceived as a part of the intersection ofJohanson's Minimalkalkül and Lewis's modal calculus S4.It uses twoprimitive inference-schemas:(?)The following are its axiom-schemas(14 in number):A1 A(?)AA2(B(?)C)(?)[A(?)(B(?)C)]A3 A(?)(A(?)B)(?)BA4(A(?)B)(?)(B(?)C)(?)(A(?)C)A5 A(?)B(?)AA6 A(?)B(?)BA7(A(?)B)(?)(A(?)C)(?)(A(?)B(?)C)A21 A(?)A ∨ BA22 B(?)A ∨ BA23(A(?)C)(?)(B(?)C)(?)(A∨B(?)C)A24 A(?)(B∨C)(?)(A(?)B)(?)(A(?)C)A32(A(?)B)(?)B(?)A A33 A(?)(B(?)B)A34 A(?)(A(?)B)(?)BThe Basic Calculus has two important properties:(1)Let A,B,…be any given propositions.A/B is an inference of thecalculus(i.e.,fits into a primitive or derived inference-schema of thecalculus)if and only if A(?)B is a theorem of the calculus.Further,(?)is an inference of the calculus if and only if A(?)B(?)C is a theorem ofthe calculus.Similarly for three or more premises.This property is aweakened form of the“Deduction Theorem”.(2)LetΦ(X)stand for Ψ(X,C,D,…),where Ψ is constructed purelyby combination of the four primitive connectives.Then(?)is an inference-schema of the calculus.This property corresponds to whatis called the“Regularity Theorem”.(If we leave out the three axiom-schemas A24,A33,A34,both proper-ties will still belong to the resulting Ultrabasic Calculus.)If A2 is replaced byB1 B(?)(A(?)B)we get the Minimalkalkül.If,on the other hand,we add the followingtwo axiom-schemas:C1(A∨B)(?)A(?)BE1 A(?)B∨(?)Bwe get S4.Doing both things at once gives the ordinary Two-ValuedCalculus.
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  AN EXTENSION OF PRIVALOF THEOREM

  Acta Mathematica Sinica, Chinese Series. 1957, 7(1): 144-165. https://doi.org/10.12386/A1957sxxb0010

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  Let m=2n(n≥2),z_α=u_α+iu_(n+α)(α=1,…,n),and(?)be a domainof the 2n-dimensional space of u_1,…,u_(2n),and its boundary Ω be a(2n—1)-dimensional smooth orientable manifold of olass C~2 defined byF(u_1,…u_(2n))=0.(1)Let K_(2n-1)(z,(?))be a complex exterior differential form of degree 2n-1(?)whereγ(z—ξ)denotes the euclidean distance of z=(z_1,…,z_n)andξ==(ξ_1,…,ξ),namely(?)(3)andσ=(n-2)1/2πi_n.It is known that if f(z)is regular in the domain(?)and on its boun-dary,then we have the Cauchy formula~([1])f(w)=σ∫_πf(z)K_(2n-1)(z,w),x∈(?)Since the exterior differential operatorD=sum from k=1 to 2n(?)du~k,if we set(?) then it is obviousD=d+(?)(5)First we prove the following two lemmas:Lemma 1.K_(2n-1)(z,ξ)is homologous to zero at those points z≠ξ.Lemma 2.K_(2n-1)(z,ξ)is invariant under the unitary transformation.We then obtainTheorem 1.If z_0 is a point on Ω,then the principle value of theintegral of K(2n-1,)(z, z_0) on Ω exists,andV.P.∫_π K_(2n-1)(z,z_0)=1/2.Corollary:If f(z)is defined on Ω and satisfies the H(?)lder conditionand z_0∈Ω,then the principle value of the integral(Ⅰ)∫_Ωf(z)K_(2n-1)(z, z_0)exists.Theorem 2.If Ω is a smooth orientable manifold belongs to class C~2and f(z)is a continuous fumction of complex value defined on Ω,whichsatisfies H(?)lder condition and difines a function F(w)in(?)such thatF(w)=∫_Ωf(z)K_(2n-1)(z,w)and if W_0 is an arbitrary point on Ω,then we have(Ⅱ)F_i(W_0)=V.P∫_Ωf(z)K_(2n-1)(z, w_0)+1/2f(w_0),(Ⅲ)F_e,(W_0)=V.P∫_Ωf(z)K_(2n-1)(Z, W_0)-1/2f(W_0)where F_i(W_0)and F_e(W_0)devote the limit values of F(w)when w approachesw_0 from the inner part and the outer part of the domain(?)respectively.Theorem 3.If f(z)is regular in(?)and on its boundary,thenF_i(z_0)=f(z_0).
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  CORRECTION TO MY NOTE“CONCERNING DERIVED LIMITS AND SET SEQUENCES IN COMPLETE METRIC SPACES”

  HSU L.C.

  Acta Mathematica Sinica, Chinese Series. 1957, 7(1): 166-166. https://doi.org/10.12386/A1957sxxb0011

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