中国科学院数学与系统科学研究院期刊网

15 October 1959, Volume 9 Issue 4
    

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  • Acta Mathematica Sinica, Chinese Series. 1959, 9(4): 365-381. https://doi.org/10.12386/A1959sxxb0034
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    In this paper the uniqueness problem of the Cha-for y≠0)is considered.The domain D is bounded(?)by three curves showing in the figure,where T_1 and T_2are characteristics defined by the equation dx~2+Kdy~2=O,T_3 is a continuous curve.Let the coordinate of P be(Xo/yo)and the minimum and maximum abscissas ofT_3 be x_1 and x_2.When y<0,let 1+2(K/k_y)=f(y)and(?)Let in(n=0,1,2)be the least positive roots of the following equations:(?)Where δ=0 or 1 according to x_0+2Y(?) Finally,let y_1=0 if f(y)>0 for all y0≤y<0,otherwise let y1 be the upperbound of values y in the interval yo≤y<0 satisfying f(y)<0.Theorem.If y1<0 and there exists a positive numberεand an integer n(n=0,1,2)such that the following relation holds for yo≤y≤y1:(?)and if u is a quasi-regular solution which vanishes on T_2+T_3,then u=0 in D.The example for gas dynamical problem shows that this theorem is better than theresult of [1] and [2].The method of proof of the theorem is to consider the sum of the energy integral(?)dxdy=0 and the zero integral(?)(Pu~2)+
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(4): 382-388. https://doi.org/10.12386/A1959sxxb0035
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  • Acta Mathematica Sinica, Chinese Series. 1959, 9(4): 389-412. https://doi.org/10.12386/A1959sxxb0036
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    A fundamental view point of intuitionistic logicians is that the law of exclude middleis not admissable while not deniable(Cf.“the absurdity of absurdity of the law of excludemiddle”).It was A.H.(?)who first formalized such a system,and thencame A.Heyting.The two systems developed by them are not identical.The formersystem(the system J)possesses the following property:If we transform an assertedproposition in the traditional two-valued system(the system M)by doubly negating eachof its components,then the resulting proposition is deducible in the system J(the con-verse is obviously true).The latter system(the system H)possesses the property:If aproposition is deducible in the system M,then its double negation is deducible in the.system H;and if the negation of a proposition is deducible in M,then the same is truefor H;and conversely.From the point of view of Brouwer,the system H is moresatisfactory.Hence,although criticized by some logicians,the system H is consideredthe sole intuitionsistic system so far developed.However,there are many other logical systems which bear the same relation witlthe system M as the system H does.Such systems will be called intuitionistic systems,In the present paper we give many of such systems.In discussing them we find thatsome of them(e.g.the systems I3 and I4)are more satisfactory than H(see §2).Intuitionistic systems will be generalized to co-denial systems(see§3).Parallelly wedevelop pseudo-modal systems and co-△ systems(see§4).As was found by J.Johanson,the system J is N-generalizable.By means of theconcepts N-generalizability and N-introducibility we discuss some important properties ofvarious logical systems.(see §1).Beside obtaining thus various properties of some important systems,we prove thefollowing theorems:Among the intuitionistic systems,the co-denial systems,the pseudomodal systems orthe co-△ systems,there exists at most one which is N-generalizable and N-introducible.Every intuitionistic system contains(in a certain sense)the whole asserted proposi-tions of the systme M,yet none of them contains also all the rules of procedure of thesystem M(except,of course,the system M itself).
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(4): 442-445. https://doi.org/10.12386/A1959sxxb0039
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  • Acta Mathematica Sinica, Chinese Series. 1959, 9(4): 446-454. https://doi.org/10.12386/A1959sxxb0040
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    The object of the present paper is to develop the theory of conjugate nets in ann-dimensional projective space S_n by utilizing Cartan's method of exterior forms.HsiungChun-Chih has demonstrated that the two tangents at a generic point of a conjugate netN_x intersect a fixed hyperplane at two points which describe in turn two conjugate netsand stand for Laplace transforms to each other.A generalized theorem with a simpleproof has been obtained by P.O.Bell,but there is no improvement concerning the secondtheorem of Hsiung:The point of intersection of the tangent plane at a generic point ofN_x with any fixed subspace S_n—2 of n—2 dimensions describes a conjugate net in S_n-2.In the present paper we merely consider the general ease where the associate Laplacesequence of the conjugate net N~x is neither periodic nor degenerate,so that a movingframe{A_1 A_2…A_(n+1)}can be attached to a generic point A_1 of the net N_x(A_1),suchthat…,A_5 A_3, A_1,A_2,A4,…constitute the Laplace sequence.Suppose that n≥2k≥4,and S′_kand S″_k denote the k-dimensional osculating spaces of the corresponding net curvesat A_3 and A_1 respectively.If we take two points X and Y respectively,in S′_k and S″_kin such a way that the tangent plane of the surface(X)[(Y)]at X [Y] passes throughY [X],then X and Y must describe two conjugate nets which are Laplace transformsto each other,and the determination of such points X and Y depends upon 2k arbitraryfunctions of one argument.The above result not only furnishes a natural generalization of Bell's theorem,butalso contains a special case where X and Y are respectively the points of intersection of S′_kand S″_k with any fixed space S_(n-k)of n—k dimensions.Obviously,even this parti-cular case may be seen as an extension of the second theorem of Hsiung.Moreover,the:last part of our theorem also gives a generalization of a former result of the presentauthor,namely,when k=1,the determination of Levy transforms of a conjugate netdepends upon two arbitrary functions of one argument.The above consideration leads us to generalize the notion of the conjugate as well asharmonic relation between a conjugate net and a rectilinear congruence.If we take apoint X in the osculating space S′_k,for example,of the curve u at the point A_3,suchthat X describes a conjugate net X(u,u),then the net X(u,v)is said to be conjugateof the k th species to the congruence T_(A3A1).According to this definition the ordinaryconjugate relation is of the first species,since the point X lies on the corresponding rayof the congruence.In the last case it is known that the Laplace transform Y of the netX must lie on the corresponding Laplace transform of the congruence,and in consequence,the congruence T_(XY)is harmonic to a conjugate net.We can now extend this fact to thecase of conjugate relations of the k th species and show,in fact,that the Laplace trans-form Y of X(u,v)along the direction u must lie in,the osculating space S″_k of thecurve u at the point A_1.Thus we reach the general notion of the harmonic relation ofthe kth species between the conjugate net(A_1)and the rectilinear congruence T_(XY).
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(4): 455-467. https://doi.org/10.12386/A1959sxxb0041
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    We consider system of linear differential equations with constant coefficients(?)Liapounoff proved that:if real parts of all roots of the characteristic equation(?)arare negative,then,for any given negative definite m-th homogeneous polynomialu(x_1…,x_n),there exists an unique positive definite m-th homogemeous polynomialV(x_1,…,x_n),which satisfies the equation(?)In this paper,we shall give the explict form of Liapounoff function V.V can bewritten in the form of a sum of square terms and its coefficients are functions of thefollowing Routh-Hurwitz determinants:(?)We introduce the following symbols:1°Take the n—th order determinant(?),replaceits j-th column by(?)and denote it by M~(i)(x_1,…,x_n),then take a minor ofM~(i)(x_1…,x_n),the elements of this minor are situated on v_1,…-th columns andV1,…,Vk-th rows(v1<…
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(4): 468-474. https://doi.org/10.12386/A1959sxxb0042
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    By the reduced product,S_∞~n,of a sphere,S~n,we mean the CW-complex(?)where e~(rn)is attached to the(r-1)n-skeleton,(S_∞~n)~((r-1)n),of S_∞~n by the secondary productaccording to[1].In[1]and[7]it has been independently proved that(?)here we consider the complex(?)being attached to(S_∞~n)~((r-1)n)bythe same map as e~(rn).Then(?)Let g:S~P→(S_∞~n)~(rn)represent an element{g}of(?)we mean an injection and(?)(?)we mean a projection.If(?),there is a homotopy F:S~p×I→(S_∞~n)~(rn)∪e′~(rn)suchthat(?)and(?).This homotopy supplies anelement of(?).With this view point the anthordefines bomomorphisms(?) (1)such that Н_(ρ-1)is defined upon(?)uponК_(j+1)~(-1)(0),j+1,…,p-2.Then he proves the followingTheorem.To each element α of(?)either there is an element β of(?)such that α=Еβ,Е being the suspension homomorphism,or there is one of the2(p-1)homomorphisms in(1)denoted by G so that G is defined on α and G(α)≠0.If p=2 the anthor shows that К_1=0,meanwhile Н_1 is actually the Hopf invariantdefined by G.W.Whitehead.If p=3 the anthor shows that К_2=К_1=0 and Н_2,Н_i areexplicitly expressed as follows:(?)
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(4): 475-493. https://doi.org/10.12386/A1959sxxb0043
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    For any finite simplex complex Κ let(?)be its reduced two-fold product consistingof all cells σ×τ of the product complex Κ×Κ for which σ and τ have no vertices ofΚ in common.The permutation σ×τ(?)τ×σ is of period two and has no fixedcells so that special groups ~ρH′((?))may be defined according to the theory of P.A.Smith about periodic transformations,where ρ is either d=1-t or s=1+t,t beingthe chain map t(σ×τ)=(-1)~(dim σ dim τ(τ×σ).A continuous mapping f of the space|Κ|of K in a Euclidean space R~N of dimen.sion N is called an imbedding if f is topological.It is said to be a linear map of Κ inR~N if J is linear on each simplex of Κ.Two linear maps f,g of Κ in R~N will be saidto be linearly isotopic if there exist a simplicial subdivision M′of the product complexΜ=Κ×Ι(Ιbeing the interval[0,1])in R~N and a linear map F of M′in R~N suchthat F/|Κ|×(0)≡f,F/|Κ|×(1)≡g,and for each t ∈Ι,F/|Κ|×(t)gives animbedding of|Κ|in R~N.Let ρ~N=1+(-1)~(N-)t,then for any two linear imbeddingsf,g of Κ in R~N(oriented in a definite manner)a certain class(?)may be defined the vanishing of which is a necessary condition for f,g to be linearlyisotopic.It turns out that in the“critical”case 2 dim Κ+1=N(dim Κ>1),this condition is not only necessary but also sufficient,which will be studied more in detaili na succeeding paper.The present paper introduces the necessary concepts,gives definitionsof the classes(?)and studies their elementary properties of which we shall mentionthe following one:For any two linear imbeddings f,g of a finite simplicial complex Κ in a Euclideanspace R~N definitely oriented we have(?)in which(?)is a certain class associated to any linear imbedding ofΚ in R~N(oriented).It is also defined in this paper and(?)iseasily seen to be necessary for two linear imbeddings f,g of K in R~N to be linearly isotopic.The classes(?)have the advantage that they may also be defined for any topolo-gical imbedding of any topological space in R~N and the identity of these classes for twotopological imbeddings is again necessary for these imbeddings to be topologically isotopicHowever,the appearance of the factor 2 in the left hand of(1)shows that in the caseof linear imbeddings of complexes the classes(?)would be more efficient than(?)in studying their linear isotopy.It is in fact the classes(?),but not theclasses(?),which furnish the necessary and sufficient condition for the linear isotopy inthe critical case as mentioned above.For a more detailed abstract of the theory of linear isotopy see also[2].
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(4): 494-502. https://doi.org/10.12386/A1959sxxb0044
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    As usual,we use M/M/n to denote such a queuing process:the arrival and servicetime are assumed negative exponentially distributed with means λ~(-1) and μ~(-1) respectively,the number of servers is n.Let pk(t) be the chance that at time t there are k customerspresent including those being served.In this paper we prove thatTheorem 1.For k≥n,we have(?)where a is the root ofnμx~2-(λ+u+nμ)x+λ=0with minimum absolutely value,and(?)Theorem 2.For n=1,2 and 3,the limits(?)p_k(t) k=0,1,2,…exist.