ON INTERPOLATION BY S_3~1(△_mn~(1))(I)

§0 Introduction Let △_(mn)~(1) be a subdivision of D: [a,b]x[c,d] (Fig.1). l_1,_2be lengths of [a,b] and [c,d] respectively, and h_1=l_1/m, h_2=l_2/n,l=max(l_1,l_2). Suppose that S∈S1/3(△_(mn)~(1)). In §1-§5 we consider thefollowing interpolation problem:     

Characterizations of Best L_p(1≤p≤+ ∞) Approximation With Restricted Ranges of Derivatives

Let [a,b] be a compact set containing at least n + 1 points. If Φ_n = span(φ_1, φ_2,…, φ_n) is an n-dimensional subspace of L,[a, b] m φ_1~(s), φ_2~(s)…, φ_n~(s)exist (s≥0is an integer) and have a maximal linearly independent subset which is an extendedChebyshev system of order rs on [a,b], then write     

Several Results on Systems of Residue Classes

Let (m,n) and a(n) denote the g.c.d, of m, n and the residue class {x∈Z∶x≡α (mod n)} respectively. Any period of the characteristic function ofkU a_i(n_i) is called a covering period of {a_i(n_i)}_(i-1)~k.i-ITheorem Let A = {a_i(n_i)}_(i-1)~k. be a disjoint system (i. e. a_I(n_I,...,a_k(n_k) are pairwise disjoint). Let [n_I,...,n_k] (the I.c.m. of n_1,...,n_k) have the prime faetorization [n_1,...,n_k] = Πp_i~ai and T = Πp_iβi(β_i≥0 be the smallest positive covering period of A. Then     

关于S_2~1(Δ_(mn)~((2)))的一个恒等式

本文将得到关于S_2~1(Δ_(mn)~(2))的一个恒等式,它对某些带限制的S_2~1(Δ_(mn)~(2))问题,提供了一个方便工具。 矩形域D_i[0,l_1]×[0,l_2],它的Δ_(mn)~(2)剖分是熟知的(见下图所示),在图上标了某些记号,这是讨论中需要的。记h_1=l_1/m,h_2=l_2/n,且节点(ih_1,jh_2)将简记为(i,j)。     

∑(a/(b c))~(1/2)<1 2(3~(1/2)/3)的初等证明

本刊文 [1 ]中用导数方法证明了 :在△ ABC中 ,有　　 ∑ ab c<1 2 33 . (1 )本文给出一个初等的证明 .证明　由对称性 ,不妨设 a≥ b≥ c=1 ,易知 a b≥ 2 ,a 相似文献   

A GENERALIZATION OF SIMPSON''''S RULE

Let TA(f)=integral form n= to 1/2(P_~n(x) + P_b~n(x))dx and let TM(f)=integral form n= to P_((+b)/2)~(n+1)(x)dx, where P_c~n denotes theTaylor polynomial to f at c of order n, where n is even. TA and TM are reach generalizations of theTrapezoidal rule and the midpoint rule, respectively. and are each exact for all polynomial of degree ≤n+1.We let L(f) = αTM(f) + (1-α)TA(f), where α =(2~(n+1)(n+1))/(2~(n+1)(n+1)+1), to obtain a numerical integrationrule L which is exact for all polynomials of degree≤n+3 (see Theorem l). The case n = 0 is just the classicolSimpson's rule. We analyze in some detail the case n=2, where our formulae appear to be new. By replacingP_(+b)/2)~(n+1)(x) by the Hermite cabic interpolant at a and b. we obtain some known formulae by a different ap-proach (see [1] and [2]). Finally we discuss some nonlinear numerical integration rules obtained by takingpiecewise polynomials of odd degree, each piece being the Taylor polynomial off at a and b. respectively. Ofcourse all of our formulae can be compounded over subintervals of [a, b].     

On Sums of Exponents of Factoring Integers

Let n=p_1~(β_1)p_2~(β_2)…p_t~(β_t)be the prime factorization of n.Define h(n)=min(β_1,β_2,…,β_t)and H(n)=max(β_1,β_2,…,β_t).For convenience take h(1)=1 and H(1)=1.P.Erdos suggested that it is likely that sum from i=1 to n h(i)=n c n~(1/2) o(n)~(1/2),where c is a posi-tive constant.This conjecture was proved by Ivan Niven[1].In[1],it is proved that     

On the bilinear square Fourier multiplier operators and related multilinear square functions

Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm(f1, f2)(x) =(∫_0~∞︱∫_((Rn)2)e~(2πix·(ξ1+ξ2))m(tξ1, tξ2)?f1(ξ1)?f2(ξ2)dξ1dξ2︱~2(dt)/t) ~(1/2).Let s be an integer with s ∈ [n + 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏_i~2=1ω_i~(p/pi) and each ω_i is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L~(p1)(ω_1) × L~(p2)(ω_2) to L~p(ν_ω) if p0 p1, p2 ∞ with 1/p = 1/p1 + 1/p2. Moreover,if p0 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L~(p1)(ω_1) × L~(p2)(ω_2) to L~(p,∞)(ν_ω). The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.     

BOUNDEDNESS OF COMMUTATORS OF FRACTIONAL INTEGRATION ON HERZ-TYPE SPACES

1 IntroductionLgt b E BMO(Rn) and 11 be a fractional integration with 0 < I < n. The commutator[b, II] generated by b and 11 is defined by[b, II]f(x) = b(x)llf(x) ~ II(sf)(x).S.Chanillolz] stated that the operator [b, II] is a bounded operator from LPI (R") on LPZ(R")for 1/Pz = 1/pl -- l/n and 1 < pl < n/l. Recently, Lu Shanzhen and Yang Dachun[9] studiedthe commutator [b,ll] on Herz spaces and a new class of Herz-type Hardy spaces introducedby them and established the correspondi…     

CORRECTION TO “ON THE PROBLEM OF BEST CONVERGENCE RATES OF DENSITY ESTIMATES

The assertion of Th.1 in[1]should be replaced bylimsup n→∞ a_nn~(k/(2k m)=∞.(A)Since the proof of Th.1 in[1]is somewhat in error,we give here a sketch ofproof of(A).Choose f∈C_ka with f(x)≥a>0 for ‖x‖≤ε>0,and define h_δ(x)=f(x) e_(kδ)(x),where e_(kδ)(x),as well as d and C_(kα)~(n)(d) to appear in the following,are thesame as in[1].Choose ■>0 so that h_δ∈C_(kα) for δ∈(0,■).For each δ in(0,■),thereexists an integer n such that h_δ∈C_(kα)~(n)(d).Hence an integer N can be found such that     

ON A CONJECTURE CONCERNING THE NON-EXISTENCE OF LIMIT CYCLES FOR QUADRATIC DIFFERENTIAL SYSTEMS

Aconjecturewasgivenin[l]asfollows:Ifforthesystemoftypeill(i.e.,6/0):wehavewhereW=(n b)(l n)'--a'(6 ZI n),thenthereisnolicestcycle(LC,forabbreviation)aroundO(0,0).Noticethatcondition(2)canbedividedintothefollowingfoursub-cases:i)m=0,a(b ZI)/0,n)a=0,m(l n)/0,iii)W=0,WI/0,tv)m--sa=2a2 n(l Zn)=0,a/0,WI/0.(3)Non-existenceofLCunderi)orn)hasbeenprovedin[2]515,Theorem15.1and15.2;andthatunderWI/0,6=0hasbeenprovedin[2]514.Ifiniii)m(l n)>0then[3]hasprovedthenon-existenceofLCaroundObytheDulacfun…     

等比定理的前提不可疏忽

众所周知等比定理是这样的:a/b=c/d=…=m/n,若b+d+…+n≠0(a+c+…+m)/(b+d+…+n)=a/b。其中条件b+d+…+n≠0极为重要。在b+d+…+n=0时就不能使用上述的等比定理。例如:已知a/b=b/c=c/d=d/a,求(a+b+c+d)/(a+b+c-d)的值。如果盲目套用等比定理,将得到其值为2:     

Precise Rates in the Generalized Law of the Iterated Logarithm in R~m

Let {X, X_n, n ≥ 1} be a sequence of i.i.d. random vectors with EX =(0,..., 0)_(m×1) and Cov(X, X) = σ~2 Ⅰ_m, and set S_n =∑_(i=1)~n X_i, n ≥ 1. For every d 0 and a_n =o((log log n)~(-d)), the article deals with the precise rates in the genenralized law of the iterated logarithm for a kind of weighted infinite series of P(|S_n| ≥(ε + a_n)σn~(1/2)(log log n)~d).     

Ramsey Number r(C_(2m 1), K_n) With Large n

This is an announcement that r(C2m 1, Kn) ＜ c(m) ( ) 1/m has been proved.The Ramsey number r(H, Kn) is the smallest integer N such that every H-free graph onN vertices has independence number at least n. The study of Ramsey number r(Ck, Kn) wasinitiated by Bondy and Erd s[2]. They proved that for any fixed n, r(Ck, Kn) = (k - 1)(n - 1) 1if k n2 - 1, and r(Ck, Kn) kn2. For fixed k 3, it is difficult to obtain a satisfied bound ofr(Ck, Kn) for n → ∞ . The bound of Bondy and Erd s w…     

Morphic环的扩张

A ring R is called left morphic, if for any a ∈ R, there exists b ∈ R such that 1R(a) = Rb and 1R(b) = Ra. In this paper, we use the method which is different from that of Lee and Zhou to investigate when R[x, σ]/(xn) is (left) morphic and when the ideal extension E(R, V) is (left) morphic. It is mainly shown that: (1) If σis an automorphism of a division ring R, then S = R[x,σ]/(xn) (n ＞ 1) is a special ring. (2) If d, m are positive integers and n = dm, then E(/n, mZn) is a morphic ring if and only if gcd(d, m) = 1.     

Ramsey Number With Largen

This is an announcement that r(C2m+1, Kn) ≤ c(m) has been proved. The Rarnsey number r(H, Kn) is the smallest integer N such that every H-free graph on N vertices has independence number at least n. The study of Ramsey number r(Ck, Kn) was initiated by Bondy and Erdos[2]. They proved that for any fixed n, r(Ck, Kn) = (k - 1)(n - 1) + 1if k≥n2-1, and r(Ck, Kn)≤kn2. For fixed k≥3, it is difficult to obtain a satisfied bound of r(Ck,Kn) for n →∞. The bound of Bondy and Erdos was improved as r(Ck, Kn)≤c(k)n1+1/m,where m = [(k - 1)/2] by Erdos, Faudree, Rousseau and Schelp[4]. For even cycle, a more refined     

High-dimensional D. H. Lehmer problem over short intervals

Letk be a positive integer and n a nonnegative integer,0 λ1,...,λk+1 ≤ 1 be real numbers and w =(λ1,λ2,...,λk+1).Let q ≥ max{[1/λi ]:1 ≤ i ≤ k + 1} be a positive integer,and a an integer coprime to q.Denote by N(a,k,w,q,n) the 2n-th moment of(b1··· bk c) with b1··· bk c ≡ a(mod q),1 ≤ bi≤λiq(i = 1,...,k),1 ≤ c ≤λk+1 q and 2(b1+ ··· + bk + c).We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N(a,k,w,q,n),which generalized the result of Zhang.Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N(a,k,w,q,n) in the case ofw =(1/2,1/2,...,1/2) and n = 0.In order to show our result is close to the best possible,the mean-square value of N(a,k,q) φk(q)/2k+2and the mean value weighted by the high-dimensional Cochrane sum are studied too.     

Universal sums of three quadratic polynomials

Let a,b,c,d,e and f be integers with a≥ c≥ e> 0,b>-a and b≡a(mod 2),d>-c and d≡c(mod 2),f>-e and f≡e(mod 2).Suppose that b≥d if a=c,and d≥f if c=e.When b(a-b),d(c-d) and f(e-f) are not all zero,we prove that if each n∈N={0,1,2,...} can be written as x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 with x,y,z∈N then the tuple(a,b,c,d,e,f) must be on our list of 473 candidates,and show that 56 of them meet our purpose.When b∈[0,a),d∈[0,c) and f∈[0,e),we investigate the universal tuples(a,b,c,d,e,f) over Z for which any n∈N can be written as x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 with x,y,z∈Z,and show that there are totally 12,082 such candidates some of which are proved to be universal tuples over Z.For example,we show that any n∈N can be written as x(x+1)/2+y(3y+1)/2+z(5z+1)/2 with x,y,z∈Z,and conjecture that each n∈N can be written as x(x+1)/2+y(3y+1)/2+z(5z+1)/2 with x,y,z∈N.     

On the primitive divisors of the recurrent sequence u_(n+1)=(4cos~2(2π/7)-1)u_n-u_(n-1) with applications to group theory

Consider the sequence of algebraic integers un given by the starting values u0=0,u1=1 and the recurrence u_(n+1)=(4cos~2(2π/7)-1)u_n-u_(n-1).We prove that for any n ■{1,2,3,5,8,12,18,28,30}the n-th term of the sequence has a primitive divisor in Z[2 cos(2π/7)].As a consequence we deduce that for any sufficiently large n there exists a prime power q such that the groupcan be generated by a pair x,y with χ~2=y~3=(xy)~7=1 and the order of the commutator[x,y]is exactly n.The latter result answers in affirmative a question of Holt and Plesken.     

用等比性质解题

<正>等比性质如果a/b=c/d=……=m/n(b+d+……+n≠0),那么(a+c+……+m)/(b+d+……+n)=a/b.应用时注意以下三点:(1)注意准确记忆性质,关键是记住性质成立的条件即b+d+……+n≠0.(2)注意掌握性质的应用,性质的应用有两层意思:一是性质本身的应用,二是性质证明方法的应用.