Estimates of the initial coefficients in the classes of univalent functions

Let S be the class of functions, regular and univalent in the circle ¦Z¦ < 1. Assume that D_n (), n=2,3,..., are defined by the expansion, ¦z¦<1,-11. In the paper one obtains sharp estimates for D₄() in the class S_R of functions from S with real coefficients C₂,C₃,... for all –1</1. In particular, as a consequence, one obtains sharp estimates for the coefficients C_3k+1 in the class S^K/R of K-symmetric functions from S_R for all K=2,3,... For k=2, the last result strengthens a result of Leeman.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 166–183, 1983.     

The Müntz-Szasz problem

A condition is obtained on the placement of point _n (in some sense, the final point) with which completeness of the system of functionsexp (– _n x), Re_n>0, in spaces L^p, 1p<2. is equivalent to divergence of the series re_n(1+¦_n¦²)^–1.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 91–103, January, 1978.Deceased.     

The solubility of certain diophantine inequalities

Let ₁, ₂, ..., _s bes non-zero real numbers not all of the same sign and not all in rational ratio, andk be a natural number; letD(k) be the leasts for which the inequality < (maxn _j )^– has infinitely many solutions. In this paper, we give some new estimations ofD(k) fork 6.Supported by the National Natural Science Foundation of China     

Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

Nonselfadjoint spectral problems for linear pencilsN-λP of ordinary differential operators with λ-linear boundary conditions: Completeness results

We study nonselfadjoint spectral problems for ordinary differential equationsN(y)–P(y)=0 with -linear boundary conditions where the orderp of the differential operatorP is less than the ordern ofN. The present paper addresses the question of the completeness of the eigenfunctions and associated functions in the Sobolev spacesW ₂ ^k (0,1) fork=0,1,...,n. To this end we associate a pencil – of operators acting fromL ₂(0,1) to the larger spaceL ₂x(0,1) ⁿ with the given problem. We establish completeness results for normal problems in certain finite codimensional subspaces ofW ₂ ^k (0,1) which are characterized by means of Jordan chains in 0 of the adjoint of the compact operator = ^–1.Dedicated to Professor Heinz Langer on the occasion of his 60th birthday     

On the convergence of Hunter's quadrature rule for Cauchy principal value integrals

Hunter's (n+1)-point quadrature rule for the approximate evaluation of the Cauchy principal value integralf ¹ _–1 (w(x)f(x)/(x – ))dx, –1<<1, is based on approximatingf by the polynomial which interpolatesf at the point and then zeros of the orthogonal polynomialp _n generated by the weight functionw. Sufficient conditions are given to ensure the convergence of a suitably chosen subsequence of the quadrature rules to the integral, whenf is Hölder continuous on [–1,1].     

Algebraic polynomial bases of space LP

Let (_n) be a system, close to the orthonormal complete system (x _n). An estimate is obtained for the deviation of the system {f_n}, obtained from {_n} by Schmidt's method, from the system {x_n}. This estimate is used to show that, in any L_P(–1,1), withp (1,4/3] [4,), and for any >e¦4 = i,13..., there exists an orthogonal algebraic system (P _n (x)) _n=0 , forming a basis in L_P and such that _n = degP _n (x) n for n>n_o(p,).Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 223–230, February, 1978.     

The minimum independence number for designs

Fort=2,3 andk2t–1 we prove the existence oft–(n,k,) designs with independence numberC _,k n ^{(k–t)/(k–1)} (ln n) ^1/(k–1) . This is, up to the constant factor, the best possible.Some other related results are considered.Supported by NSF Grant DMS-9011850     

An extreme value theory for long head runs

Summary For an infinite sequence of independent coin tosses withP(Heads)=p(0,1), the longest run of consecutive heads in the firstn tosses is a natural object of study. We show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum ofn(1-p) i.i.d. exponential random variables. These results are extended to the case of the longest head run interrupted byk tails. The mean length of this run is shown to be log(n)+klog(n)+(k+1)log(1–p)–log(k!)+k+/–1/2+ r₁(n)+ o(1) where log=log_1/p , =0.577 ... is the Euler-Mascheroni constant, =ln(1/p), andr ₁(n) is small. The variance is ₂/6²+1/12 +r ₂(n)+ o(1), wherer ₂(n) is again small. Upper and lower class results for these run lengths are also obtained and extensions discussed.This work was supported by a grant from the System Development Foundation     

A technique for constructing divisible difference sets

An (m, n, k, ₁,₂) divisible difference set in a groupG of ordermn relative to a subgroupN of ordern is ak-subsetD ofG such that the list {xy^–1:x, y D} contains exactly ₁ copies of each nonidentity element ofN and exactly ₂ copies of each element ofG N. It is called semi-regular ifk > ₁ and k²=mn₂. We develop a method for constructing a divisible difference set as a product of a difference set and a relative difference set or a difference set and a subset ofG which we call a relative divisible difference set. The method results in several parametrically new families of semi-regular divisible difference sets.     

Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.     

Some limit results for lag sums of independent,non-i.i.d., random variables

Summary Let X _k be an independent, but not necessarily i.i.d., sequence of random variables. Suppose EX _k 0 and that _Nk ² = E(S _N –S _N–k )² where S _N =X ₁+...+X_N. We obtain conditions under which and obtain other related results. We assume either that P{¦X _k ¦t}Mt^–r for all k and all t>0, or that E(expX _k )b< for all k and all &#x2266;t₀. The results are like earlier results for the i.i.d. case, but the method of proof is entirely different.The research of both authors was supported by National Science Foundation Grant No. MCS 81-00740     

Commutator Subgroups of the Extended Hecke Groups \bar H(\lambda _q )

Hecke groups H(_q) are the discrete subgroups of generated by S(z) = –(z+ _q)^–1and T(z) = –1/z. The commutator subgroup of H(_q), denoted by H(_q), is studied in [2]. It was shown that H(_q) is a free group of rank q– 1.Here the extended Hecke groups obtained by adjoining to the generators of H(_q) are considered. The commutator subgroup of is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the H(_q) case, the index of H(_q) is changed by q, in the case of this number is either 4 for qodd or 8 for qeven.     

On the lowest eigenvalue of the Laplacian for the intersection of two domains

IfA andB are two bounded domains in _n and (A), (B) are the lowest eigenvalues of – with Dirichlet boundary conditions then there is some translate,B _x, ofB such that (AB _x)<(A)+(B). A similar inequality holds for .There are two corollaries of this theorem: (i) A lower bound for sup _x {volume (AB _x)} in terms of (A), whenB is a ball; (ii) A compactness lemma for certain sequences inW ^1,p ( _n ).Work partially supported by U.S. National Science Foundation grant PHY-8116101 A01. AMS(MOS) Classification: 35P15     

Problem of the maximum of the product of the conformal radii of nonoverlapping domains

Let c_k, k=1,...,4, be arbitrary distinct points of . LetD be the family of all systems of simply connected domains in. By R(D_k, c_k) we denote the conformal radius of the domain D_k, relative to the point c_k. We prove that in the familyD one has the sharp inequality, (1) where a=(+i)/(–1), being the cross-ratio of the points c₁,c₂, c₃, c₄: E(–1, 1,a) is the continuum of least capacity containing the points –1,1,a. An explicit expression for capE(–1,1,a.) in terms of elliptic Jacobi functions has been obtained earlier by the author [Tr. Mat. Inst. Akad. Nauk SSSR,94, 47–65, 1968]. On the basis of the well-known properties of continua of least capacity, one shows that the largest value of the right-hand side of (1) is attained for a=± i3 and it is equal to 4^–8/3·3². One gives all the configurations for which equality prevails in the obtained estimates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 131–145, 1980.     

The caratheodory number for thek-core

Thek-core of the setS ⁿ is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core _{kS, S} ⁿ implies the existence of a subsetT S such thatx core_kT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812.     

Elementarteiler von Inzidenzmatrizen symmetrischer Blockpläne

By a study of the integral code generated by the rows of the incidence matrix and its extention the following results are obtained: Let d ₁,...,d _V(d ₁|d ₂,d ₂|d ₃...) be the elementary divisors of the incidence matrix of a symmetric (v,n+, ) design. Then d _v=(n+)n/g.c.d. (n, ). Moreover, if p is a prime such that p|n, p and if x _p denotes the p-part of x, then (d _id_v+2–i) _p =n _p for 2iv. For projective planes it can be shown that d ₁=···=d _3n–2=1, hence and . The paper also contains some results about elementary divisors of incidence matrices G satisfying the condition G ^tG=nI+ J.

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Elementarteiler von Inzidenzmatrizen symmetrischer Blockpläne

     

Minimal coverings and maximal packings of (k−1)-subsets by k-subsets

This paper studies the asymptotic behavior of functions M(n, k, k–1, ) and m(n, k, k–1, ), equal to the respective cardinalities of the minimal -covering and maximal -packing of all (k–1)-subsets of the n-element set of its k-subsets. It is shown that, if sequence k=k(n) is such that k(n)/n 0 as then m(n, k, k–1, ).( _k–1 ⁿ ).k¹, and if as n , thenM(n,k,k–1,).( _k–1 ⁿ ).k^–1. A consequence of these results is the validity of the Erdös-Hanani conjecture concerning the asymptotic behavior of functions M(n, k, k–1, 1) and m(n, k, k–1, 1).Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 565–571, April, 1977.In conclusion, the author wishes to thank A. A. Sapozhenko under whose direction the present work was achieved.     

Solutions d'équations elliptiques semi-linéaires singulières

Résumé Récemment T.Kusano et C. A.Swanson ont étudié l'équation elliptique semilinéaire u+j(x)u ^– =0,x R ⁿ,n 3et 0Ils ont montré l'existence d'une solution positive globale se comportant comme ¦x¦ ^2–n pour ¦x¦ grand. Nous démontrons l'existence et l'unicité d'une telle solution pour >/ 0.

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Summary T.Kusano and C. A.Swanson have recently studied the semilinear elliptic equationu+j(x)u ^– =0,x R ⁿ,n3 and 0< < 1. They have shown the existence of a global positive solution behaving like ¦x¦^2–n for large ¦x¦. We establish the existence and uniqueness of such a solution when > 0.

     

ON A CLASS OF BESICOVITCHFUNCTIONS TO HAVE EXACT BOX DIMENSION： A NECESSARY AND SUFFICIENT CONDITION

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.