A characteristic property of spheres

Summary We prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k₁ ≥ ... ≥ k_n denote its principle curvatures. Let φ(ξ₁, ..., ξ_n) be such that . Then if φ(k₁, ..., k_n)=const on S and S is subject to some additional general conditions (those(II ₀) or(II) n^o 1), S is a sphere. To Enrico Bompiani on his scientific Jubilee     

Rate of decay of weakly lacunary series

Let function f(z) ≠ 0 be analytic in the unit disk and have sparse nonzero Taylor coefficients. Then the rate of decay of the function f as x → 1 − 0 depends on the rate of sparseness of its nonzero Taylor coefficients. In this paper, we consider the case f(z) = $ \sum\nolimits_{k = 0}^\infty {a_k z^{n_k } } $ \sum\nolimits_{k = 0}^\infty {a_k z^{n_k } } , where n _k ≥ A ₀(k + 2) ^p logb(k + 2).     

Computing commutator length in free groups

We study commutator length in free groups. (By a commutator lengthcl(g) of an element g in a derived subgroup G′ of a group G we mean the least natural number k such that g is a product of k commutators.) A purely algebraic algorithm is constructed for computing commutator length in a free group F₂ (Thm. 1). Moreover, for every element z ε F′₂ and for any natural m, the following estimate derives:cl(z^m) ≥ (ms(z) + 6)/12, where s(z) is a nonnegative number defined by an element z (Thm. 2). This estimate is used to compute commutator length of some particular elements. By analogy with the concept of width of a derived subgroup known in group theory, we define the concept of width of a derived subalgebra. The width of a derived subalgebra is computed for an algebra P of pairs, and also for its corresponding Lie algebra P_L. The algebra of pairs arises naturally in proving Theorem 2 and enjoys a number of interesting properties. We state that in a free group F_2k with free generators a₁, b₁, ..., a_k, b_k, k εN, every natural m satisfiescl(([a₁, b₁] ... [a_k, b_k])^m)=[(2 − m)/2] + mk. For k=1, this entails a known result of Culler. The notion of a growth function as applied to a finitely generated group G is well known. Associated with a derived subgroup of F₂ is some series depending on two variables which bears information not only on the number of elements of prescribed length but also on the number of elements of prescribed commutator length. A number of open questions are formulated. Supported by RFFR grant No. 98-01-00699. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 395–440, July–August, 2000.     

Carathéodory balls and norm balls of the domainH={(z1,z2)∈C2∶|z1|+|z2|<1}

LetH be the domain inC ² defined byH={Z=(z ₁,z ₂):║Z║₁=│z║₁│+│z║₂│<1}. LetC _H(z,w) be the Carathéodory distance ofH,z,w∈H. The Carathéodory ballB _C(z_C,α;H) with centerz _C,z_C∈H, and radius α, 0<α<1, is defined byB _c(z_C,α;H)={z∶C_H(z,z_C)<arc tanh α}. The norm ballB _N(z_N,r) with centerz _N,z_N∈H, and radiusr, 0<r<1-‖z _N‖₁, is defined byB _N(z_N,r)={z∶ ‖z−z_N‖₁<r}. Theorem:The only Carathéodory balls of H which are also norm balls are those with their center at the origin.     

Strictly Hermitian positive definite functions

LetH be any complex inner product space with inner product <·,·>. We say thatf: ℂ→ℂ is Hermitian positive definite onH if the matrix

Asymptotics of Recurrence Relation Coefficients,Hankel Determinant Ratios,and Root Products Associated with Laurent Polynomials Orthogonal with Respect to Varying Exponential Weights

Let Λ^ℝ denote the linear space over ℝ spanned by z ^k , k∈ℤ. Define the real inner product 〈⋅,⋅〉 _L :Λ^ℝ×Λ^ℝ→ℝ, , N∈ℕ, where V satisfies: (i) V is real analytic on ℝ∖{0}; (ii) lim  _{| x |→∞}(V(x)/ln (x ²+1))=+∞; and (iii) lim  _{| x |→0}(V(x)/ln (x ⁻²+1))=+∞. Orthogonalisation of the (ordered) base with respect to 〈⋅,⋅〉 _L yields the even degree and odd degree orthonormal Laurent polynomials (OLPs) : φ _2n (z)=∑ _k=−n ⁿ ξ _k ⁽²ⁿ⁾ z ^k , ξ _n ⁽²ⁿ⁾>0, and φ _2n+1(z)=∑ _k=−n−1 ⁿ ξ _k ⁽²ⁿ⁺¹⁾ z ^k , ξ _−n−1⁽²ⁿ⁺¹⁾>0. Associated with the even degree and odd degree OLPs are the following two pairs of recurrence relations: z φ _2n (z)=c _2n ^♯ φ _2n−2(z)+b _2n ^♯ φ _2n−1(z)+a _2n ^♯ φ _2n (z)+b _2n+1 ^♯ φ _2n+1(z)+c _2n+2 ^♯ φ _2n+2(z) and z φ _2n+1(z)=b _2n+1 ^♯ φ _2n (z)+a _2n+1 ^♯ φ _2n+1(z)+b _2n+2 ^♯ φ _2n+2(z), where c ₀ ^♯ =b ₀ ^♯ =0, and c _2k ^♯ >0, k∈ℕ, and z ⁻¹ φ _2n+1(z)=γ _2n+1 ^♯ φ _2n−1(z)+β _2n+1 ^♯ φ _2n (z)+α _2n+1 ^♯ φ _2n+1(z)+β _2n+2 ^♯ φ _2n+2(z)+γ _2n+3 ^♯ φ _2n+3(z) and z ⁻¹ φ _2n (z)=β _2n ^♯ φ _2n−1(z)+α _2n ^♯ φ _2n (z)+β _2n+1 ^♯ φ _2n+1(z), where β ₀ ^♯ =γ ₁ ^♯ =0, β ₁ ^♯ >0, and γ _2l+1 ^♯ >0, l∈ℕ. Asymptotics in the double-scaling limit N,n→∞ such that N/n=1+o(1) of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios associated with the real-valued, bi-infinite strong moment sequence , and the products of the (real) roots of the OLPs are obtained by formulating the even degree and odd degree OLP problems as matrix Riemann-Hilbert problems on ℝ, and then extracting the large-n behaviours by applying the non-linear steepest-descent method introduced in (Ann. Math. 137(2):295–368, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277–337, [1995]) and (Int. Math. Res. Not. 6:285–299, [1997]).      

Sojourn time of almost semicontinuous integral-valued processes in a fixed state

Let ξ(t) be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps

Positive linear operators generated by analytic functions

Let φ be a power series with positive Taylor coefficients {a _k } _k=0^∞ and non-zero radius of convergence r ≤ ∞. Let ξ _x , 0 ≤ x < r be a random variable whose values α _k , k = 0, 1, …, are independent of x and taken with probabilities a _k x ^k /φ(x), k = 0, 1, …. The positive linear operator (A _φ f)(x):= E[f(ξ _x )] is studied. It is proved that if E(ξ _x ) = x, E(ξ _x ²) = qx ² + bx + c, q, b, c ∈ R, q > 0, then A _φ reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1.     

Planar wiener inverses

Let B(Z₁, z₂) be a power series and C(z₁, z₂) be the least mean-square inverse approximation of B among polynomials of a fixed order. This paper discusses conditions under which $\text{B}\text{?}\text{(z}{}_1\text{, z}{}_2\text{)}$, the least mean-square inverse polynomial approximation of C, does not vanish on the unit bicylinder.     

A Ramsey-Sperner theorem

Letn≥k≥1 be integers and letf(n, k) be the smallest integer for which the following holds: If ℱ is a family of subsets of ann-setX with |ℱ|<f(n,k) then for everyk-coloring ofX there existA B ∈ ℱ,A∈B, A⊂B such thatB-A is monochromatic. Here it is proven that for a fixedk there exist constantsc _k andd _k such that and ask→∞. The proofs of both the lower and the upper bounds use probabilistic methods.     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

Properties of generalised juxtapolynomials

GivenF(z),f ₁(z), ..,f _n(z) defined on a finite point setE, and givenB — the set of generalised polynomials Σ _k ^=1/n a _kf_k(z) — the definition of a juxtapolynomial is extended in the following manner: for a fixedλ(0<λ≦1),f(z) εB is called a generalizedλ-weak juxtapolynomial toF(z) onE if and only if there exists nog(z) εB for whichg(z)=F(z) wheneverf(z)=F(z) and |g(z)−F(z) |<λ|f(z)−F(z)| wheneverf(z)≠F(z). The properties of suchf(z) are investigated with particular attention given to the real case. This note is an extension of a part of the author’s M.Sc. Thesis under the supervision of Prof. B. Grünbaum to whom the author wishes to express his sincerest appreciation. The author also wishes to thank Dr. J. Lindenstrauss for his valuable remarks in the preparation of this paper.     

New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

NON-EMPTY CROSS-2-INTERSECTING FAMILIES OF SUBSETS

Let Cdenote the set of all k-subests of an n-set.Assume Alohtain in Ca,and A lohtain in (A,B) is called a cross-2-intersecting family if |A B≥2 for and A∈A,B∈B.In this paper,the best upper bounds of the cardinalities for non-empty cross-2-intersecting familles of a-and b-subsets are obtained for some a and b,A new proof for a Frankl-Tokushige theorem[6] is also given.     

Geometric progressions in sumsets over finite fields

Given two sets A, B í \Bbb F_q^d{\cal A}, {\cal B}\subseteq {\Bbb F}_q^d , the set of d dimensional vectors over the finite field \Bbb F_q{\Bbb F}_q with q elements, we show that the sumset A+B = {a+b | a ? A, b ? B}{\cal A}+{\cal B} = \{{\bf a}+{\bf b}\ \vert\ {\bf a} \in {\cal A}, {\bf b} \in {\cal B}\} contains a geometric progression of length k of the form vΛ ^j , where j = 0,…, k − 1, with a nonzero vector v ? \Bbb F_q^d{\bf v} \in {\Bbb F}_q^d and a nonsingular d × d matrix Λ whenever # A # B 3 20 q^2d-2/k\# {\cal A} \# {\cal B} \ge 20 q^{2d-2/k} . We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic varieties.     

Property $${(\hbar)}$$ and cellularity of complete Boolean algebras

A complete Boolean algebra \mathbbB{\mathbb{B}}satisfies property ((^h/_2p)){(\hbar)}iff each sequence x in \mathbbB{\mathbb{B}}has a subsequence y such that the equality lim sup z _n = lim sup y _n holds for each subsequence z of y. This property, providing an explicit definition of the a posteriori convergence in complete Boolean algebras with the sequential topology and a characterization of sequential compactness of such spaces, is closely related to the cellularity of Boolean algebras. Here we determine the position of property ((^h/_2p)){(\hbar)}with respect to the hierarchy of conditions of the form κ-cc. So, answering a question from Kurilić and Pavlović (Ann Pure Appl Logic 148(1–3):49–62, 2007), we show that ${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}is not a theorem of ZFC and that there is no cardinal \mathfrakk{\mathfrak{k}}, definable in ZFC, such that ${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}is a theorem of ZFC. Also, we show that the set { k: each k-cc c.B.a. has ((^h/_2p) ) }{\{ \kappa : {\rm each}\, \kappa{\rm -cc\, c.B.a.\, has}\, (\hbar ) \}}is equal to [0, \mathfrakh){[0, \mathfrak{h})}or [0, \mathfrak h]{[0, {\mathfrak h}]}and that both values are consistent, which, with the known equality {k: each c.B.a. having  ((^h/_2p) ) has the k-cc } = [\mathfrak s, ￥){{\{\kappa : {\rm each\, c.B.a.\, having }\, (\hbar )\, {\rm has\, the}\, \kappa {\rm -cc } \} =[{\mathfrak s}, \infty )}}completes the picture.     

Geometry of rectangular block triangular matrices

Let D be any division ring, and let T（mi,ni,k） be the set of k × k （k ≥ 2） rectangular block triangular matrices over D. For A, B ∈ T（mi,ni,k）, if rank（A - B） = 1, then A and B are said to be adjacent and denoted by A -B. A map T ： T（mi,ni,k） -〉 T（mi,ni,k） is said to be an adjacency preserving map in both directions if A - B if and only if φ（A） φ（B）. Let G be the transformation group of all adjacency preserving bijections in both directions on T（mi,ni,k）. When m1,nk ≥ 2, we characterize the algebraic structure of G, and obtain the fundamental theorem of rectangular block triangular matrices over D.     

Moments of an exponential functional of random walks and permutations with given descent sets

The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial Y = 1 + ξ₁ + ξ₁ξ₂ + ξ₁ξ₂ξ₃ + ⋯ of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables μ _k = E(ξ ^k ) < 1 with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle. This revised version was published online in August 2006 with corrections to the Cover Date.     

Modular curves and bases for the spaces of cuspidal modular forms

Let Γ⊂SL ₂(ℝ) be a Fuchsian group of the first kind. For a character χ of Γ→ℂ^× of finite order, we define the usual space S _m (Γ,χ) of cuspidal modular forms of weight m≥0. For each ξ in the upper half–plane and m≥3, we construct cuspidal modular forms Δ _k,m,ξ,χ ∈S _m (Γ,χ) (k≥0) which represent the linear functionals f?\fracd^kfdz^k|_z=xf\mapsto\frac{d^{k}f}{dz^{k}}|_{z=\xi} in terms of the Petersson inner product. We write their Fourier expansion and use it to write an expression for the Ramanujan Δ-function. Also, with the aid of the geometry of the Riemann surface attached to Γ, for each non-elliptic point ξ and integer m≥3, we construct a basis of S _m (Γ,χ) out of the modular forms Δ _k,m,ξ ,χ (k≥0). For Γ=Γ ₀(N), we use this to write a matrix realization of the usual Hecke operators T _p for S _m (N,χ).     

A sieve result for Farey fractions

This paper is concerned with the sieve problem for Farey fractions (i.e., rational numbers with denominators less thanx) lying in an interval (λ₁, λ₂). An asymptotic formula for the sifting function is derived under the assumption that (λ₁, λ₂)x→∞ asx→∞. Two applications of this result are made. In the first one, the value distribution of the vector η(m/n)=(ξ(m), ξ(n)) is considered; here, fork=p ₁ p ₂...p _s ,p ₁≥p ₂>-..., ξk)_is defined by ξ(k)=(logp ₁/logk, logp ₂/logk,..., logp _s /logk, 0, ...); allp _i are prime numbers. It is shown that the limit distribution is π×π, where π is the Poisson-Dirichlet distribution. The asymptotical behavior of finite-dimensional distributions of ξ(k) for natural numbers was studied by Billingsley, Knuth, Trabb Pardo, Vershik, and others; the result of weak convergence to the Poisson-Dirichlet distribution appears in Donnelly and Grimmett. The second application is concerned with the density of sets {m/n: f(m/n)=a}, wheref is a function with the almost squareful kernel. Supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2600, Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 108–127, January–March, 1999. Translated by V. Stakénas