Frobenius problem for semigroups \mathsf{S}(d_{1},d_{2},d_{3})

We find the matrix representation of the set Δ(d ³), where d ³=(d ₁,d ₂,d ₃), of integers that are unrepresentable by d ₁,d ₂,d ₃ and develop a diagrammatic procedure for calculating the generating function Φ(d ³;z) for the set Δ(d ³). We find the Frobenius number F(d ³), the genus G(d ³), and the Hilbert series H(d ³;z) of a graded subring for nonsymmetric and symmetric semigroups and enhance the lower bounds of F(d ³) for symmetric and nonsymmetric semigroups.      

Analytic representations in the three-dimensional Frobenius problem

We consider the Diophantine problem of Frobenius for the semigroup , where d ³ denotes the triple (d ₁,d ₂,d ₃), gcd (d ₁,d ₂,d ₃)=1. Based on the Hadamard product of analytic functions, we find the analytic representation of the diagonal elements a _kk (d ³) of Johnson’s matrix of minimal relations in terms of d ₁, d ₂, and d ₃. With our recent results, this gives the analytic representation of the Frobenius number F(d ³), genus G(d ³), and Hilbert series H(d ³;z) for the semigroups . This representation complements Curtis’s theorem on the nonalgebraic representation of the Frobenius number F(d ³). We also give a procedure for calculating the diagonal and off-diagonal elements of Johnson’s matrix.      

Buffons Nadelproblem und verwandte probleme behandelt mit der Theorie der Gleichverteilung: Teil II

We denote with PC ^m the m-dimensional complex projective space, with U the unitary group acting on it with z _i(j=0, 1,..., m) the homogenous coordinates of a point [z] of PC ^m and assume that the z _i are normalized such that z ₀z₀ +...+z _mz_m=1. Furthermore we denote the U-invariant metric on PC ^m with d. We consider now a uniformly distributed sequence ([z] _k ; k=1,2,...) of points on PC ^m and study the sequence (d ^l([z] _k , [z]₀)), l0, [z]₀ a fixed point. We prove with the help of the theory of uniform distribution properties of this sequence. We consider furthermore a dual sequence suggested by the theory of H. Weyl and L. V. Ahlfors on meromorphic curves.     

Characterizations of Arveson's Hardy Space

A Hardy-type space H ² _d in the unit ball B_d of C^d , which was recently introduced by Arveson [W. Arveson (1998). Subalgebras of C*-algebras III: multivariable operator theory. Acta Math., 181, 159-228.], is appropriate for the operator theory of d-contractions. In this article, it is proved that H ² _d actually coincides with a Hardy-Sobolev space. This yields almost immediately some of the related results obtained in [W. Arveson (1998). Subalgebras of C*-algebras III: multivariable operator theory. Acta Math., 181, 159-228.], including the facts that H ² _d is not associated with any measure on C ^d ; and that the corresponding algebra of multipliers M ? H ^∞(B_d ) and the inclusion is proper. Finally, a function-theoretic version of von Neumann's inequality for the d-contractions is presented.     

Restricted partition functions as Bernoulli and Eulerian polynomials of higher order

Explicit expressions for restricted partition function W(s,d^m) and its quasiperiodic components W_j(s,d^m) (called Sylvester waves) for a set of positive integers d^m = {d₁, d₂, ..., d_m} are derived. The formulas are represented in a form of a finite sum over Bernoulli and Eulerian polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of finite groups is discussed. 2000 Mathematics Subject Classification Primary—11P81; Secondary—11B68, 11B37 The research was supported in part (LGF) by the Gileadi Fellowship program of the Ministry of Absorption of the State of Israel.     

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces HL²(\mathbb B^d, m_l){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dm_l(z) = c_l(1-|z|²)^l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.     

Randomness-optimal oblivious sampling

We present the first efficient oblivious sampler that uses an optimal number of random bits, up to an arbitrary constant factor bigger than 1. Specifically, for any α>0, it uses (1+α)(m+log γ⁻¹) random bits to output d=poly(ϵ⁻¹, log γ⁻¹, m) sample points z₁,…,z_d∈{0, 1}^m such that for any function f: {0, 1}^m→[0, 1], Pr [|(1/d)∑_i=1^df(z_i)− E f|≤ϵ]≥1−γ. Our proof is based on an improved extractor construction. An extractor is a procedure which takes as input the output of a defective random source and a small number of truly random bits, and outputs a nearly random string. We present the first optimal extractor, up to constant factors, for defective random sources with constant entropy rate. We give applications to constructive leader election and reducing randomness in interactive proofs. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 345–367 (1997)     

Symmetries on almost symmetric numerical semigroups

The notion of an almost symmetric numerical semigroup was given by V. Barucci and R. Fröberg in J. Algebra, 188, 418–442 (1997). We characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. We give a criterion for H ^? (the dual of M) to be an almost symmetric numerical semigroup. Using these results we give a formula for the multiplicity of an opened modular numerical semigroup. Finally, we show that if H ₁ or H ₂ is not symmetric, then the gluing of H ₁ and H ₂ is not almost symmetric.     

Convergence of ray sequences of Padé approximants for 2 f 1(a, 1; c; z), (c > a > 0)

The Padé table of ₂ F ₁(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ? Z^-, the denominator polynomial Q _mn (z) in the [m/n] Padé approximant P _mn (z)/Q _mn (z) for ₂ F ₁(a, 1; c; z) and the remainder term Q _mn (z)₂ F ₁(a, 1; c; z)-P_mn (z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of P_mn (z)/Q_mn (z) lie on the cut (1,∞). We deduce that the sequence of approximants P_mn (z)/Q_mn (z) converges to ₂ F ₁(a, 1; c; z) as m → ∞, n/m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc |z| < 1 for c > a > 0.     

Expression for restricted partition function through Bernoulli polynomials

Explicit expressions for restricted partition function W(s,d ^m ) and its quasiperiodic components W _j (s,d ^m ) (called Sylvester waves) for a set of positive integers d ^m ={d ₁,d ₂,…,d _m } are derived. The formulas are represented in a form of a finite sum over Bernoulli polynomials of higher order with periodic coefficients.      

Almost everywhere convergence of Riesz means on noncompact symmetric spaceSL(3,H)/Sp(3)

LetG/K be the noncompact Riemannian symmetric spaceSL(3,H)/Sp(3). We shall prove in this paper that forf∈L ^p(SL(3,H)/Sp(3)), 1≤p≤2, the Riesz means of orderz off with respect to the eigenfunctions expansion of Laplace operator almost everywhere converge tof for Rez >δ(n,p). The critical index δ(n,p) is the same as in the classical Stein's result for Euclidean space, and as in the noncompact symmetric spaces of rank one and of complex type. Partially supported by National Natural Science Foundation of China     

On the Kronecker pairing for mixed cusp forms

LetG⊃PSL(2,R) be a Fuchsian group of the first kind with no elements of finite order, and letS ^2m V be the 2m-fold symmetric power of the standard representationV ofSL(2,R) on C². We determine the value of the Kronecker pairing between the canonical image of a mixed cusp formf of type (2,2m) inH ¹(G, S ^2m V) and a cycleg ⊗Q _g ^m inH ₁ (G, (S ^2m V)^*) for eachg inG, whereQ _g ^m is an element of (S ^2m V)^* associated tog, m and a monodromy representation ofG.     

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.     

Some extremal results in cochromatic and dichromatic theory

For a graph G, the cochromatic number of G, denoted z(G), is the least m for which there is a partition of the vertex set of G having order m. where each part induces a complete or empty graph. We show that if {G_n} is a family of graphs where G_n has o(n² log²(n)) edges, then z(G_n) = o(n). We turn our attention to dichromatic numbers. Given a digraph D, the dichromatic number of D is the minimum number of parts the vertex set of D must be partitioned into so that each part induces an acyclic digraph. Given an (undirected) graph G, the dichromatic number of G, denoted d(G), is the maximum dichromatic number of all orientations of G. Let m be an integer; by d(m) we mean the minimum size of all graphs G where d(G) = m. We show that d(m) = θ(m² ln²(m)).     

Structure of relatively free dimonoids

Loday introduced the notion of a dimonoid and constructed the free dimonoid. The paper concerns the variety theory of dimonoids. We recall and summarize the results obtained by Loday, the author as well as others on the structure of some relatively free dimonoids. This part of the paper should be viewed as a survey, however we include some new ideas in the last sections. Namely, we construct a free (?z;rs)-dimonoid, a free (rs;rz)-dimonoid, a free rs-dimonoid, a free (rb;rs)-dimonoid, a free (rs;rb)-dimonoid and characterize the least (?z;rs)-congruence, the least (rs;rz)-congruence, the least rs-congruence, the least (rb;rs)-congruence and the least (rs;rb)-congruence on the free dimonoid. We also establish that the automorphism groups of constructed free algebras are isomorphic to the symmetric group and the semigroups of the free rs-dimonoid are anti-isomorphic.     

Sylvester Waves in the Coxeter Groups

A new recursive procedure of the calculation of partition numbers function W(s, d ^m ) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W(s, d ^m ) and their periods (G) for the irreducible Coxeter groups and a list for the first twelve symmetric group _m are presented. A least common multiple (m) of the series of the natural numbers 1,2,...,m plays a role in the period ( _m ) of W(s, d ^m) in _m .     

Generalized Bounded Variation and Inserting Point Masses

Let d μ be a probability measure on the unit circle and d ν be the measure formed by adding a pure point to d μ. We give a formula for the Verblunsky coefficients of d ν following the method of Simon. Then we consider d μ ₀, a probability measure on the unit circle with ℓ ² Verblunsky coefficients (α _n (d μ ₀)) _n=0^∞ of bounded variation. We insert m pure points z _j into d μ ₀, rescale, and form the probability measure d μ _m . We use the formula above to prove that the Verblunsky coefficients of d μ _m are in the form , where the c _j ’s are constants of norm 1 independent of the weights of the pure points and independent of n; the error term E _n is in the order of o(1/n). Furthermore, we prove that d μ _m is of (m+1)-generalized bounded variation—a notion that we shall introduce in the paper. Then we use this fact to prove that lim  _n→∞ φ _n ^*(z,d μ _m ) is continuous and is equal to D(z,d μ _m )⁻¹ away from the pure points.      

Bands of k-Archimedean semigroups

In this paper we define the radical ϱ _k (k∈Z ⁺) of a relation ϱ on an arbitrary semigroup. Also, we define various types of k-regularity of semigroups and various types of k-Archimedness of semigroups. Using these notions we describe the structure of semigroups in which ρ _k is a band (semilattice) congruence for some Green’s relation.     

Sequentially equidistant points in metric spaces

A sequence (z ₀,z ₁,z ₂,, ...,z _n, z _n+1) of points fromp=z ₀ toq=z _n+1 in a metric spaceX is said to besequentially equidistant ifd(z _i−1,z _i)=d(z _i,z _i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR ^m ,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the positive results. Supported in part by NSF Grant DMS-8701666.