THE EXTREME POINTS OF SEVERAL CLASSES OF ANALYTIC FUNCTIONS

1Intr0ducti0nLetAden0tethesetofallfunctionsanalyticinA={z:Izl<1}.LetB={W:WEAandIW(z)l51}.Aisalocallyconvexlineaztop0l0gicalspacewithrespecttothetopologyofuniformconvergenceon`c0mpact8ubsetsofA-LetTh(c1,'tc.-1)={p(z):p(z)EA,Rop(z)>0,p(z)=1 clz czzz ' c.-lz"-l 4z" ',wherecl,',cn-1areforedcomplexconstants}.LetTh,.(b,,-..,b,-,)={p(z):P(z)'EAwithReP(z)>Oandp(z)=1 blz ' b.-lz"-l 4z" '-,wherebl,-'-jbu-1areffeedrealconstantsanddkarerealnumbersf0rk=n,n 1,'--}-LetTu(l1,'i'tI.-1)={…     

Global inverse spectral problems for rational matrix functions

Given rational matrix functions ψ₁(λ) = I_m + C₁(λI_n1 − A₁)⁻¹B₁ and ψ₂(λ) = I_m + C₂(λI_n2 − A₂)⁻¹B₂ which are analytic and invertible on the unit circle, we characterize in terms of the operators A₁,B₁,C₁,A₂,B₂,C₂ when there exists a single rational matrix function W(λ) = I_m + C(λI_n − A)⁻¹B such that WH²_m^⊥ = ψ ₁H²_m^⊥and WH²_m = ψ₂H²_m. When this is the case, we give explicit formulae for A,B,C in terms of A₁,B₁,C₁,A₂,B₂,C₂. Applications include Wiener-Hopf factorization, J- inner-outer factorization, and coprime factorization. The results on J-inner-outer factorization have application to a model reduction problem for discrete time linear systems.     

Minimal free resolutions of monomial curves in P3k

Minimal free resolutions for prime ideals with generic zero (tn3,tn3?n10tn11,tn3?n20tn2, tn31), n1<n2<n3 positive integers, (n1,n2,n3)=1, are determined.     

Weighted rational approximation in the complex plane

Given a triple (G, W, γ) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ(z), which is analytic in G, by weighted rational functions W^m_i+n_i(z)R_mi, n_i(z)_{i = 0}^∞, where R_mi, n_i(z) = P_mi(z)/Q_ni(z) with deg P_mi ≤ m_i and deg Q_ni ≤ n_i for all i ≥ 0 and where m_i + n_i → ∞ as i → ∞ such that lim m_i/[m_i + n_i] = γ. Our main result is a necessary and sufficient condition for such an approximation to be valid. Applications of the result to various classical weights are also included.     

The Erd?s-Ginzburg-Ziv theorem for dihedral groups

Let n≥23 be an integer and let D_2n be the dihedral group of order 2n. It is proved that, if g₁,g₂,…,g_3n is a sequence of 3n elements in D_2n, then there exist 2n distinct indices i₁,i₂,…,i_2n such that g_i1g_i2?g_i2n=1. This result is a sharpening of the famous Erd?s-Ginzburg-Ziv theorem for G=D_2n.     

An addition theorem for finite Abelian groups

If g₁, g₂, …, g_2n?1 is a sequence of 2n ? 1 elements in an Abelian group G of order n, it is known that there are n distinct indices i₁, i₂, …, i_n such that 0 = g_i1 + g_i2 + ? + g_in. In this paper a suitably general condition on the sequence is given which insures that every element g in G has a representation g = g_i1 + g_i2 + ? + g_in as the sum of n terms of the sequence.     

On cyclic decompositions of the complete graph into the 2-regular graphs

The symbol C(m₁ ⁿ ₁m₂ ⁿ ₂...m_s ⁿ _s) denotes a 2-regular graph consisting ofn _i cycles of lengthm _i , i=1, 2,…,s. In this paper, we give some construction methods of cyclic(K _v ,G)-designs, and prove that there exists a cyclic(K _v , G)-design whenG=C((4m ₁) ⁿ ₁(4m ₂) ⁿ ₂...(4m _s ) ⁿ _s andv ≡ 1 (mod 2¦G¦).     

An explicit expression for the cost of a class of Huffman trees

Let T_n denote a binary tree with n terminal nodes V={υ₁,…,υ_n} and let l_i denote the path length from the root to υ_i. Consider a set of nonnegative numbers W={w₁,…,w_n} and for a permutation π of {1,…,n} to {1,…,n}, associate the weight w_i to the node υ_π(i). The cost of T_n is defined as C(T_n∣W)=Min_π∑ⁿ_i=1w_il_π(i).A Huffman tree H_n is a binary tree which minimizes C(T_n∣W) over all possible T_n. In this note, we give an explicit expression for C(H_n∣W) when W assumes the form: w_i=k for i=1,…,n?m; w_i=x for i=n?m+1,…,n. This simplifies and generalizes earlier results in the literature.     

Enumeration of sequences of given specification according to rises,falls and maxima

Letn = (a₁.a₂ … a_N) denote a sequence of integers a_i={1.2.…n}. A rise is a part a_i.a_i+1 with a_i <a_i+1: a fall is a pair with a_ia_i+1: a level is a pair with a_i = a_i+1. A maximum is a triple a_i-1.a_ia_i+1 with a_i-1?a_i.a_i?a_i+1. If e_i is the number of a_j?n witha_j = i, then [e₁…e_n] is called the specification of n. In addition, a conventional rise is counted to the left of a₁ and a conventional fall to the right of a_N: ifa₁?a₂, then a₁ is counted as a conventional maximum, similarly if a_N-1 ? a_N thena_N is a conventional maximum. Simon Newcomb's problem is to find the number of sequences n with given specification and r rises; the refined problem of determining the number of sequences of given specification with r rises and s falls has also been solved recently. The present paper is concerned with the problem of finding the number of sequences of given specification with r rises, s falls. λ levels and λ maxima. A generating function for this enumerant is obtained as the quotient of two continuants. In certain special cases this result simplifies considerably.     

A note on ��- and ��-modifications

The notions of ??(?? ₁,?? ₂), ??(?? ₁,?? ₂) and (??(?? ₁,?? ₂),??(?? ₁,?? ₂))-continuity were introduced in [2]. In this paper, the characterization of the continuity is investigated, and we introduce the (??(?? ₁,?? ₂),??(?? ₁,?? ₂))-continuity on generalized topological spaces. Finally, we investigate the relations between (??(?? ₁,?? ₂),??(?? ₁,?? ₂))-continuity and (??(?? ₁,?? ₂),??(?? ₁,?? ₂))-continuity on generalized topological spaces     

Interpolation orbits in couples of Lp spacesOrbites d'interpolation pour les couples d'espaces Lp

We consider linear operators T mapping a couple of weighted L_p spaces {L_p0(U₀), L_p1(U₁)} into {L_q0(V₀),L_q1(V₁)} for any 1?p₀, p₁, q₀, q₁?∞, and describe the interpolation orbit of any a∈L_p0(U₀)+L_p1(U₁) that is we describe a space of all {Ta}, where T runs over all linear bounded mappings from {L_p0(U₀),L_p1(U₁)} into {L_q0(V₀),L_q1(V₁)}. We show that interpolation orbit is obtained by the Lions–Peetre method of means with functional parameter as well as by the K-method with a weighted Orlicz space as a parameter. To cite this article: V.I. Ovchinnikov, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 881–884.     

Lattice point theory in polyhedra

The exact order of the remainder term is determined in the formula for the number of lattice points in the region α₁∥u₁ + b₁∥ + α₂∥u₂ + b₂∥ + … + α_r∥u_r + b_r∥ ≤ x in dependence on the arithmetical properties of the coefficients α₁, α₂,…, α_r.     

Local approximations for maximum partial subgraph problem

We deal with MAXH₀-FREE PARTIAL SUBGRAPH. We mainly prove that 3-locally optimum solutions achieve approximation ratio (δ₀+1)/(B+2+ν₀), where B=max_v∈Vd_G(v), δ₀=min_v∈V(H0)d_H0(v) and ν₀=(|V(H₀)|+1)/δ₀. Next, we show that this ratio rises up to 3/(B+1) when H₀=K₃. Finally, we provide hardness results for MAXK₃-FREE PARTIAL SUBGRAPH.     

Graham’s pebbling conjecture on products of many cycles

A pebbling move on a connected graph G consists of removing two pebbles from some vertex and adding one pebble to an adjacent vertex. We define f_t(G) as the smallest number such that whenever f_t(G) pebbles are on G, we can move t pebbles to any specified, but arbitrary vertex. Graham conjectured that f₁(G×H)≤f₁(G)f₁(H) for any connected G and H. We define the α-pebbling number α(G) and prove that α(C_pj×?×C_p2×C_p1×G)≤α(C_pj)?α(C_p2)α(C_p1)α(G) when none of the cycles is C₅, and G satisfies one more criterion. We also apply this result with G=C₅×C₅ by showing that C₅×C₅ satisfies Chung’s two-pebbling property, and establishing bounds for f_t(C₅×C₅).     

Some new sets and decompositions of A I-R-continuity, α-I-continuity, continuity via idealization

The concepts of I-R closed set, A _I-R-set, α _I M ₁-set, α _I M ₂-set, α _I N ₁-set, α _I N ₂-set, α _I N ₃-set, α _I N ₄-set and α _I N ₅-set are introduced via idealization. New decompositions of some weaker forms of continuity are obtained by using these sets.     

An estimator for the mean of an exponential distribution with random right-censoring

Let (Y₁, Z₁),…,(Y_N, Z_N) be i.i.d. pairs of independent random variables such that Y_i is exponentially distributed with unknown mean 1/λ and Z_i has an unknown distribution function F. Let X_i ≔ min(Y_i, Z_i). Under certain assumptions on F an estimator T_N(X₁,…,X_N) for 1/λ is constructed which is consistent and asymptotically normal.     

A balancing strategy

Suppose x₁, x₂,…, is a sequence of vectors in R^k, 6X_n6⩽1, where 6(x₁,…,x_k)6 = max_j|x_j|. An algorithm is given for choosing a corresponding sequence ε₁, ε₂,…, of numbers, ε_n = ±1, so that 6ε₁x₁+ … +ε_nx_n6 remains small.     

Properties of Regular Uniform k—Stratified Graphs

A regular graph G = (V, E) is a k-stratified graph if V is partitioned into V₁, V₂, …, V_k subsets called strata. The stratification splits the degree d_v ∀v ϵ V into k-integers d_v1, d_v2, …, d_vk each one corresponding to a stratum. If d_v1 = d_v2 = … = d_vk ∀v ϵ V then G is called regular uniform k-stratified, RU^k_s(n, d) where n is the cardinality of the vertex set in each stratum and d is the degree of every vertex in each stratum. For every k, the class RU^k_s(n, d) has a unique graph generator class RU^l_s(n, d) derived by decomposition of graphs in RU^k_s(n, d). We investigate the minimization of the cardinality of V, the colorability, vertex coloring and the diameter of the graphs in the class. We also deal with complexity questions concerning RU^k_s(n, d). Some well-known computer network models such as barrel shifters and hypercubes are shown to belong in RU^k_s(n, d).     

The interval number of a complete multipartite graph

The interval number of a graph G, denoted i(G), is the least positive integer t for which G is the intersection graph of a family of sets each of which is the union of at most t closed intervals of the real line $\text{R}$. Trotter and Harary showed that the interval number of the complete bipartite graph K(m,n) is $\text{?}\text{(mn + 1)}\text{(m + n)}\text{?}$. Matthews showed that the interval number of the complete multipartite graph K(n₁,n₂,…,n_p) was the same as the interval number of K(n₁,n₂) when n₁ = n₂ = ? = n_p. Trotter and Hopkins showed that i(K(n₁,n₂,…,n_p)) ≤ 1 + i(K(n₁,n₂)) whenever p ≥ 2 and n₁≥n₂≥ ? ≥n_p. West showed that for each n ≥ 3, there exists a constant c_n so that if p ≥ c_n,n₁ = n²?n ?1, and n₂ = n₃ = ? n_p = n, then i(K(n₁,n₂,…,n_p) = 1 + i(K(n₁, n₂)). In view of these results, it is natural to consider the problem of determining those pairs (n₁,n₂) with n₁ ≥ n₂ so that i(K(n₂,…,n_p)) = i(K(n₁,n₂)) whenever p ≥ 2 and n₂ ≥ n₃ ≥ ? ≥ n_p. In this paper, we present constructions utilizing Eulerian circuits in directed graphs to show that the only exceptional pairs are (n² ? n ? 1, n) for n ≥ 3 and (7,5).     

A trichotomy for a class of equivalence relations

Let Xn, n ∈ N be a sequence of non-empty sets, ψn : Xn2 → IR+. We consider the relation E = E((Xn, ψn)n∈N) on ∏n∈N Xn by (x, y) ∈ E((Xn, ψn)n∈N) <=>Σn∈Nψn(x(n), y(n)) < +∞. If E is an equiv- alence relation and all ψn, n ∈ N, are Borel, we show a trichotomy that either IRN/e1≤B E, E1≤B E, or E≤B E0. We also prove that, for a rather general case, E((Xn, ψn)n∈N) is an equivalence relation iff it is an ep-like equivalence relation.