Hirzebruch Functional Equations and Krichever Complex Genera

As is well known, the two-parameter Todd genus and the elliptic functions of level d define n-multiplicative Hirzebruch genera if d divides n + 1. Both cases are special cases of the Krichever genera defined by the Baker–Akhiezer function. In the present paper, the inverse problem is solved. Namely, it is proved that only these properties define n-multiplicative Hirzebruch genera among all Krichever genera for all n.

     

On the parity of crossing numbers

For an integer n ? 1, a graph G has an n-constant crossing number if, for any two good drawings ? and ?′ of G in the plane, μ(?) ≡ μ(?′) (mod n), where μ(?) is the number of crossings in ?. We prove that, except for trivial cases, a graph G has n-constant crossing number if and only if n = 2 and G is either K_p or K_q,r, where p, q, and r are odd.     

Additivity of multiplicative maps on triangular rings

In this paper we shall give a unified technique in the discussion of the additivity of n-multiplicative automorphisms, n-multiplicative derivations, n-elementary surjective maps, and Jordan multiplicative surjective maps on triangular rings.     

Investigations in the Theory of q-Additive and q-Multiplicative Functions. I

The mean values of g ₁(a ₁ n)...g _k(a _k n) are investigated where g ₁...g _k are q-multiplicative functions the absolute values of which are 1. Some applications are given.     

On Entire Functions which Share One Value CM with their First Derivative

If ? is a non-constant entire function which is sharing two distinct values a, b ∈ ? with ?′, then a result of Mues and Steinmetz states that ?′ = ?. In this note we consider the case that ? and ?′ share only one value counting multiplicity, where appropriate restrictions on the growth of ? are assumed.     

On Overspill Principles and Axiom Schemes for Bounded Formulas

We study the theories I?_n, L?_n and overspill principles for ?_n formulas. We show that IE_n ? L?_n ? I?_n, but we do not know if I?_n L?_n. We introduce a new scheme, the growth scheme Crγ, and we prove that L?_n ? Cr?_n? I?_n. Also, we analyse the utility of bounded collection axioms for the study of the above theories. Mathematics Subject Classification: 03F30, 03H15.     

Finite Groups with p-Multiplicative Probabilistic Zeta Function

We prove that a finite group G is p-soluble if and only if the Dirichlet polynomial P _G (s) is p-multiplicative. Moreover, we show that one can recover the set of prime divisors of the order of G from the knowledge of P _G (s).     

Efficient Communication by Phone Calls

We are given the question: how many phone calls are needed for n people to pool all their information in a succession of k-person party line phone calls? The question was proposed by Erdös for the special case K = 2. We prove here the result that [n ? 2/K ? 1] + [n ? 1 /k] + 1 calls are required if 1 ≤ n ≤ k², while 2[n ? 2/k ? 1] are required for n > k².     

On digraphs without antidirected cycles

Let t(n) denote the greatest number of arcs in a diagraph of orders n which does not contain any antidrected cycles. We show that [16/5(n ? 1)] ≤ t(n) ≤ 1/2 (n ? 1) for n ≥ 5. Let t_r (n) denote the corresponding quantity for r-colorable digraphs. We show that [16/5(n ? 1)] ≤ t₅(n) ≤ t₆(n) ≤ 10/3(n ? 1) for n ≥ 5 and that t₄(n) = 3(n ? 1) for n ≥ 3.     

Averaging of variational inequalities for the Laplacian with nonlinear restrictions along manifolds

In this article, we consider variational inequalities arising, e.g., in modelling diffusion of substances in porous media. We assume that the media fills a domain Ω_? of ? ⁿ with n?≥?3. We study the case where the number of cavities is large and they are periodically distributed along a (n???1)-dimensional manifold. ? is the period while ?^α is the size of each cavity with α?≥?1; ? is a parameter that converges towards zero. Moreover, we also assume that the nonlinear process involves a large parameter ?^?κ with κ?=?(α???1)(n???1). Passing to the scale limit, and depending on the value of α, the effective equation or variational inequality is obtained. In particular, we find a critical size of the cavities when α?=?κ?=?(n???1)/(n???2). We also construct correctors which improve convergence for α?≥?(n???1)/(n???2).     

Bachelor latin squares with large indivisible plexes

In a latin square of order n , a k ‐plex is a selection of kn entries in which each row, column, and symbol occurs k times. A 1 ‐plex is also called a transversal. A k ‐plex is indivisible if it contains no c ‐plex for 0 < c < k . We prove that, for all n ≥ 4 , there exists a latin square of order n that can be partitioned into an indivisible ? n / 2 ?‐plex and a disjoint indivisible ? n / 2 ?‐plex. For all n ≥ 3 , we prove that there exists a latin square of order n with two disjoint indivisible ? n / 2 ?‐plexes. We also give a short new proof that, for all odd n ≥ 5 , there exists a latin square of order n with at least one entry not in any transversal. Such latin squares have no orthogonal mate. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:304‐312, 2011     

On self-complementation

We prove that, with very few exceptions, every graph of order n, n ? 0, 1(mod 4) and size at most n ? 1, is contained in a self-complementary graph of order n. We study a similar problem for digraphs.     

Spanning triangulations in graphs

We prove that every graph of sufficiently large order n and minimum degree at least 2n/3 contains a triangulation as a spanning subgraph. This is best possible: for all integers n, there are graphs of order n and minimum degree ?2n/3? ? 1 without a spanning triangulation. © 2005 Wiley Periodicals, Inc. J Graph Theory     

On the -equation¶over pseudoconvex Kähler manifolds

The Picard variety Pic⁰(? ⁿ ) of a complex n-dimensional torus? ⁿ is the group of all holomorphic equivalence classes of topologically trivial holomorphic (principal) line bundles on ? ⁿ . The total space of a topologically trivial holomorphic (principal) line bundle on a compact K?hler manifold is weakly pseudoconvex. Thus we can regard Pic⁰(? ⁿ ) as a family of weakly pseudoconvex K?hler manifolds. We consider a problem whether the Kodaira's -Lemma holds on a total space of holomorphic line bundle belonging to Pic⁰(? ⁿ ). We get a criterion for this problem using a dynamical system of translations on Pic⁰(? ⁿ ). We also discuss the problem whether the -Lemma holds on strongly pseudoconvex K?hler manifolds or not. Using the result of ColColţoiu, we find a 1-convex complete K?hler manifold on which the -Lemma does not hold. Received: 11 June 1999 / Revised version: 22 November 1999     

Some unavoidable subdigraphs of tournaments

Let H(n, i) be a simple (n ? 1)-path v₁ → v₂ → …? → v_n with an additional arc v₁v_i (3 ? i ? n). We prove that for each n and i (3 ? i ? n), with few exceptions, every n-tournament T_n contains a copy of H(n, i).     

On the decomposition of Kn into complete m-partite graphs

Graham and Pollak [3] proved that n ?1 is the minimum number of edge-disjoint complete bipartite subgraphs into which the edges of K_n can be decomposed. Using a linear algebraic technique, Tverberg [2] gives a different proof of that result. We apply his technique to show that for “almost all n,” ? (n + m ?3)/(m ?1) ? is the minimum number of edge-disjoint complete m-partite subgraphs in a decomposition of K_n.     

Regular factors in K1,n free graphs

A graph is said to be K_1,n-free, if it contains no K_1,n as an induced subgraph. We prove that for n ? 3 and r ? n ?1, if G is a K_1,n-free graph with minimum degree at least (n²/4(n ?1))r + (3n ?6)/2 + (n ?1)/4r, then G has an r-factor (in the case where r is even, the condition r ? n ?1 can be dropped).     

ANEW RELATION-COMBINING THEOREM AND ITS APPLICATION

Let ?ⁿ denote the set of all formulas ?x₁…?x_n[P(x₁, …,x_n) = 0], where P is a polynomial with integer coefficients. We prove a new relation-combining theorem from which it follows that if ?ⁿ is undecidable over N, then ?²ⁿ⁺² is undecidable over Z.     

Regular reals

Say that α is an n‐strongly c. e. (n‐strongly computably enumerable) real if α is a sum of n many strongly c. e. reals, and that α is regular if α is n‐strongly c. e. for some n. Let Sⁿ be the set of all n‐strongly c. e. reals, Reg be the set of regular reals and CE be the set of c. e. reals. Then we have: S¹ ? S² ? · · · ? Sⁿ ? · · · ? ? Reg ? CE . This gives a hierarchy of the c. e. reals. We also study the regularity of the d. c. e. reals. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimum vertex‐diameter‐2‐critical graphs

We prove that the minimum number of edges in a vertex‐diameter‐2‐critical graph on n ≥ 23 vertices is (5n ? 17)/2 if n is odd, and is (5n/2) ? 7 if n is even. © 2005 Wiley Periodicals, Inc. J Graph Theory