Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">d</Emphasis></Subscript>

Let Y be a smooth, connected, projective complex curve. In this paper, we study the Hurwitz spaces which parameterize branched coverings of Y whose monodromy group is a Weyl group of type D _d and whose local monodromies are all reflections except one. We prove the irreducibility of these spaces when and successively we extend the result to curves of genus g ≥ 1.     

Irreducible Components of Hurwitz Spaces of Coverings with Two Special Fibers

In this paper we prove new results of irreducibility for Hurwitz spaces of coverings whose monodromy group is a Weyl group of type B _d and whose local monodromies are all reflections except two.     

A note on the triangle conjecture

We prove some particular cases of the following conjecture of Perrin and Schützenberger, known as “the triangle conjecture.” Let A = {a, b} be a two-letter alphabet, d a positive integer and let B_d = {aⁱba^j| 0 ? i + j ? d}. If X ? B_d is a code, then |X| ? d + 1.     

Necessary and sufficient consistency conditions for a recursive kernel regression estimate

A recursive kernel estimate ∑_{i = 1}ⁿ Y_iK⧸(x − X_i)h_i)⧸∑_{j = 1}ⁿ K((x − X_j)⧸h_j) of a regression m(x) = E{Y|X = x} calculated from independent observations (X₁, Y₁),…, (X_n, Y_n) of a pair (X, Y) of random variables is examined. ForE|Y|^{1 + δ} < ∞, δ > 0, the estimate is weakly pointwise consistent for almost all (μ) x ∈ R^d, μ is the probability measure of X, if and only if∑_i−1ⁿ h_i^d I_{hi > ɛ } ⧸ ∑_{j = 1}ⁿ h_j^d → 0 as n → ∞, all ɛ > 0, and∑_{i = 1}^∞ h_i^d = ∞, d is the dimension of X. For E|Y|^{1 + δ} < ∞, δ > 0, the estimate is strongly pointwise consistent for almost all (μ) x ∈ R^d, if and only if the same conditions hold. ForE|Y|^{1 + δ} < ∞, δ > 0, weak and strong consistency are equivalent. Similar results are given for complete convergence.     

On additive representation function of general sequences

Let 0 ≦ a ₁ < a ₂ < ? be an infinite sequence of integers and let r ₁(A, n) = |(i;j): a _i + a _j = n, i ≦ j|. We show that if d > 0 is an integer, then there does not exist n ₀ such that d ≦ r ₁ (A, n) ≦ d + [√2d + ½] for n > n ₀.     

Lower bounds on the size of spheres of permutations under the Chebychev distance

Lower bounds on the number of permutations p of {1, 2, . . . , n} satisfying |p _i ? i| ?? d for all i are given.     

Simultaneous rational approximation to the successive powers of a real number

Let n be an integer ≥ 1 and let θ be a real number which is not an algebraic number of degree ≤ [n/2]. We show that there exist ? > 0 and arbitrary large real numbers X such that the system of linear inequalities |x₀| ≤ X and |x₀θ^j − x_j| ≤ ?X^−1/[n/2] for 1 < j < n, has only the zero solution in rational integers x₀,…, x_n. This result refines a similar statement due to H. Davenport and W. M. Schmidt, where the upper integer part [n/2] is replaced everywhere by the integer part [n/2]. As a corollary, we improve slightly the exponent of approximation to 0 by algebraic integers of degree n + 1 over Q obtained by these authors.     

Removable singularity of the polyharmonic equation

Let x₀∈Ω⊂Rⁿ, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω?{x₀} has a removable singularity at x₀ if and only if as |x−x₀|→0 for n≥3 and as |x−x₀|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x−x₀|^2m−n) as |x−x₀|→0 for n≥3 and |u(x)|=o(|x−x₀|^2m−2log(|x−x₀|⁻¹)) as |x−x₀|→0 for n=2.     

On a characterization of irreducibility of a non-negative matrix

Let A denote an n×n matrix with all its elements real and non-negative, and let r_i be the sum of the elements in the ith row of A, i=1,…,n. Let B=A?D(r₁,…,r_n), where D(r₁,…,r_n) is the diagonal matrix with r_i at the position (i,i). Then it is proved that A is irreducible if and only if rank B=n?1 and the null space of B^T contains a vector d whose entries are all non-null.     

Optimal labelling of a product of two paths

If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.P_n will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. P_m × P_n will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(P_m × P_n) = min(m, n).     

An extremal problem for sets with applications to graph theory

Let X₁, …, X_n be n disjoint sets. For 1 ? i ? n and 1 ? j ? h let A_ij and B_ij be subsets of X_i that satisfy |A_ij| ? r_i and |B_ij| ? s_i for 1 ? i ? n, 1 ? j ? h, $\text{(∪}{}_{\mathrm{i}}\text{A}{}_{\mathrm{ij}}\text{) ∩ (∪}{}_{\mathrm{i}}\text{B}{}_{\mathrm{ij}}\text{) = ?}$ for 1 ? j ? h, $\text{(∪}{}_{\mathrm{i}}\text{A}{}_{\mathrm{ij}}\text{) ∩ (∪}{}_{\mathrm{i}}\text{B}{}_{\mathrm{il}}\text{) ≠ ?}$ for 1 ? j < l ? h. We prove that $\text{h?}\text{Π}\text{i=1}\text{n}\text{r}{}_{\mathrm{i}}\text{+s}{}_{\mathrm{i}}\text{r}{}_{\mathrm{i}}$. This result is best possible and has some interesting consequences. Its proof uses multilinear techniques (exterior algebra).     

Arithmetic-progression-weighted subsequence sums

Let G be an abelian group, let s be a sequence of terms s ₁, s ₂, …, s _n ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let $$W \odot S = \left\{ {w_1 s_1 + \cdots + w_n s_n :w_i a term of W,w_i \ne w_j for i \ne j} \right\},$$ which is a particular kind of weighted restricted sumset. We show that |W ⊙ S| ≥ min{|G| ? 1, n}, that W ⊙ S = G if n ≥ |G| + 1, and also characterize all sequences S of length |G| with W ⊙ S ≠ G. This result then allows us to characterize when a linear equation $$a_1 x_1 + \cdots + a_r x_r \equiv \alpha mod n,$$ where α, a ₁, …, a _r ∈ ? are given, has a solution (x ₁, …, x _r ) ∈ ? ^r modulo n with all x _i distinct modulo n. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $G \cong C_{n_1 } \oplus C_{n_2 }$ (where n ₁ |n ₂ and n ₂ ≥ 3) having k distinct terms, for any k ε [3, min{n ₁ + 1, exp(G)}]. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence.     

A generalization of Fan’s results: Distribution of cycle lengths in graphs

Fan [G. Fan, Distribution of cycle lengths in graphs, J. Combin. Theory Ser. B 84 (2002) 187-202] proved that if G is a graph with minimum degree δ(G)≥3k for any positive integer k, then G contains k+1 cycles C₀,C₁,…,C_k such that k+1<|E(C₀)|<|E(C₁)|<?<|E(C_k)|, |E(C_i)−E(C_i−1)|=2, 1≤i≤k−1, and 1≤|E(C_k)|−|E(C_k−1)|≤2, and furthermore, if δ(G)≥3k+1, then |E(C_k)|−|E(C_k−1)|=2. In this paper, we generalize Fan’s result, and show that if we let G be a graph with minimum degree δ(G)≥3, for any positive integer k (if k≥2, then δ(G)≥4), if d_G(u)+d_G(v)≥6k−1 for every pair of adjacent vertices u,v∈V(G), then G contains k+1 cycles C₀,C₁,…,C_k such that k+1<|E(C₀)|<|E(C₁)|<?<|E(C_k)|, |E(C_i)−E(C_i−1)|=2, 1≤i≤k−1, and 1≤|E(C_k)|−|E(C_k−1)|≤2, and furthermore, if d_G(u)+d_G(v)≥6k+1, then |E(C_k)|−|E(C_k−1)|=2.     

A differentiable classification of certain locally free actions of Lie groups

Let L be a Lie group and let M be a compact manifold with dimension dim(L) + 1. Let Φ be a locally free action of L on M having class C ^r with r ≥ 2. Let R be the radical of L and let χ₁, . . ., χ _n be the characters of the adjoint action of {itR}. Finally, let Δ be the modular function of R. Under the assumption that none of the identities Δ×|χ _i | = |χ _j |^α hold for any α ∈ [0, 1], one shows that Φ is the restriction to L of a locally free and transitive C ^r action of a larger Lie group. A second result is the existence of a unique Φ-invariant probability measure on {itM}; that measure is induced by a C ^r?1 nonsingular volume form. What makes that theorem all the more interesting is that certain of the Lie groups under consideration are not amenable.     

The algebraic independence of the sum of divisors functions

Let σj(n)=_∑d|ndj be the sum of divisors function, and let I be the identity function. When considering only one input variable n, we show that the set of functions is algebraically independent. With two input variables, we give a non-trivial identity involving the sum of divisors function, prove its uniqueness, and use it to prove that any perfect number n must have the form n=rσ(r)/(2r−σ(r)), with some restrictions on r. This generalizes the known forms for both even and odd perfect numbers.     

Sperner families over a subset

Let |X| = n > 0, |Y| = k > 0, and Y ? X. A family A of subsets of X is a Sperner family of X over Y if A₁A₂ for every pair of distinct members of A and every member of A has a nonempty intersection with Y. The maximum cardinality f(n, k) of such a family is determined in this paper. $\text{f(n,k)=(}\text{n}\text{[n}\text{2]}\text{)?(}\text{?k}\text{[n}\text{2]}\text{)}$.     

On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V→A induces an edge labeling f^∗:E→A defined by f^∗(xy)=f(x)+f(y). For i∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

On positional games

Let {A_i} be a family of sets and let S = ∩_iA_i. By a positional game we shall mean a game played by two players on {A_i}. The players alternately pick elements of S and that player wins who fist has all the elements of one of the A_i. This paper deals with almost disjoint hypergraphs only, i.e., |A_i∪A_j| ? 1 if i ≠ j. Let $\text{M}{}^1\text{(n)}$ be the smallest integer for which there is an almost disjoint n-uniform hypergraph $\text{|T| = M}{}^1\text{(n)}$, so that the first player has a winning strategy. It is shown that $\text{lim}{}_{\mathrm{n}}\text{[M}{}^1\text{(n)]}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}\text{= 4}$, which was conjectured by Erdös. The same method is applied to prove a conjecture of Hales and Jewett on r-dimensional tick-tack-toe if r is large enough. Finally we prove that for an arbitrary almost disjoint n-uniform hypergraph the second player has such a strategy that the first player unable to win in his mth move if m < (2 ? ?)ⁿ.     

Functional characterization of Vasil’ev invariants

A cover of a manifold X is called an r-cover if any r points of X belong to a set in the cover. Let X and Y be two smooth manifolds, let Emb(X, Y) be the family of smooth embeddings X → Y, let M be an Abelian group, and let F: Emb(X, Y) → M be a functional. One says that the degree of F does not exceed r if for each finite open r-cover {U _i } _i∈I ; of X there exist functionals F _i : Emb(U _i , Y) → M, i ∈ I, such that for each a ∈ Emb(X, Y) one has