Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations

For the nth order differential equation, y(n)=f(x,y,y^′,…,y(n−1)), we consider uniqueness implies existence results for solutions satisfying certain nonlocal (k+2)-point boundary conditions, 1?k?n−1. Uniqueness of solutions when k=n−1 is intimately related to uniqueness of solutions when 1?k?n−2. These relationships are investigated as well.     

On the modular sumset partition problem

A sequence m₁≥m₂≥?≥m_k of k positive integers isn-realizable if there is a partition X₁,X₂,…,X_k of the integer interval [1,n] such that the sum of the elements in X_i is m_i for each i=1,2,…,k. We consider the modular version of the problem and, by using the polynomial method by Alon (1999) [2], we prove that all sequences in Z/pZ of length k≤(p−1)/2 are realizable for any prime p≥3. The bound on k is best possible. An extension of this result is applied to give two results of p-realizable sequences in the integers. The first one is an extension, for n a prime, of the best known sufficient condition for n-realizability. The second one shows that, for n≥(4k)³, an n-feasible sequence of length k isn-realizable if and only if it does not contain forbidden subsequences of elements smaller than n, a natural obstruction forn-realizability.     

Generalizations of a lemma of Freudenburg

Let k be an algebraically closed field of characteristic zero and ℘ a prime ideal in k[X]?k[X₁,…,X_n]. Let g∈k[X] and d?1. If for all 1?|α|?d the derivatives ∂^αg belong to ℘, then there exists c∈k such that g−c∈℘(d+1), the d+1th symbolic power of ℘. In particular, if ℘ is a complete intersection it follows that g−c∈℘d+1.     

Convolution identities and lacunary recurrences for Bernoulli numbers

We extend Euler's well-known quadratic recurrence relation for Bernoulli numbers, which can be written in symbolic notation as n(B₀+B₀)=−nBn−1−(n−1)Bn, to obtain explicit expressions for n(Bk+Bm) with arbitrary fixed integers k,m?0. The proof uses convolution identities for Stirling numbers of the second kind and for sums of powers of integers, both involving Bernoulli numbers. As consequences we obtain new types of quadratic recurrence relations, one of which gives B6k depending only on B2k,B2k+2,…,B4k.     

On the dimension of global sections of adjoint bundles for polarized 3-folds and 4-folds

Let (X,L) be a polarized manifold of dimension n defined over the field of complex numbers. In this paper, we treat the case where n=3 and 4. First we study the case of n=3 and we give an explicit lower bound for h⁰(K_X+L) if κ(X)≥0. Moreover, we show the following: if κ(K_X+L)≥0, then h⁰(K_X+L)>0 unless κ(X)=−∞ and h¹(O_X)=0. This gives us a partial answer of Effective Non-vanishing Conjecture for polarized 3-folds. Next for n=4 we investigate the dimension of H⁰(K_X+mL) for m≥2. If n=4 and κ(X)≥0, then a lower bound for h⁰(K_X+mL) is obtained. We also consider a conjecture of Beltrametti-Sommese for 4-folds and we can prove that this conjecture is true unless κ(X)=−∞ and h¹(O_X)=0. Furthermore we prove the following: if (X,L) is a polarized 4-fold with κ(X)≥0 and h¹(O_X)>0, then h⁰(K_X+L)>0.     

Galois actions on torsion points of one-dimensional formal modules

Let F be a local non-Archimedean field with ring of integers o and uniformizer ?, and fix an algebraically closed extension k of the residue field of o. Let X be a one-dimensional formal o-module of F-height n over k. By the work of Drinfeld, the universal deformation X of X is a formal group over a power series ring R₀ in n−1 variables over Wo(k). For h∈{0,…,n−1} let Uh⊂Spec(R₀) be the locus where the connected part of the associated ?-divisible module X[?^∞] has height h. Using the theory of Drinfeld level structures we show that the representation of π₁(Uh) on the Tate module of the étale quotient is surjective.     

A new result on Davenport constant

Let G be a finite abelian group of order n and Davenport constant D(G). Let S=⁰h(S)_∏g∈Ggvg(S)∈F(G) be a sequence with a maximal multiplicity h(S) attained by 0 and t=|S|?n+D(G)−1. Then 0∈k_∑(S) for every 1?k?t+1−D(G). This is a refinement of the fundamental result of Gao [W.D. Gao, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100-103].     

Topology of random clique complexes

In a seminal paper, Erd?s and Rényi identified a sharp threshold for connectivity of the random graph G(n,p). In particular, they showed that if p?logn/n then G(n,p) is almost always connected, and if p?logn/n then G(n,p) is almost always disconnected, as n→∞.The clique complexX(H) of a graph H is the simplicial complex with all complete subgraphs of H as its faces. In contrast to the zeroth homology group of X(H), which measures the number of connected components of H, the higher dimensional homology groups of X(H) do not correspond to monotone graph properties. There are nevertheless higher dimensional analogues of the Erd?s-Rényi Theorem.We study here the higher homology groups of X(G(n,p)). For k>0 we show the following. If p=n^α, with α<−1/k or α>−1/(2k+1), then the kth homology group of X(G(n,p)) is almost always vanishing, and if −1/k<α<−1/(k+1), then it is almost always nonvanishing.We also give estimates for the expected rank of homology, and exhibit explicit nontrivial classes in the nonvanishing regime. These estimates suggest that almost all d-dimensional clique complexes have only one nonvanishing dimension of homology, and we cannot rule out the possibility that they are homotopy equivalent to wedges of a spheres.     

Graphs with maximum size and lower bounded girth

For integers n≥4 and ν≥n+1, let ex(ν;{C₃,…,C_n}) denote the maximum number of edges in a graph of order ν and girth at least n+1. The {C₃,…,C_n}-free graphs with order ν and size ex(ν;{C₃,…,C_n}) are called extremal graphs and denoted by EX(ν;{C₃,…,C_n}). We prove that given an integer k≥0, for each n≥2log₂(k+2) there exist extremal graphs with ν vertices, ν+k edges and minimum degree 1 or 2. Considering this idea we construct four infinite families of extremal graphs. We also see that minimal (r;g)-cages are the exclusive elements in EX(ν₀(r,g);{C₃,…,C_g−1}).     

Properties of connected graphs having minimum degree distance

The degree distance of a connected graph, introduced by Dobrynin, Kochetova and Gutman, has been studied in mathematical chemistry. In this paper some properties of graphs having minimum degree distance in the class of connected graphs of order n and size m≥n−1 are deduced. It is shown that any such graph G has no induced subgraph isomorphic to P₄, contains a vertex z of degree n−1 such that G−z has at most one connected component C such that |C|≥2 and C has properties similar to those of G.For any fixed k such that k=0,1 or k≥3, if m=n+k and n≥k+3 then the extremal graph is unique and it is isomorphic to K₁+(K_1,k+1∪(n−k−3)K₁).     

Type-B generalized triangulations and determinantal ideals

For n≥3, let Ω_n be the set of line segments between the vertices of a convex n-gon. For j≥2, a j-crossing is a set of j line segments pairwise intersecting in the relative interior of the n-gon. For k≥1, let Δ_n,k be the simplicial complex of (type-A) generalized triangulations, i.e. the simplicial complex of subsets of Ω_n not containing any (k+1)-crossing.The complex Δ_n,k has been the central object of many papers. Here we continue this work by considering the complex of type-B generalized triangulations. For this we identify line segments in Ω_2n which can be transformed into each other by a 180^°-rotation of the 2n-gon. Let F_n be the set Ω_2n after identification, then the complex D_n,k of type-B generalized triangulations is the simplicial complex of subsets of F_n not containing any (k+1)-crossing in the above sense. For k=1, we have that D_n,1 is the simplicial complex of type-B triangulations of the 2n-gon as defined in [R. Simion, A type-B associahedron, Adv. Appl. Math. 30 (2003) 2-25] and decomposes into a join of an (n−1)-simplex and the boundary of the n-dimensional cyclohedron. We demonstrate that D_n,k is a pure, k(n−k)−1+kn dimensional complex that decomposes into a kn−1-simplex and a k(n−k)−1 dimensional homology-sphere. For k=n−2 we show that this homology-sphere is in fact the boundary of a cyclic polytope. We provide a lower and an upper bound for the number of maximal faces of D_n,k.On the algebraical side we give a term order on the monomials in the variables X_ij,1≤i,j≤n, such that the corresponding initial ideal of the determinantal ideal generated by the (k+1) times (k+1) minors of the generic n×n matrix contains the Stanley-Reisner ideal of D_n,k. We show that the minors form a Gröbner-Basis whenever k∈{1,n−2,n−1} thereby proving the equality of both ideals and the unimodality of the h-vector of the determinantal ideal in these cases. We conjecture this result to be true for all values of k<n.     

Power integral bases for Selmer-like number fields

The Selmer trinomials are the trinomials f(X)∈{Xn−X−1,Xn+X+1|n>1 is an integer} over Z. For these trinomials we show that the ideal C=(f(X),f^′(X))Z[X] has height two and contains the linear polynomial (n−1)X+n. We then give several necessary and sufficient conditions for D[X]/(f(X)D[X]) to be a regular ring, where f(X) is an arbitrary polynomial over a Dedekind domain D such that its ideal C has height two and contains a product of primitive linear polynomials. We next specialize to the Selmer-like trinomials bXn+cX+d and bXn+cXn−1+d over D and give several more such necessary and sufficient conditions (among them is that C is a radical ideal). We then specialize to the Selmer trinomials over Z and give quite a few more such conditions (among them is that the discriminant Disc(Xn−X−1)=±(nn−(1−n)ⁿ⁻¹) of Xn−X−1 is square-free (respectively Disc(Xn+X+1)=±(nn+(1−n)ⁿ⁻¹) of Xn+X+1 is square-free)). Finally, we show that nn+(1−n)ⁿ⁻¹ is never square-free when n≡2 (mod 3) and n>2, but, otherwise, both are very often (but not always) square-free.     

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.     

Higher-order recurrences for Bernoulli numbers

Euler's well-known nonlinear relation for Bernoulli numbers, which can be written in symbolic notation as n(B₀+B₀)=−nBn−1−(n−1)Bn, is extended to n(Bk₁+?+Bkm) for m?2 and arbitrary fixed integers k₁,…,km?0. In the general case we prove an existence theorem for Euler-type formulas, and for m=3 we obtain explicit expressions. This extends the authors' previous work for m=2.     

On minimal Sturmian partial words

Partial words, which are sequences that may have some undefined positions called holes, can be viewed as sequences over an extended alphabet A_?=A∪{?}, where ? stands for a hole and matches (or is compatible with) every letter in A. The subword complexity of a partial word w, denoted by p_w(n), is the number of distinct full words (those without holes) over the alphabet that are compatible with factors of length n of w. A function f:N→N is (k,h)-feasible if for each integer N≥1, there exists a k-ary partial word w with h holes such that p_w(n)=f(n) for all n such that 1≤n≤N. We show that when dealing with feasibility in the context of finite binary partial words, the only affine functions that need investigation are f(n)=n+1 and f(n)=2n. It turns out that both are (2,h)-feasible for all non-negative integers h. We classify all minimal partial words with h holes of order N with respect to f(n)=n+1, called Sturmian, computing their lengths as well as their numbers, except when h=0 in which case we describe an algorithm that generates all minimal Sturmian full words. We show that up to reversal and complement, any minimal Sturmian partial word with one hole is of the form aⁱ?a^jba^l, where i,j,l are integers satisfying some restrictions, that all minimal Sturmian partial words with two holes are one-periodic, and that up to complement, ?(a^N−1?)^h−1 is the only minimal Sturmian partial word with h≥3 holes. Finally, we give upper bounds on the lengths of minimal partial words with respect to f(n)=2n, showing them tight for h=0,1 or 2.     

Direct images of bundles under Frobenius morphism

Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→X ₁ be the relative Frobenius morphism. For any vector bundle W on X, we prove that instability of F _* W is bounded by instability of W⊗T^ℓ(Ω¹ _X ) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F _* W being stable whenever W is stable. Dedicated to Professor Zhexian Wan on the occasion of his 80th birthday.     

A process of runs and its convergence to the brownian motion

Let X₁,X₂,… be i.i.d. random variables with a continuous distribution function. Let R₀=0, R_k=min{j>R_k?1, such that X_j>X_j+1}, k?1. We prove that all finite-dimensional distributions of a process $\text{W}{}^{\mathrm{(n)}}\text{(t)=}\text{(R}{}_{\mathrm{[nt]}}\text{?2[nt])}\text{2}\text{3}\text{n}\text{, t ? [0,1]}$, converge to those of the standard Brownian motion.     

Removable matchings and hamiltonian cycles

For a graph G, let σ₂(G) denote the minimum degree sum of two nonadjacent vertices (when G is complete, we let σ₂(G)=∞). In this paper, we show the following two results: (i) Let G be a graph of order n≥4k+3 with σ₂(G)≥n and let F be a matching of size k in G such that G−F is 2-connected. Then G−F is hamiltonian or G≅K₂+(K₂∪K_n−4) or ; (ii) Let G be a graph of order n≥16k+1 with σ₂(G)≥n and let F be a set of k edges of G such that G−F is hamiltonian. Then G−F is either pancyclic or bipartite. Examples show that first result is the best possible.     

On the simpleness of zeros of Stokes multipliers

The aim of this paper is to discuss the simpleness of zeros of Stokes multipliers associated with the differential equation -Φ^″(X)+W(X)Φ(X)=0, where W(X)=X^m+a₁X^m-1+?+a_m is a real monic polynomial. We show that, under a suitable hypothesis on the coefficients a_k, all the zeros of the Stokes multipliers are simple.