Cells in any simple polygon formed by a planar point set

Let P be a finite point set in general position in the plane. We consider empty convex subsets of P such that the union of the subsets constitute a simple polygon S whose dual graph is a path, and every point in P is on the boundary of S. Denote the minimum number of the subsets in the simple polygons S's formed by P by f_p(P), and define the maximum value of f_p(P) by F_p(n) over all P with n points. We show that ⌈(4n-17)/15⌉?F_p(n)?⌊n/2⌋.     

Analytic Extension of Non Quasi - Analytic Whitney Jets of Beurling Type

Let (M_r)_r∈_?0 be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi - analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function f on ?ⁿ belonging to the non quasi - analytic (M_r)-class of Beurling type, there is an element g of the same class which is analytic on ?,ⁿ F and such that D^αf(x) = D^αg(x) for every α ∈ ?ⁿ₀ and x ∈ F.     

Linear transformations which preserve fixed rank

Let M_{m, n}(F) denote the set of all m×n matrices over the algebraically closed field F. Let T denote a linear transformation, T:M_{m, n}(F)→M_{m, n}(F). Theorem: If max(m, n)?2k?2, k?1, and T preserves rank k matrices [i.e.?(A)=k implies ?(T(A))=k], then there exist nonsingular m×m and n×n matrices U and V respectively such that either (i) T:A→UAV for all A?M_{m, n}(F), or (ii) m=n and T:A→UA^tV for all A?M_n(F), where A^t denotes the transpose of A.     

Sp(2N)-covers for self-contragredient supercuspidal representations of GL(N)

Let F be a non-archimedean local field of odd residual characteristic. Let (J,τ) be a maximal simple type in GLN(F) for the inertial class [GLN(F),π]_GLN(F) of a self-contragredient supercuspidal irreducible representation π of GLN(F). Identify GLN(F) to the standard Siegel Levi subgroup in Sp2N(F). We construct, in Sp2N(F), a type for the inertial class [GLN(F),π]_Sp2N(F), as a Sp2N(F)-cover of (J,τ), strongly related to the GL2N(F)-cover of (J×J,τ⊗τ) in GL2N(F) constructed by Bushnell and Kutzko and which induces to a simple type in GL2N(F). In the process, we show that if τ has positive level, then the maximal simple type (J,τ) may be attached to a simple stratum [A,n,0,β] where the field F[β] is a quadratic extension of F[β²], and to a simple character θ in C(A,0,β) Galois conjugate of its inverse.     

On a Heilbronn-type problem

Let F be a convex figure with area |F| and let G(n,F) denote the smallest number such that from any n points of F we can get G(n,F) triangles with areas less than or equal to |F|/4. In this article, to generalize some results of Soifer, we will prove that for any triangle T, G(5,T)=3; for any parallelogram P, G(5,P)=2; for any convex figure F, if S(F)=6, then G(6,F)=4.     

Analytic extension of non quasi-analytic Whitney jets of Roumieu type

Let (M_r)_{r∈? 0} be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi-analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function ? on ?ⁿ belonging to the non quasi-analytic (M_r)-class of Roumieu type, there is an element g of the same class which is analytic on ?ⁿ F and such that D^α ?(x) = D^αg(x) for every σ ∈ ?₀ ⁿ SBAP and x ∈ F.     

A new class of multiset Wilf equivalent pairs

We say the pair of patterns (σ,τ) is multiset Wilf equivalent if, for any multiset M, the number of permutations of M that avoid σ is equal to the number of permutations of M that avoid τ. In this paper, we find a large new class of multiset Wilf equivalent pairs, namely, the pair (σ_n-2(n-1)n, σ_n-2n(n-1)), for n?3 and σ_n-2 a permutation of {1^x₁,2^x₂,…,(n-2)^x_n-2}. It is the most general multiset Wilf equivalence result to date.     

On the transversal number and VC-dimension of families of positive homothets of a convex body

Let F be a family of positive homothets (or translates) of a given convex body K in Rⁿ. We investigate two approaches to measuring the complexity of F. First, we find an upper bound on the transversal number τ(F) of F in terms of n and the independence number ν(F). This question is motivated by a problem of Grünbaum [L. Danzer, B. Grünbaum, V. Klee, Helly’s theorem and its relatives, in: Proc. Sympos. Pure Math., vol. VII, Amer. Math. Soc., Providence, RI, 1963, pp. 101-180]. Our bound is exponential in n, an improvement from the previously known bound of Kim, Nakprasit, Pelsmajer and Skokan [S.-J. Kim, K. Nakprasit, M.J. Pelsmajer, J. Skokan, Transversal numbers of translates of a convex body, Discrete Math. 306 (18) (2006) 2166-2173], which was of order nⁿ. By a lower bound, we show that the right order of magnitude is exponential in n.Next, we consider another measure of complexity, the Vapnik-?ervonenkis dimension of F. We prove that vcdim(F)≤3 if n=2 and is infinite for some F if n≥3. This settles a conjecture of Grünbaum [B. Grünbaum, Venn diagrams and independent families of sets, Math. Mag. 48 (1975) 12-23]: Show that the maximum dual VC-dimension of a family of positive homothets of a given convex body K in Rⁿ is n+1. This conjecture was disproved by Naiman and Wynn [D.Q. Naiman, H.P. Wynn, Independent collections of translates of boxes and a conjecture due to Grünbaum, Discrete Comput. Geom. 9 (1) (1993) 101-105] who constructed a counterexample of dual VC-dimension . Our result implies that no upper bound exists.     

Sums of commutators in non-commutative Banach function spaces

Let M be a II_∞-factor and denote by τ its normal faithful semi-finite trace. For any rearrangement invariant Köthe function space X on [0,+∞[, let X(M,τ) be the associated non-commutative Banach function space. This paper is concerned with ideals in M of the form I_X(M,τ)=M∩X(M,τ) that are contained in L^p(M,τ) for some p>0. It is proved that an element T in I_X(M,τ) is a finite sum of commutators of the form [A,B] with A∈I_X(M,τ) and B∈M if and only if the function belongs to X, where ν_T is the Brown spectral measure of T and t→λ_t(T) is the non-increasing rearrangement of the function λ→|λ| with respect to ν_T. This extends to general Banach function spaces a result obtained by Kalton for quasi-Banach ideals of compact operators and implies that the Dixmier's trace of a quasi-nilpotent element in L^1,∞(M,τ) is always zero.     

Monotonicity of the minimum cardinality of an identifying code in the hypercube

Let M_t(n) denote the minimum cardinality of a t-identifying code in the n-cube. It was conjectured that for all n?2 and t?1 we have M_t(n)?M_t(n+1). We prove this inequality for t=1.     

A criterion for generating full matrix algebras

Let M_n(F) be the algebra of n×n matrices over a field F, and let A∈M_n(F) have characteristic polynomial c(x)=p₁(x)p₂(x)?p_r(x) where p₁(x),…,p_r(x) are distinct and irreducible in F[x]. Let X be a subalgebra of M_n(F) containing A. Under a mild hypothesis on the p_i(x), we find a necessary and sufficient condition for X to be M_n(F).     

A problem on conjugacy of matrices

Let F be an infinite field and n?12. Then the number of conjugacy classes of the upper triangular nilpotent matrices in M_n(F) under action by the subgroup of GL_n(F) consisting of all the upper triangular matrices is infinite.     

A simplified proof for the limit of a tower over a cubic finite field

Recently Bezerra, Garcia and Stichtenoth constructed an explicit tower F=(Fn)_n?0 of function fields over a finite field Fq³, whose limit λ(F)=limn→∞N(Fn)/g(Fn) attains the Zink bound λ(F)?2(q²−1)/(q+2). Their proof is rather long and very technical. In this paper we replace the complex calculations in their work by structural arguments, thus giving a much simpler and shorter proof for the limit of the Bezerra, Garcia and Stichtenoth tower.     

Sur les entiers dont la somme des chiffres est moyenne

The goal of this paper is to study sets of integers with an average sum of digits. More precisely, let g be a fixed integer, s(n) be the sum of the digits of n in basis g. Let f:N→N such that, in any interval [g^ν,g^ν+1[, f(n) is constant and near from (g-1)ν/2. We give an asymptotic for the number of integers n<x such that s(n)=f(n) and we prove that for every irrational α the sequence (αn) is equidistributed mod 1, for n satisfying s(n)=f(n).     

The minimal dimension of maximal commutative subalgebras of full matrix algebras

Let F be a field and A a maximal commutative subalgebra of the full matrix algebra M_n(F). It is shown that dim A > (2n)^{$\text{2}\text{3}$} ? 1. It is also shown that if the radical of A has cube zero, then dim A ? [3n^{$\text{2}\text{3}$} ? 4], and that this result is best possible for infinitely many natural numbers n.     

On averaging Frankl's conjecture for large union-closed-sets

Let F be a union-closed family of subsets of an m-element set A. Let n=|F|?2 and for a∈A let s(a) denote the number of sets in F that contain a. Frankl's conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element a∈A with n−2s(a)?0. Strengthening a result of Gao and Yu [W. Gao, H. Yu, Note on the union-closed sets conjecture, Ars Combin. 49 (1998) 280-288] we verify the conjecture for the particular case when m?3 and n?m²−²m/2. Moreover, for these “large” families F we prove an even stronger version via averaging. Namely, the sum of the n−2s(a), for all a∈A, is shown to be non-positive. Notice that this stronger version does not hold for all union-closed families; however we conjecture that it holds for a much wider class of families than considered here. Although the proof of the result is based on elementary lattice theory, the paper is self-contained and the reader is not assumed to be familiar with lattices.     

On the number of antipodal bicolored necklaces

Let ν(2n) be the number of antipodal bicolored necklaces with 2n pearls. In this note, we find the first two terms of the asymptotic expansion of ν(2n). As a byproduct of this result, we also show that the sequence (ν(2n)) _n≥1 is non-holonomic, i.e., it satisfies no linear recurrence of a fixed finite order k with polynomial coefficients.     

A sufficient condition for the genus of an amalgamated 3-manifold not to go down

Let M i be a connected, compact, orientable 3-manifold, F i a boundary component of M i with g(F i ) 2, i = 1, 2, and F 1 ≌ F 2 . Let  : F 1 → F 2 be a homeomorphism, and M = M 1 ∪  M 2 , F = F 2 = (F 1 ). Then it is known that g(M ) g(M 1 ) + g(M 2 ) - g(F ). In the present paper, we give a sufficient condition for the genus of an amalgamated 3-manifold not to go down as follows: Suppose that there is no essential surface with boundary (Q i1 , Q i ) in (M i1 , F i ) satisfying χ(Q i ) 3 - 2g(M i ), i = 1, 2. Then g(M ) = g(M 1 ) + g(M 2 ) - g(F ).