The pinched Veronese is Koszul

In this paper we prove that the coordinate ring of the pinched Veronese, k[X ³,X ² Y,XY ²,Y ³,X ² Z,Y ² Z,XZ ²,YZ ²,Z ³], is Koszul. The result is obtained by combining the use of a flat deformation induced by a distinguished weight together with a generalization of the notion of Koszul filtrations.     

Some Infinite Families of Locally Partial Geometries Generalizing the Classical Laguerre Planes

Starting with a subgeometry Ω embedded in a β-dimensional projective space PG(β, q), β 1, we construct inductively a series of rank n residually connected geometries Γ^{(n, β, Ω)}, n β, by putting Γ^{(β, β, Ω)} = Ω and extending Γ^{(n - 1, β, Ω)} with a partial geometry.     

On certain infinite extensions of the rationals with Northcott property

A set of algebraic numbers has the Northcott property if each of its subsets of bounded Weil height is finite. Northcott’s Theorem, which has many Diophantine applications, states that sets of bounded degree have the Northcott property. Bombieri, Dvornicich and Zannier raised the problem of finding fields of infinite degree with this property. Bombieri and Zannier have shown that \mathbb Q_ab^(d){{\mathbb Q}_{ab}^{(d)}} , the maximal abelian subfield of the field generated by all algebraic numbers of degree at most d, is such a field. In this note we give a simple criterion for the Northcott property and, as an application, we deduce several new examples, e.g. \mathbb Q(2^1/d₁,3^1/d₂,5^1/d₃,7^1/d₄,11^1/d₅,?){{\mathbb Q}(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},\ldots)} has the Northcott property if and only if 2^1/d₁, 3^1/d₂, 5^1/d₃, 7^1/d₄, 11^1/d₅{2^{1/d_1}, 3^{1/d_2}, 5^{1/d_3}, 7^{1/d_4}, 11^{1/d_5}} , . . . tends to infinity.     

A survey of orthogonal arrays of strength two

ＡＳＵＲＶＥＹＯＦＯＲＴＨＯＧＯＮＡＬＡＲＲＡＹＳＯＦＳＴＲＥＮＧＴＨＴＷＯＬＩＵＺＨＡＮＧＷＥＮ（刘璋温）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ．ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｔｉｃｅｓ．Ｂｅｉｊｉｎｇ１０００...     

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups II

Let p ₁, p ₂, p ₃ be primes. This is the second article in a series of three on the (p ₁, p ₂, p ₃)-generation of the finite projective special unitary and linear groups PSU₃(p ⁿ ), PSL₃(p ⁿ ), where we say a noncyclic group is (p ₁, p ₂, p ₃)-generated if it is a homomorphic image of the triangle group T _{p 1, p 2, p 3} . This paper is concerned with the case where p ₁ = 2 and p ₂ = p ₃. We determine for any prime p ₂ the prime powers p ⁿ such that PSU₃(p ⁿ ) (respectively, PSL₃(p ⁿ )) is a quotient of T = T _{2, p 2, p 2} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU₃(p ⁿ )) (respectively, Hom(T, PSL₃(p ⁿ ))) is surjective as p ⁿ tends to infinity.     

On numerical experiments with symmetric semigroups generated by three elements and their generalization

We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87−4k) and S(9,3+9k,85−9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r₁²,r₁r₂+r₁²k,r₃-r₁²k)\mathsf{S}(r_{1}^{2},r_{1}r_{2}+r_{1}^{2}k,r_{3}-r_{1}^{2}k), k∈ℤ, r ₁,r ₂,r ₃∈ℤ⁺, r ₁≥2 and gcd(r ₁,r ₂)=gcd(r ₁,r ₃)=1, and calculate their universal Frobenius number Φ(r ₁,r ₂,r ₃) for the wide range of k providing semigroups be symmetric. We show that this type of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r₁²,r₃-r₁²k}\{r_{1}^{2},r_{3}-r_{1}^{2}k\} for sporadic values of k and find these values by solving the quadratic Diophantine equation.     

Matrices dont deux lignes quelconque coïncident dans un nombre donne de positions communes

Le nombre maximal de lignes de matrices seront désignées par:

1. (a) R(k, λ) si chaque ligne est une permutation de nombres 1, 2,…, k et si chaque deux lignes différentes coïncide selon λ positions;

2. (b) S⁰(k, λ) si le nombre de colonnes est k et si chaque deux lignes différentes coïncide selon λ positions et si, en plus, il existe une colonne avec les éléments y₁, y₂, y₃, ou y₁ = y₂ ≠ y₃;

3. (c) T⁰(k, λ) si c'est une (0, 1)-matrice et si chaque ligne contient k unités et si chaque deux lignes différentes contient les unités selon λ positions et si, en plus, il existe une colonne avec les éléments 1, 1, 0.

La fonction T⁰(k, λ) était introduite par Chvátal et dans les articles de Deza, Mullin, van Lint, Vanstone, on montrait que T⁰(k, λ) max(λ + 2, (k − λ)² + k − λ + 1). La fonction S⁰(k, λ) est introduite ici et dans le Théorème 1 elle est étudiée analogiquement; dans les remarques 4, 5, 6, 7 on donne les généralisations de problèmes concernant T⁰(k, λ), S⁰(k, λ), dans la remarque 9 on généralise le problème concernant R(k, λ). La fonction R(k, λ) était introduite et étudiée par Bolton. Ci-après, on montre que R(k, λ) S⁰(k, λ) T⁰(k, λ) d'où découle en particulier: R(k, λ) λ + 2 pour λ k + 1 − (k + 2)^1/2; R(k, λ) = 0(k²) pour k − λ = 0(k); R(k, λ) (k − 1)² − (k + 2) pour k 1191.     

On the number of solutions of the generalized Ramanujan-Nagell equation D1X2 + DM2 = 2N+2

Let D₁, D₂ be coprime odd integers with min (D₁, D₂) > 1, and let N (D₁, D₂) denote the number of positive integer solutions (x, m, n) of the equation D₁x²+D^m₂ = 2ⁿ⁺². In this paper, we prove that N (D₁, D₂) ≤ 2 except for N (3, 5) = N (5, 3) = 4 and N (13, 3) = N (31, 97) = 3.     

Differential Invariants of Second Order ODEs,I

This paper is devoted to differential invariants of equations

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups III

Let p₁, p₂, p₃ be primes. This is the final paper in a series of three on the (p₁, p₂, p₃)-generation of the finite projective special unitary and linear groups PSU ₃(pⁿ), PSL ₃(pⁿ), where we say a noncyclic group is (p₁, p₂, p₃)-generated if it is a homomorphic image of the triangle group T_{p1, p2, p3} . This article is concerned with the case where p₁ = 2 and p₂ ≠ p₃. We determine for any primes p₂ ≠ p₃ the prime powers pⁿ such that PSU ₃(pⁿ) (respectively, PSL ₃(pⁿ)) is a quotient of T = T_{2, p2, p3} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU ₃(pⁿ)) (respectively, Hom(T, PSL ₃(pⁿ))) is surjective as pⁿ tends to infinity.     

Upper bounds on the sum of powers of the degrees of a simple planar graph

For a simple planar graph G and a positive integer k, we prove the upper bound 2(n ? 1)^k + 4^k(n ? 4) + 2·3^k ? 2((δ + 1)^k ? δ^k)(3n ? 6 ? m) on the sum of the kth powers of the degrees of G, where n, m, and δ are the order, the size, and the minimum degree of G, respectively. The bound is tight for all m with 0?3n ? 6 ? m≤?n/2? ? 2 and δ = 3. We also present upper bounds in terms of order, minimum degree, and maximum degree of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:112‐123, 2011     

Asymptotic properties of Cramér-Smirnov statistics—A new approach

Let Y₁,…, Y_n be independent identically distributed random variables with distribution function F(x, θ), θ = (θ′₁, θ′₂), where θ_i (i = 1, 2) is a vector of p_i components, p = p₁ + p₂ and for θI, an open interval in ^p, F(x, θ) is continuous. In the present paper the author shows that the asymptotic distribution of modified Cramér-Smirnov statistic under H_n: θ₁ = θ₁₀ + n^−1/2γ, θ₂ unspecified, where γ is a given vector independent of n, is the distribution of a sum of weighted noncentral χ₁² variables whose weights are eigenvalues of a covariance function of a Gaussian process and noncentrality parameters are Fourier coefficients of the mean function of the Gaussian process. Further, the author exploits the special form of the covariance function by using perturbation theory to obtain the noncentrality parameters and the weights. The technique is applicable to other goodness-of-fit statistics such as U² [G. S. Watson, Biometrika 48 (1961), 109–114].     

On the Cardinalities of the Row spaces of Non-full Rank Boolean Matrices

n -2 integers 2 ^{n -2}+2 ^{n -3}+2 ^s, where s=0,1,2,..., n-3, in the interval (2 ^{n -2}+2 ^{n -3},2 ^{n -1}] such that these integers are the cardinalities of row spaces R(A) of non-full rank Boolean matrices A of order n. We also show that for each s, where s=0,1,2,..., n-3, there exists A epsilon B _n such that A is non-full rank and the cardinality of R(A) equals 2 ^{n -2}+2 ^{n -3}+2 ^s.     

Coordinate gap and bidualily

For a varietyX inP ⁿ, we define a numberb ₂, called coordinate gap number,and prove thatb ₂>2 only ifX is not reflexive. Then, for a smooth surface inP ³, we obtain a concrete sufficient and necessary condition forb ₂>2, which enables us to discuss the biduality of surfaces inP ³.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

Approximation to continuous functions by Cesàro means of double Fourier series and conjugate series

We study the rate of uniform approximation to continuous functions ƒ(x, y), 2π-periodic in each variable, in Lipschitz classes Lip(α, β) and in Zygmund classes Z(α, β), 0 < α, β 1, by Cesàro means σ_mn^γδ(ƒ) of positive orders of the rectangular partial sums of double Fourier series. The rate of uniform approximation to the conjugate functions ^1,0, ^0,1 and ^1,1 by the corresponding Cesàro means is also discussed in detail. The difference between the classes Lip(α, β) and Z(α, β), similar to the one-dimensional case, appears again when max(α, β) = 1. (Compare Theorems 2 and 3 with Theorems 4 and 5.) One surprising result is the following: The uniform approximation rate by σ_mn^{γδ 1,0} to ^1,0 is worse in general than that by σ_mn^{γδ 1,1} to ^1,1 for ƒ ε Lip(1, 1). In fact, the appearance of an extra factor [log(n + 2)]² in the former case is unavoidable (see Theorem 6). All approximation rates we obtain, with one exception, are shown to be exact. Two conjectures are also included.     

A degree nine surface with 39 triple points

Starting from a cubic coneF ₃(x ₀,x ₁,x ₂,x ₃)=0 in the projective space, having the vertex as unique singularity and in particular position with respect to the fundamental tetrahedron, we construct the degree nine projective surfaceF ₃ (x ₀ ³,x ₁ ³,x ₂ ³,x ₃ ³)=0 endowed with 39 ordinary triple points and no other singularities. Al mio Maestro Dionisio Gallarati e alla passione per la ricerca a cui mi iniziò col problema del massimo numero di singolarità.     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.     

Bifurcation in the stable manifold of a chemostat with general polynomial yields

A three‐dimensional chemostat with nth‐ and mth‐order polynomial yields, instead of the particular ones such as A+BS, A+BS², A+BS³, A+BS⁴, A+BS² + CS³, and A+BSⁿ, is proposed. The existence of limit cycles in the two‐dimensional stable manifold, the Hopf bifurcation, and the stability of the periodic solution created by the bifurcation is proved. Copyright © 2009 John Wiley & Sons, Ltd.