A remark on blocking sets of almost Rédei type

In [1] it is shown that ifB is a minimum size blocking set in PG(2,p),p prime, such that there is a lineL meetingB in at least s|B|–p–p+45/20 points, thenB/L consists ofp, p + 1 orp + 2 points. It has been known for a long time that for anyp > 2 there is a unique example with ¦B/L¦= p [2]. In [3] the authors prove that ¦B/L¦ =p + 1 can only occur whenp 7. Here we show that if¦B/L¦= p+2, thenp = 3, 5 or 7, and all examples are classical ones. Besides combinatorial arguments we use polynomials over finite fields and a formula that generalizes the Newton formulae relating power sums and elementary symmetric polynomials.Research of the author was supported by OTKA Grants F-016302, T-019367 and COST Grant 3314/96.     

On π 2 almost geodesic mappings of almost Hermitian manifolds

We consider a class of almost geodesic mappings, namely, almost geodesic mappings of class π ₂, and obtain conditions under which almost Hermitian manifolds admit almost geodesic mappings of class π ₂. We prove that an almost Hermitian manifold admits a π ₂-mapping with respect to a Riemannian connection if and only if it is an NK-manifold. We obtain a condition on the defining form ψ of any nontrivial π ₂(e)-mapping under which a properNK-structure is taken to a proper NK-structure.     

Which sets are images of disk-algebra functions?

We give a characterization of those compact sets in the plane with finitely many holes that are images of disk-algebra functions. We also show that the image of the closed unit disk via a polynomial is, in general, not polynomially convex.     

Easily determining which urns are ‘favorable’

The optimal sampling strategy for an urn, containing known numbers of plus and minus ones, can be simply described with the use of an empirically justified rule, based upon what appears to be a legitimate third-order asymptotic expansion of “the optimal stopping boundary” as the urn size goes to infinity performs exceedingly well. There is a known first-order asymptotic expansion due to Shepp. The reader is invited to try to justify a second-order asymptotic expansion of a type described by Chernoff and Petkau. The evidence presented in its support is very persuasive.     

Hausdorff’s forgotten proof that almost all numbers are normal

Mathematische Semesterberichte - In 1914, Felix Hausdorff published an elegant proof that almost all numbers are simply normal in base 2. We generalize this proof to show that almost all...     

Quartic graphs which are Bakry-Émery curvature sharp

We give a classification of all connected quartic graphs which are (infinity) curvature sharp in all vertices with respect to Bakry-Émery curvature. The result is based on a computer classification by F. Gurr and L. Watson May and a combinatorial case by case investigation.     

Terms which are Mal’cev modulo some functions

We derive consequences from the existence of a term which satisfies Mal’cev identities (characterizing permutability) modulo two functions F and G from admissible relations to admissible relations. We also provide characterizations of varieties having a Mal’cev term modulo F and G. Received September 21, 2006; accepted in final form March 20, 2007.     

On commutative rings which are strongly prüfer

Abstract

As defined by Nicholson [Nicholson, W. K. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229:269–278] an element of a ring R is clean if it is the sum of a unit and an idempotent, and a subset A of R is clean if every element of A is clean. It is shown that a semiprimitive Gelfand ring R is clean if and only if Max(R) is zero-dimensional; if and only if for each M ∈ Max(R), the intersection all prime ideals contained in M is generated by a set of idempotents. We also give several equivalent conditions for clean functional rings. In fact, a functional ring R is clean if and only if the set of clean elements is closed under sum; if and only if every zero-divisor is clean; if and only if; R has a clean prime ideal.     

Directional approximation of functions of two variables in the spaces ℒ p (R2) and ℒ p (R + 2 )

Let r ∈ N, α, t ∈ R, x ∈ R ², f: R ² → C, and denote $ \Delta _{t,\alpha }^r (f,x) = \sum\limits_{k = 0}^r {( - 1)^{r - k} c_r^k f(x_1 + kt\cos \alpha ,x_2 + kt\sin \alpha ).} $ In this paper, we investigate the relation between the behavior of the quantity $ \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n (t)dt} } \right\|_{p,G} , $ as n → ∞ (here, E ? R, G ∈ {R ², R ₊ ² }, and ψ _n ∈ L ₁(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: $ \omega _{r,\alpha } (f,h)_{p,G} = \mathop {\sup }\limits_{0 \leqslant t \leqslant h} \left\| {\Delta _{t,\alpha }^r (f)} \right\|_{p,G} . $ Here is one of the results obtained. Theorem 1. Let E and A be intervals in R ₊ such that A ? E, f ∈ L _p (G), α ∈ [0, 2π] when G =R ² and α ∈ [0, π/2] when G = R ₊ ² Denote Δ _{n, k} = ∫ _A t ^k ψ _n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ _{m, r} > 0, Δ _{m, r + 1} < ∞, and $ \mathop {\lim }\limits_{n \to \infty } \frac{{\Delta _{n,r + 1} }} {{\Delta _{n,r} }} = 0,\mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \int\limits_{E\backslash A} {\Psi _n = 0} , $ then the relations $ \mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n dt} } \right\|_{p,G} \leqslant K, \mathop {\sup }\limits_{t \in (0,\infty )} t^r \omega _{r,\alpha } (f,t)_{p,G} \leqslant K $ are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and $ \sigma _{n,\alpha } (f,x) = \frac{2} {{\pi n}}\int\limits_{R_ + } {\Delta _{t,\alpha }^1 (f,x)} \left( {\frac{{\sin \frac{{nt}} {2}}} {t}} \right)^2 dt. $ Then the relations $ \mathop {\underline {\lim } }\limits_{n \to \infty } \frac{{\pi n}} {{\ln n}}\left\| {\sigma _{n,\alpha } (f)} \right\|_{p,G} \leqslant K Let r ∈ N, α, t ∈ R, x ∈ R ², f: R ² → C, and denote In this paper, we investigate the relation between the behavior of the quantity as n → ∞ (here, E ⊂ R, G ∈ {R ², R ₊²}, and ψ _n ∈ L ₁(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: Here is one of the results obtained. Theorem 1. Let E and A be intervals in R ₊ such that A ⊂ E, f ∈ L _p (G), α ∈ [0, 2π] when G =R ² and α ∈ [0, π/2] when G = R ₊² Denote Δ _{n, k} = ∫ _A t ^k ψ _n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ _{m, r} > 0, Δ _{m, r + 1} < ∞, and then the relations are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and Then the relations and are equivalent. Original Russian Text ? N.Yu. Dodonov, V.V. Zhuk, 2008, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2008, No. 2, pp. 23–33.     

ℒ2-cohomologie des espaces stratifiésdes espaces stratifiés

The first results relating intersection homology with ℒ²-cohomology were found by Cheeger, Goresky and MacPhersson (cf.[4] and [5]). The first spaces considered were the compact stratified pseudomanifolds with isolated singularities. Later, Nagase extended this result to any compact stratified spaceA possessing a Cheeger type riemannian metric μ (cf. [12]). The proof of the isomorphism uses the axiomatic caractérisation of the intersection homology of [2]. In this work we show how to realize this isomorphism by the usual integration of differential forms on simplices. The main tool used is the blow up of A into a smooth manifold, introduced in [2]. We also prove that any stratified space possesses a Cheeger type riemannian metric.

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Allocation de recherche de la DGICYT-Spain     

Ab initio generic structures which are superstable but not ω-stable

Let L be a countable relational language. Baldwin asked whether there is an ab initio generic L-structure which is superstable but not ω-stable. We give a positive answer to his question, and prove that there is no ab initio generic L-structure which is superstable but not ω-stable, if L is finite and the generic is saturated.     

Solvable extensions of naturally graded quasi-filiform Leibniz algebras of second type ℒ1 and ℒ3

For sequences of naturally graded quasi-filiform Leibniz algebras of second type ?¹ and ?³ introduced by Camacho et al., all possible right and left solvable indecomposable extensions over the field ? are constructed so that these algebras serve as the nilradicals of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program established to classify solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

Harder–Narasimhan Filtrations which are not split by the Frobenius maps

We will produce a smooth projective scheme X over ?, a rank 2 vector bundle V on X with a line subbundle L having the following property. For a prime p, let F _p be the absolute Fobenius of X _p , and let L _p ???V _p be the restriction of L???V. Then for almost all primes p, and for all t?≥?0, $(F_p^*)^t L_P \subset (F_p^*)^t V_p$ is a non-split Harder-Narasimhan filtration. In particular, $(F_p^*)^t V_p$ is not a direct sum of strongly semistable bundles for any t. This construction works for any full flag veriety G/B, with semisimple rank of G?≥?2. For the construction, we will use Borel–Weil–Bott theorem in characteristic 0, and Frobenius splitting in characteristic p.     

On Cullen numbers which are both Riesel and Sierpiński numbers

We describe an algorithm to determine whether or not a given system of congruences is satisfied by Cullen numbers. We use this algorithm to prove that there are infinitely many Cullen numbers which are both Riesel and Sierpiński. (Such numbers should be discarded if you are searching prime numbers with Proth?s theorem.)     

Solvable extensions of the naturally graded quasi-filiform Leibniz algebra of second type ℒ2

For a sequence of the naturally graded quasi-filiform Leibniz algebra of second type ?² introduced by Camacho, Gómez, González and Omirov, all possible right and left solvable indecomposable extensions over the field ? are constructed so that the algebra serves as the nilradical of the corresponding solvable Leibniz algebras we find in the paper.     

On the number of divisors which are values of a polynomial

Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let

A Dedekind σ-Complete Banach Lattice into which all Bounded Operators are Regular

We give an answer to a question of Abramovich–Wickstead asked in [Quart. J. Math. Oxford (2), 44 (1993), 257–270] under the axiom ‘measurable cardinal exists’. Mathematics Subject Classification (2000): Primary. 47B60