A priori estimate of gradient of a solution of a certain differential inequality and quasiconformal mappings

We prove a global estimate for the gradient of the solution of the Poisson differential inequality |Δu(x)| ≤ a|Du(x)|² + b, x ∈ B ⁿ , where a, b < ∞ and $u|_{S^{n - 1} } \in C^{1,\alpha } (S^{n - 1} ,\mathbb{R}^m )$ . If m = 1 and $a \le (n + 1)/({\left| u \right|_\infty }4n\sqrt n )$ , then |Du| is a priori bounded. This generalizes some similar results due to S. Bernstein [4] and E. Heinz [10] for the plane. An application of these results yields the main result, namely that a quasiconformal mapping of the unit ball onto a domain with C ² smooth boundary satisfying the Poisson differential inequality is Lipschitz continuous. This extends some results of the author, Mateljevi?, and Pavlovi? from the complex plane to ? ⁿ .     

On a variant of Rado’s selection lemma and its equivalence with the Boolean prime ideal theorem

We establish that, in ZF (i.e., Zermelo–Fraenkel set theory minus the Axiom of Choice AC), the statement RLT: Given a set I and a non-empty set \({\mathcal{F}}\) of non-empty elementary closed subsets of 2 ^I satisfying the fip, if \({\mathcal{F}}\) has a choice function, then \({\bigcap\mathcal{F} \ne \emptyset}\) , which was introduced in Morillon (Arch Math Logic 51(7–8):739–749, 2012), is equivalent to the Boolean Prime Ideal Theorem (see Sect. 1 for terminology). The result provides, on one hand, an affirmative answer to Morillon’s corresponding question in Morillon (2012) and, on the other hand, a negative answer—in the setting of ZFA (i.e., ZF with the axiom of extensionality weakened to permit the existence of atoms)—to the question in Morillon (2012) of whether RLT is equivalent to Rado’s selection lemma.     

L p -theory for second-order elliptic operators with unbounded coefficients

Second-order elliptic operators with unbounded coefficients of the form ${Au := -{\rm div}(a\nabla u) + F . \nabla u + Vu}$ in ${L^{p}(\mathbb{R}^{N}) (N \in \mathbb{N}, 1 < p < \infty)}$ are considered, which are the same as in recent papers Metafune et?al. (Z Anal Anwendungen 24:497–521, 2005), Arendt et?al. (J Operator Theory 55:185–211, 2006; J Math Anal Appl 338: 505–517, 2008) and Metafune et?al. (Forum Math 22:583–601, 2010). A new criterion for the m-accretivity and m-sectoriality of A in ${L^{p}(\mathbb{R}^{N})}$ is presented via a certain identity that behaves like a sesquilinear form over L ^p ×?L ^p'. It partially improves the results in (Metafune et?al. in Z Anal Anwendungen 24:497–521, 2005) and (Metafune et?al. in Forum Math 22:583–601, 2010) with a different approach. The result naturally extends Kato’s criterion in (Kato in Math Stud 55:253–266, 1981) for the nonnegative selfadjointness to the case of p ≠?2. The simplicity is illustrated with the typical example ${Au = -u\hspace{1pt}'' + x^{3}u\hspace{1pt}' + c |x|^{\gamma}u}$ in ${L^p(\mathbb{R})}$ which is dealt with in (Arendt et?al. in J Operator Theory 55:185–211, 2006; Arendt et?al. in J Math Anal Appl 338: 505–517, 2008).     

Generation and Enumeration of Some Classes of Interval Orders

In this paper we consider the class of interval orders, recently considered by several authors from both an algebraic and an enumerative point of view. According to Fishburn’s Theorem (Fishburn J Math Psychol 7:144–149, 1970), these objects can be characterized as posets avoiding the poset 2?+?2. We provide a recursive method for the unique generation of interval orders of size n?+?1 from those of size n, extending the technique presented by El-Zahar (1989) and then re-obtain the enumeration of this class, as done in Bousquet-Melou et al. (2010). As a consequence we provide a method for the enumeration of several subclasses of interval orders, namely AV(2?+?2, N), AV(2?+?2, 3?+?1), AV(2?+?2, N, 3?+?1). In particular, we prove that the first two classes are enumerated by the sequence of Catalan numbers, and we establish a bijection between the two classes, based on the cardinalities of the principal ideals of the posets.     

Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes of Besov- and Sobolev-Type

We derive ? ^r (μ)-bracketing metric and sup-norm metric entropy rates of bounded subsets of general function spaces defined over ? ^d or, more generally, over Borel subsets thereof, by adapting results of Haroske and Triebel (Math. Nachr. 167, 131–156, 1994; 278, 108–132, 2005). The function spaces covered are of (weighted) Besov, Sobolev, Hölder, and Triebel type. Applications to the theory of empirical processes are discussed. In particular, we show that (norm-)bounded subsets of the above mentioned spaces are Donsker classes uniformly in various sets of probability measures.     

A Short Note on Essentially Σ1 Sentences

Guaspari (J Symb Logic 48:777–789, 1983) conjectured that a modal formula is it essentially Σ₁ (i.e., it is Σ₁ under any arithmetical interpretation), if and only if it is provably equivalent to a disjunction of formulas of the form ${\square{B}}$ . This conjecture was proved first by A. Visser. Then, in (de Jongh and Pianigiani, Logic at Work: In Memory of Helena Rasiowa, Springer-Physica Verlag, Heidelberg-New York, pp. 246–255, 1999), the authors characterized essentially Σ₁ formulas of languages including witness comparisons using the interpretability logic ILM. In this note we give a similar characterization for formulas with a binary operator interpreted as interpretability in a finitely axiomatizable extension of IΔ ₀  + Supexp and we address a similar problem for IΔ ₀  + Exp.     

A remark on normal forms and the “upside-down” I-method for periodic NLS: Growth of higher Sobolev norms

We study growth of higher Sobolev norms of solutions of the onedimensional periodic nonlinear Schr?dinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish $${\left\| {u(t)} \right\|_{{H^s}}} \le {(1 + \left| t \right|)^{a(s - 1) + }}$$ with ?? = 1 for a general power nonlinearity. In the quintic case, we obtain the above estimate with ?? = 1/2 via the space-time estimate due to Bourgain [4, 5]. In the cubic case, we compute concretely the terms arising in the first few steps of the normal form reduction and prove the above estimate with ?? = 4/9. These results improve the previously known results (except for the quintic case). In the Appendix, we also show how Bourgain??s idea in [4] on the normal form reduction for the quintic nonlinearity can be applied to other powers.     

Bounds of Singular Integrals on Weighted Hardy Spaces and Discrete Littlewood–Paley Analysis

We apply the discrete version of Calderón??s reproducing formula and Littlewood?CPaley theory with weights to establish the $H^{p}_{w} \to H^{p}_{w}$ (0<p<??) and $H^{p}_{w}\to L^{p}_{w}$ (0<p??1) boundedness for singular integral operators and derive some explicit bounds for the operator norms of singular integrals acting on these weighted Hardy spaces when we only assume w??A _??. The bounds will be expressed in terms of the A _q constant of w if q>q _w =inf?{s:w??A _s }. Our results can be regarded as a natural extension of the results about the growth of the A _p constant of singular integral operators on classical weighted Lebesgue spaces $L^{p}_{w}$ in Hytonen et al. (arXiv:1006.2530, 2010; arXiv:0911.0713, 2009), Lerner (Ill.?J.?Math. 52:653?C666, 2008; Proc. Am. Math. Soc. 136(8):2829?C2833, 2008), Lerner et?al. (Int.?Math. Res. Notes 2008:rnm 126, 2008; Math. Res. Lett. 16:149?C156, 2009), Lacey et?al. (arXiv:0905.3839v2, 2009; arXiv:0906.1941, 2009), Petermichl (Am. J. Math. 129(5):1355?C1375, 2007; Proc. Am. Math. Soc. 136(4):1237?C1249, 2008), and Petermichl and Volberg (Duke Math. J. 112(2):281?C305, 2002). Our main result is stated in Theorem?1.1. Our method avoids the atomic decomposition which was usually used in proving boundedness of singular integral operators on Hardy spaces.     

A Characterization of the Finite Veronesean by Intersection Properties

A combinatorial characterization of the Veronese variety of all quadrics in PG(n, q) by means of its intersection properties with respect to subspaces is obtained. The result relies on a similar combinatorial result on the Veronesean of all conics in the plane PG(2, q) by Ferri [Atti Accad. Naz. Lincei Rend. 61(6), 603?C610 (1976)], Hirschfeld and Thas [General Galois Geometries. Oxford University Press, New York (1991)], and Thas and Van Maldeghem [European J. Combin. 25(2), 275?C285 (2004)], and a structural characterization of the quadric Veronesean by Thas and Van Maldeghem [Q. J. Math. 55(1), 99?C113 (2004)].     

Generalized absolute Hausdorff summability of orthogonal series

For 1≦k≦2 and a sequence $\gamma :={\{\gamma(n)\}}_{n=1}^{\infty}$ that is quasi β-power monotone decreasing with ${\beta>1-\frac{1}{k}}$ , we prove the |A,γ| _k summability of an orthogonal series, where A is either a regular or Hausdorff matrix. For ${\beta>-\frac{3}{4}}$ , we give a necessary and sufficient condition for |A,γ| _k summability, where A is Hausdorff matrix. Our sufficient condition for ${\beta>-\frac{3}{4}}$ is weaker than that of Kantawala [1], ${\beta>-\frac{1}{k}}$ for |E,q,γ| _k summability; and of Leindler [4], β>?1 for |C,α,γ| _k , ${\alpha<\frac{1}{4}}$ . Also, our result generalizes the result of Spevakov [6] for |E,q,1|₁ summability.     

Degree Sum Conditions for Cyclability in Bipartite Graphs

We denote by G[X, Y] a bipartite graph G with partite sets X and Y. Let d _G (v) be the degree of a vertex v in a graph G. For G[X, Y] and ${S \subseteq V(G),}$ we define ${\sigma_{1,1}(S):=\min\{d_G(x)+d_G(y) : (x,y) \in (X \cap S,Y) \cup (X, Y \cap S), xy \not\in E(G)\}}$ . Amar et al. (Opusc. Math. 29:345–364, 2009) obtained σ _1,1(S) condition for cyclability of balanced bipartite graphs. In this paper, we generalize the result as it includes the case of unbalanced bipartite graphs: if G[X, Y] is a 2-connected bipartite graph with |X| ≥ |Y| and ${S \subseteq V(G)}$ such that σ _1,1(S) ≥ |X| + 1, then either there exists a cycle containing S or ${|S \cap X| > |Y|}$ and there exists a cycle containing Y. This degree sum condition is sharp.     

On topological Morse theory

Starting from the concept of Morse critical point, introduced in [A. Ioffe and E. Schwartzman, J. Math. Pures Appl. (9), 75 (1996), 125?C153], we propose a purely topological approach to Morse theory.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

NIP for some pair-like theories

Generalising work of Berenstein, Dolich and Onshuus (Preprint 145 on MODNET Preprint server, 2008) and Günayd?n and Hieronymi (Preprint 146 on MODNET Preprint server, 2010), we give sufficient conditions for a theory T _P to inherit N I P from T, where T _P is an expansion of the theory T by a unary predicate P. We apply our result to theories, studied by Belegradek and Zilber (J. Lond. Math. Soc. 78:563?C579, 2008), of the real field with a subgroup of the unit circle.     

On Characterization of Absolute Geometries

Starting from a general absolute plane A = (P, L, α, ≡) in the sense of Karzel et al. (Einführung in die Geometrie, p. 96, 1973), Karzel and Marchi introduced the notion of a Lambert–Saccheri quadrangle (L-S quadrangle) in Karzel and Marchi (Le Matematiche LXI:27–36, 2006): A quadruple (a, b, c, d) of points of P, no three collinear, is a L-S quadrangle, if ${\overline{a,d}\bot\overline{a,b}\bot\overline{b,c}\bot\overline{c,d}}$ . Denoting the foot of a on the line ${\overline{c, d}}$ with ${a^{\prime}=\{a\bot\overline{c,d}\}\cap \overline{c,d}}$ , the L-S quadrangle (a, b, c, d) is called rectangle, hyperbolic or elliptic quadrangle if ${a^{\prime}=d,\; a^{\prime}\,{\in}\, ]c,d[}$ or ${a^{\prime}\,{\notin}\, ]c,d[\cup \{d\}}$ respectively. Let LS be the set of all L-S quadrangles and LS _r , LS _h or LS _e the subset of all rectangles, hyperbolic or elliptic L-S quadrangles respectively. In Karzel and Marchi (Le Matematiche LXI:27–36, 2006) it was claimed that either LS =  LS _r or LS =  LS _h or LS =  LS _e . To this classification we add five further classifications of general absolute planes by using “distance” [defined in Karzel and Marchi (Discrete Math 308:220–230, 2008)] or the notions of “interior” and “exterior” angle, introduced in Karzel et al. (Resultate Math 51:61–71, 2007) and considering besides Lambert–Saccheri quadrangles, also triangles in particular right-angled triangles. For Lambert–Saccheri quadrangles (a, b, c, d) the relations between distances of the diagonal points (a, c) and (b, d) or between the “midpoint” ${o:=\overline{a,c}\cap\overline{b,d}}$ , and the corner points a, b, c, d give us possibilities for complete characterizations. Using triangles (a, b, c) and denoting by m and n the midpoints of (a, b) and (a, c) we classify the absolute planes by the relations between the distances |b, c| and 2|m, n|. All our main results are summarized at the end of the introduction.     

On Probability and Moment Inequalities for Supermartingales and Martingales

Burkholder’s type inequality is stated for the special class of martingales, namely the product of independent random variables. The constants in the latter are much less than in the general case which is considered in Nagaev (Acta Appl. Math. 79, 35–46, 2003; Teor. Veroyatn. i Primenen. 51(2), 391–400, 2006). On the other hand, the moment inequality is proved, which extends these by Wittle (Teor. Veroyatn. i Primenen. 5(3), 331–334, 1960) and Dharmadhikari and Jogdeo (Ann. Math. Stat. 40(4), 1506–1508, 1969) to martingales.     

Two remarks on normality preserving Borel automorphisms of \boldsymbol{\mathbb{R}{\kern-8.3pt}\mathbb{R}}^{{\bf \textit{n}}}

Let T be a bijective map on ? ⁿ such that both T and T ^???1 are Borel measurable. For any θ?∈?? ⁿ and any real n ×n positive definite matrix Σ, let N (θ, Σ) denote the n-variate normal (Gaussian) probability measure on ? ⁿ with mean vector θ and covariance matrix Σ. Here we prove the following two results: (1) Suppose $N(\boldsymbol{\theta}_j, I)T^{-1}$ is gaussian for 0?≤?j?≤?n, where I is the identity matrix and {θ _j ???θ ₀, 1?≤?j?≤?n } is a basis for ? ⁿ . Then T is an affine linear transformation; (2) Let $\Sigma_j = I + \varepsilon_j \mathbf{u}_j \mathbf{u}_j^{\prime},$ 1?≤?j?≤?n where ε _j ?>???1 for every j and {u _j , 1?≤?j?≤?n } is a basis of unit vectors in ? ⁿ with $\mathbf{u}_j^{\prime}$ denoting the transpose of the column vector u _j . Suppose N(0, I)T ^???1 and $N (\mathbf{0}, \Sigma_j)T^{-1},$ 1?≤?j?≤?n are gaussian. Then $T(\mathbf{x}) = \sum\nolimits_{\mathbf{s}} 1_{E_{\mathbf{s}}}(\mathbf{x}) V \mathbf{s} U \mathbf{x}$ a.e. x, where s runs over the set of 2 ⁿ diagonal matrices of order n with diagonal entries ±1, U, V are n ×n orthogonal matrices and { E _s } is a collection of 2 ⁿ Borel subsets of ? ⁿ such that { E _s } and {V s U (E _s )} are partitions of ? ⁿ modulo Lebesgue-null sets and for every j, $V \mathbf{s} U \Sigma_j (V \mathbf{s} U)^{-1}$ is independent of all s for which the Lebesgue measure of E _s is positive. The converse of this result also holds. Our results constitute a sharpening of the results of Nabeya and Kariya (J. Multivariate Anal. 20 (1986) 251–264) and part of Khatri (Sankhyā Ser. A 49 (1987) 395–404).     

Arbitrary torsion classes and almost free abelian groups

Using the set theoretical principle ? for arbitrary large cardinals κ, arbitrary large strongly κ-free abelian groupsA are constructed such that Hom(A, G)={0} for all cotorsion-free groupsG with |G|<κ. This result will be applied to the theory of arbitrary torsion classes for Mod-Z. It allows one, in particular, to prove that the classF of cotorsion-free abelian groups is not cogenerated by aset of abelian groups. This answers a conjecture of Göbel and Wald positively. Furthermore, arbitrary many torsion classes for Mod-Z can be constructed which are not generated or not cogenerated by single abelian groups.     

Classification of Sol lattices

Sol geometry is one of the eight homogeneous Thurston 3-geometries $${\bf E}^{3}, {\bf S}^{3}, {\bf H}^{3}, {\bf S}^{2}\times{\bf R}, {\bf H}^{2}\times{\bf R}, \widetilde{{\bf SL}_{2}{\bf R}}, {\bf Nil}, {\bf Sol}.$$ In [13] the densest lattice-like translation ball packings to a type (type I/1 in this paper) of Sol lattices has been determined. Some basic concept of Sol were defined by Scott in [10], in general. In our present work we shall classify Sol lattices in an algorithmic way into 17 (seventeen) types, in analogy of the 14 Bravais types of the Euclidean 3-lattices, but infinitely many Sol affine equivalence classes, in each type. Then the discrete isometry groups of compact fundamental domain (crystallographic groups) can also be classified into infinitely many classes but finitely many types, left to other publication. To this we shall study relations between Sol lattices and lattices of the pseudoeuclidean (or here rather called Minkowskian) plane [1]. Moreover, we introduce the notion of Sol parallelepiped to every lattice type. From our new results we emphasize Theorems 3?C6. In this paper we shall use the affine model of Sol space through affine-projective homogeneous coordinates [6] which gives a unified way of investigating and visualizing homogeneous spaces, in general.