Arrangements of planes in space

This paper presents a variety of formulas for the number of cells, faces, and edges, bounded and unbounded, that are formed by an arbitrary set of planes in $\text{R}$³. Using an elegant geometric method described in 1966 by Brousseau, we first prove a version in $\text{R}$³ of the general partition formulas established algebraically by Zaslavsky in 1975. From these formulas we deduce two families of inclusion–exclusion formulas for the counters, the first similar to formulas outlined by Roberts in 1889, the second related to formulas given by Steiner in 1826. We conclude with some non-trivial new bounds for the counters of an arbitrary arrangement in $\text{R}$³ and two specific examples.     

Ovoids and translation planes revisited

Kantor has previously described the translation planes which may be obtained by projecting sections of ovoids in ⁺(8, q)-spaces to ovoids in corresponding ⁺(6, q)-spaces. Since the Klein correspondence associates spreads in 4-dimensional vector spaces with ovoids in ⁺(6, q)-spaces, there are corresponding translation planes of order q ² and kernel containing GF(q). In this article, we revisit some of these translation planes and give some presentations of the spreads. Motivated by various properties of the planes, we study, in general, translation planes which admit certain homology groups and/or elation groups. In particular, we develop new constructions of projective planes of Lenz-Barlotti class II-1.Finally, we show how certain projective planes of order q ² of Lenz-Barlotti class II-1 may be considered equivalent to flocks of quadratic cones in PG(3, q).This work was partially supported by NSF grant DMS-8800843.     

Eine geometrische Kennzeichnung der reellen absoluten Ebenen

A new geometric characterization of the real absolute planes is presented, which is based upon few and simple axioms concerning properties of a congruence relation. The ordering properties are developed from two axioms concerning triangles and circles. We use essential results of reflection geometry in order to prove that the structures under consideration have the well-known representations.     

Invariant <Emphasis Type="Italic">f</Emphasis>-structures on naturally reductive homogeneous spaces

We study invariant metric f-structures on naturally reductive homogeneous spaces and establish their relation to generalized Hermitian geometry. We prove a series of criteria characterizing geometric and algebraic properties of important classes of metric f-structures: nearly Kähler, Hermitian, Kähler, and Killing structures. It is shown that canonical f-structures on homogeneous Φ-spaces of order k (homogeneous k-symmetric spaces) play remarkable part in this line of investigation. In particular, we present the final results concerning canonical f-structures on naturally reductive homogeneous Φ-spaces of order 4 and 5.     

On quasilinear parabolic evolution equations in weighted L p -spaces

In this paper we develop a geometric theory for quasilinear parabolic problems in weighted L _p -spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the ω-limit sets and the long-time behaviour of the solutions. These techniques are applied to a free boundary value problem. The results in this paper are mainly based on maximal regularity tools in (weighted) L _p -spaces.     

Extension Theorems for Large Scale Spaces via Coarse Neighbourhoods

We introduce the notion of (hybrid) large scale normal space and prove coarse geometric analogues of Urysohn’s Lemma and the Tietze Extension Theorem for these spaces, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which we obtain both the classical topological results and the (hybrid) large scale results as corollaries. We prove that all metric spaces are hybrid large scale normal, and characterize those locally compact abelian groups which (as hybrid large scale spaces) are hybrid large scale normal. Finally, we look at some properties of the Higson compactifications and coronas of hybrid large scale normal spaces.     

Symmetry of solutions for quasimonotone second-order elliptic systems in ordered Banach spaces

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \mathbb Rⁿ{\mathbb R^n} . The solutions take values in ordered Banach spaces E, e.g. E=\mathbb R^N{E=\mathbb R^N} ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.     

Local well-posedness for the homogeneous Euler equations

We introduce Triebel-Lizorkin-Lorentz function spaces, based on the Lorentz L^p,q-spaces instead of the standard L^p-spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of inviscid incompressible fluid in Rⁿ,n≥2. As a corollary we obtain global existence of solutions to the 2D Euler equations in the Triebel-Lizorkin-Lorentz space. For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator estimates in our spaces. The key methods of proof used are the Littlewood-Paley decomposition and the paradifferential calculus by J.M. Bony.     

Nonsingularity Conditions for Multifunctions

We discuss three different characterizations of continuity properties for general multifunctions S : R^d Rⁿ. Each of these characterizations is given by the same simple nonsingularity condition, but stated in terms of three different generalized derivatives. Two of these characterizations are known, but the third is new to this paper. We discuss how all three have immediate analogues as generalized inverse mapping theorems, and we apply our new characterization to develop a fundamental and very broad sensitivity theorem for solutions to parameterized optimization problems.     

On representing sylvester-gallai designs

A misstated conjecture in [3] leads to an interesting (1, 3) representation of the 7-point projective plane inR ⁴ where points are represented by lines and planes by 3-spaces. The corrected form of the original conjecture will be negated if there is a (1, 3) representation of the 13-point projective plane inR ⁴ but that matter is not settled.     

Convergence analysis for a family of improved super-Halley methods under general convergence condition

In this paper, we focus on the semilocal convergence for a family of improved super-Halley methods for solving non-linear equations in Banach spaces. Different from the results in Wang et al. (J Optim Theory Appl 153:779–793, 2012), the condition of Hölder continuity of third-order Fréchet derivative is replaced by its general continuity condition, and the latter is weaker than former. Moreover, the R-order of the methods is also improved. By using the recurrence relations, we prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is analyzed with the third-order Fréchet derivative of the operator satisfies general continuity condition and Hölder continuity condition.     

The Alexandroff Dimension of Digital Quotients of Euclidean Spaces

Alexandroff T ₀ -spaces have been studied as topological models of the supports of digital images and as discrete models of continuous spaces in theoretical physics. Recently, research has been focused on the dimension of such spaces. Here we study the small inductive dimension of the digital space X(W) constructed in [15] as a minimal open quotient of a fenestration W of R ⁿ . There are fenestrations of R ⁿ giving rise to digital spaces of Alexandroff dimension different from n , but we prove that if W is a fenestration, each of whose elements is a bounded convex subset of R ⁿ , then the Alexandroff dimension of the digital space X(W) is equal to n . Received December 6, 1999, and in revised form July 5, 2001, and August 31, 2001. Online publication January 7, 2002.     

On isometrically universal spaces, mappings, and actions of groups

In this paper we first consider some well-known classes of separable metric spaces which are isometrically ω-saturated (see [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005, xvi+559]) and, therefore, contain isometrically universal spaces. We put some problems concerning such spaces most of which are related with the properties of the isometrically universal Urysohn space. Furthermore, using the defined notions of isometrically universal mappings and G-spaces (which are analogies of the notion of isometrically universal spaces) we introduce the notions of an isometrically ω-saturated class of mappings and an isometrically ω-saturated class of G-spaces (in which there are “many” isometrically universal elements). We prove that all results of Sections 6.1 and 7.1 of [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005, xvi+559] can be reformulated for isometrically ω-saturated classes of spaces and G-spaces, respectively. In particular, we prove that if D and R are isometrically ω-saturated classes of spaces, then the class of all mappings with the domain in D and range in R is an isometrically ω-saturated class of mappings and, therefore, in this class there are isometrically universal elements. As a corollary of this result we have that since the class of all mappings is isometrically ω-saturated, in this class there are isometrically universal mappings. Similarly, if G is an arbitrary separable metric group and P is an isometrically ω-saturated class of spaces, then the class of all G-spaces (X,F), where X is an element of P, is an isometrically ω-saturated class of G-spaces and, therefore, in this class there are isometrically universal elements. In particular, for any separable metric group G, in the class of all G-spaces there are isometrically universal G-spaces. We also pose some problems concerning isometrically universal mappings and G-spaces some of which concern the Urysohn space.     

Continuity properties in non-commutative convolution algebras, with applications in pseudo-differential calculus

We study continuity properties for a family {s_p}_p?1 of increasing Banach algebras under the twisted convolution, which also satisfies that a∈s_p, if and only if the Weyl operator a^w(x,D) is a Schatten-von Neumann operator of order p on L². We discuss inclusion relations between the s_p-spaces, Besov spaces and Sobolev spaces. We prove also a Young type result on s_p for dilated convolution. As an application we prove that f(a)∈s₁, when a∈s₁ and f is an entire odd function. We finally apply the results on Toeplitz operators and prove that we may extend the definition for such operators.     

Convexities of metric spaces

We introduce two kinds of the notion of convexity of a metric space, called k-convexity and L-convexity, as generalizations of the CAT(0)-property and of the nonpositively curved property in the sense of Busemann, respectively. 2-uniformly convex Banach spaces as well as CAT(1)-spaces with small diameters satisfy both these convexities. Among several geometric and analytic results, we prove the solvability of the Dirichlet problem for maps into a wide class of metric spaces.      

Arithmetic of divisibility in finite models

We prove that the finite‐model version of arithmetic with the divisibility relation is undecidable (more precisely, it has Π⁰₁‐complete set of theorems). Additionally we prove FM‐representability theorem for this class of finite models. This means that a relation R on natural numbers can be described correctly on each input on almost all finite divisibility models if and only if R is of degree ≤ 0 ′. We obtain these results by interpreting addition and multiplication on initial segments of finite models with divisibility only. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Short-time behavior of the heat kernel and Weyl’s law on $${{\mathrm{RCD}}}^*(K,N)$$ spaces

In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of \({{\mathrm{RCD}}}^{*}(K,N)\)-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in \({{\mathrm{RCD}}}^*(K,N)\)-spaces. We use then these results to initiate the study of Weyl’s law in the \({{\mathrm{RCD}}}\) setting.     

Comparison theorems for diffusion processes

We prove comparison theorems for diffusion processes onR ^d. From these theorems we derive lower and upper bounds for the transition probabilities of a diffusion process. In contrast to the known estimates for fundamental solutions of parabolic equations our bounds do not depend on the moduli of continuity of the coefficients of the differential operator.     

On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

In this article, we study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri and Gérard (1999) [2] on R³, is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim (2004) [21] on the behavior of concentrating waves on manifolds.     

On the structure of a relatively free Grassmann algebra

We investigate the multiplicative and T-space structure of the relatively free algebra F ⁽³⁾ with a unity corresponding to the identity [[x ₁ , x ₂], x ₃] = 0 over an infinite field of characteristic p > 0. The highest emphasis is placed on unitary closed T-spaces over a field of characteristic p > 2. We construct a diagram containing all basic T-spaces of the algebra F ⁽³⁾, which form infinite chains of the inclusions. One of the main results is the decomposition of quotient T-spaces connected with F ⁽³⁾ into a direct sum of simple components. Also, the studied T-spaces are commutative subalgebras of F ⁽³⁾; thus, the structure of F(3) and its subalgebras can be described as modules over these commutative algebras. Separately, we consider the specifics of the case p = 2. In the Appendix, we study nonunitary closed T-spaces and the case of a field of zero characteristic.