Universal Monos in Partial Morphism Categories

In this paper the category, C\mathcal{C} with respect to a certain class D\mathcal{D} of subobjects of C\mathcal{C} is formed and the universality of monomorphisms of ${\overset{\lower0.5em\hbox{${\overset{\lower0.5em\hbox{ is investigated. The main result characterizes ${\overset{\lower0.5em\hbox{${\overset{\lower0.5em\hbox{-universality of monos, in terms of C\mathcal{C}-universality of monos and the existence of local C\mathcal{C}-implications.     

Domains with Prescribed Automorphism Group

Let G be the automorphism group of a bounded strictly pseudoconvex domain D⊂ℂ ^N with a smooth ( C^￥\mathcal{C}^{\infty} ) boundary. Let H be a closed subgroup of G. Pertaining to the question whether it is possible to realize H as the automorphism group of a strictly pseudoconvex domain D′ which is an arbitrarily small perturbation of D in C^￥\mathcal{C}^{\infty} topology, we give a partial answer by describing sufficient conditions for D and G.     

On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering

Given a set of vectors F={f ₁,…,f _m } in a Hilbert space H\mathcal {H}, and given a family C\mathcal {C} of closed subspaces of H\mathcal {H}, the subspace clustering problem consists in finding a union of subspaces in C\mathcal {C} that best approximates (is nearest to) the data F. This problem has applications to and connections with many areas of mathematics, computer science and engineering, such as Generalized Principal Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces C\mathcal {C} for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set C⁺\mathcal {C}^{+}. In infinite dimensions, however, the characterization is in terms of a new but related notion; that of contact half-spaces. As an application, the existence of best approximations from π(G)-invariant families C\mathcal {C} of unitary representations of Abelian groups is derived.     

Circular Favard Length of the Four-Corner Cantor Set

Let C_n\mathcal{C}_{n} be the n-th generation in the construction of the middle-half Cantor set. The Cartesian square K_n\mathcal{K}_{n} of C_n\mathcal{C}_{n} consists of 4 ⁿ squares of side-length 4⁻ⁿ . We drop a circle of radius r on the plane and try to estimate from below the conditional probability of this circle to intersect K_n\mathcal{K}_{n} if it already intersects a disc containing K_n\mathcal{K}_{n}. If the radius is very large  ≈4 ⁿ then clearly this should not differ too much from the usual Buffon needle probability. But it turns out that the best known lower bound (Bateman and Volberg in , 2008) persists even when the radius is much smaller than this—r>Cn ^ε suffices—and the intersection probability is at least \fracC_elognn\frac{C_{\varepsilon}\log n}{n}. This suggests that the method of Bateman and Volberg (, 2008) may be of use in proving a certain estimate for the lacunary circular maximal function from Seeger et al. (Preprint, 2005).     

A$\mathcal{T}$-algebras and extensions of A$\mathbb{T}$-algebras

Lin and Su classified A$ \mathcal{T} $ \mathcal{T} -algebras of real rank zero. This class includes all A$ \mathbb{T} $ \mathbb{T} -algebras of real rank zero as well as many C*-algebras which are not stably finite. An A$ \mathcal{T} $ \mathcal{T} -algebra often becomes an extension of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra. In this paper, we show that there is an essential extension of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra which is not an A$ \mathcal{T} $ \mathcal{T} -algebra. We describe a characterization of an extension E of an A$ \mathbb{T} $ \mathbb{T} -algebra by an AF-algebra if E is an A$ \mathcal{T} $ \mathcal{T} -algebra.     

Weak Reflections and Weak Factorization Systems

We describe a one-to-one correspondence between saturated weak factorization systems and weak reflections in categories C\mathcal{C} with finite products. This actually extends to an adjunction between the category of natural weak factorization systems on C\mathcal{C} (in the sense of Grandis and Tholen, Arch Math 42:397–408, 2006, and Garner, arXiv preprint, 2007) and the category of monads on C\mathcal{C}. Explicit comparisons are made with the parallel result of Cassidy et al. (J Aust Math Soc 38:287–329, 1985), linking factorization systems and reflective subcategories.     

Commutativity and Ideals in Pre-crystalline Graded Rings

Pre-crystalline graded rings constitute a class of rings which share many properties with classical crossed products. Given a pre-crystalline graded ring A\mathcal{A} , we describe its center, the commutant C_A(A₀)C_{\mathcal{A}}(\mathcal{A}_{0}) of the degree zero grading part, and investigate the connection between maximal commutativity of A₀\mathcal{A}_{0} in A\mathcal{A} and the way in which two-sided ideals intersect A₀\mathcal{A}_{0} .     

Automorphisms and Homotopies of Groupoids and Crossed Modules

This paper is concerned with the algebraic structure of groupoids and crossed modules of groupoids. We describe the group structure of the automorphism group of a finite connected groupoid C as a quotient of a semidirect product. We pay particular attention to the conjugation automorphisms of C, and use these to define a new notion of groupoid action. We then show that the automorphism group of a crossed module of groupoids C\mathcal{C}, in the case when the range groupoid is connected and the source group totally disconnected, may be determined from that of the crossed module of groups C_u\mathcal{C}_u formed by restricting to a single object u. Finally, we show that the group of homotopies of C\mathcal{C} may be determined once the group of regular derivations of C_u\mathcal{C}_u is known.     

Modules with Cosupport and Injective Functors

Several authors have studied the filtered colimit closure \varinjlimB\varinjlim\mathcal{B} of a class B\mathcal{B} of finitely presented modules. Lenzing called \varinjlimB\varinjlim\mathcal{B} the category of modules with support in B\mathcal{B}, and proved that it is equivalent to the category of flat objects in the functor category (B^op,Ab)(\mathcal{B}^\mathrm{op},\mathsf{Ab}). In this paper, we study the category (Mod-R)^B({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}} of modules with cosupport in B\mathcal{B}. We show that (Mod-R)^B({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}} is equivalent to the category of injective objects in (B,Ab)(\mathcal{B},\mathsf{Ab}), and thus recover a classical result by Jensen-Lenzing on pure injective modules. Works of Angeleri-Hügel, Enochs, Krause, Rada, and Saorín make it easy to discuss covering and enveloping properties of (Mod-R)^B({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}, and furthermore we compare the naturally associated notions of B\mathcal{B}-coherence and B\mathcal{B}-noetherianness. Finally, we prove a number of stability results for \varinjlimB\varinjlim\mathcal{B} and (Mod-R)^B({\mathsf{Mod}\textnormal{-}R})^{\mathcal{B}}. Our applications include a generalization of a result by Gruson-Jensen and Enochs on pure injective envelopes of flat modules.     

The Cycle Discrepancy of Three-Regular Graphs

Let G = (V, E) be an undirected graph and C(G){{\mathcal C}(G)} denote the set of all cycles in G. We introduce a graph invariant cycle discrepancy, which we define as

The Moduli Space of Complex Geodesics in the Complex Hyperbolic Plane

We consider the space M\mathcal{M} of ordered m-tuples of distinct complex geodesics in complex hyperbolic 2-space, H_\mathbbC²{\rm\bf H}_{\mathbb{C}}^{2}, up to its holomorphic isometry group PU(2,1). One of the important problems in complex hyperbolic geometry is to construct and describe the moduli space for M\mathcal{M}. This is motivated by the study of the deformation space of groups generated by reflections in complex geodesics. In the present paper, we give the complete solution to this problem.     

Latin trades in groups defined on planar triangulations

For a finite triangulation of the plane with faces properly coloured white and black, let A_W\mathcal{A}_{W} be the abelian group constructed by labelling the vertices with commuting indeterminates and adding relations which say that the labels around each white triangle add to the identity. We show that A_W\mathcal{A}_{W} has free rank exactly two. Let A_W^*\mathcal{A}_{W}^{*} be the torsion subgroup of  A_W\mathcal{A}_{W} , and A_B^*\mathcal{A}_{B}^{*} the corresponding group for the black triangles. We show that A_W^*\mathcal{A}_{W}^{*} and A_B^*\mathcal{A}_{B}^{*} have the same order, and conjecture that they are isomorphic. For each spherical latin trade W, we show there is a unique disjoint mate B such that (W,B) is a connected and separated bitrade. The bitrade (W,B) is associated with a two-colourable planar triangulation and we show that W can be embedded in  A_W^*\mathcal{A}_{W}^{*} , thereby proving a conjecture due to Cavenagh and Drápal. The proof involves constructing a (0,1) presentation matrix whose permanent and determinant agree up to sign. The Smith normal form of this matrix determines A_W^*\mathcal{A}_{W}^{*} , so there is an efficient algorithm to construct the embedding. Contrasting with the spherical case, for each genus g≥1 we construct a latin trade which is not embeddable in any group and another that is embeddable in a cyclic group.     

Regular Polytopes of Nearly Full Rank

An abstract regular polytope P\mathcal{P} of rank n can only be realized faithfully in Euclidean space \mathbbE^d\mathbb{E}^{d} of dimension d if d≥n when P\mathcal{P} is finite, or d≥n−1 when P\mathcal{P} is infinite (that is, P\mathcal{P} is an apeirotope). In case of equality, the realization P of P\mathcal{P} is said to be of full rank. If there is a faithful realization P of P\mathcal{P} of dimension d=n+1 or d=n (as P\mathcal {P} is finite or not), then P is said to be of nearly full rank. In previous papers, all the at most four-dimensional regular polytopes and apeirotopes of nearly full rank have been classified. This paper classifies the regular polytopes and apeirotopes of nearly full rank in all higher dimensions.     

Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WF_FL^q(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FL^q_(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S^(w₀)_r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that

Automorphism group and representation of a twisted multi-loop algebra

Let G be the complexification of the real Lie algebra so（3） and A = C[t1^±1, t2^±1] be the Lau-ent polynomial algebra with commuting variables. Let L：（t1, t2, 1） = G c .A be the twisted multi-loop Lie algebra. Recently we have studied the universal central extension, derivations and its vertex operator representations. In the present paper we study the automorphism group and bosonic representations ofL（t1, t2, 1）.     

Convergence and Stability of Locally ℝ N -Invariant Solutions of Ricci Flow

Valuable models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally $\mathcal{G}Valuable models for immortal solutions of Ricci flow that collapse with bounded curvature come from locally G\mathcal{G} -invariant solutions on bundles G^N\hookrightarrowM \oversetp? Bⁿ\mathcal{G}^{N}\hookrightarrow\mathcal{M}\,\overset{\pi }{\mathcal{\longrightarrow}}\,\mathcal{B}^{n} , with G\mathcal{G} a nilpotent Lie group. In this paper, we establish convergence and asymptotic stability, modulo smooth finite-dimensional center manifolds, of certain ℝ ^N -invariant model solutions. In case N+n=3, our results are relevant to work of Lott classifying the asymptotic behavior of all 3-dimensional Ricci flow solutions whose sectional curvatures and diameters are respectively O(t^-1)\mathcal{O}(t^{-1}) and O(t^1/2)\mathcal{O}(t^{1/2}) as t→∞.     

The asymptotic distribution of circles in the orbits of Kleinian groups

Let P\mathcal{P} be a locally finite circle packing in the plane ℂ invariant under a non-elementary Kleinian group Γ and with finitely many Γ-orbits. When Γ is geometrically finite, we construct an explicit Borel measure on ℂ which describes the asymptotic distribution of small circles in P\mathcal{P}, assuming that either the critical exponent of Γ is strictly bigger than 1 or P\mathcal{P} does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Γ. Our construction also works for P\mathcal{P} invariant under a geometrically infinite group Γ, provided Γ admits a finite Bowen-Margulis-Sullivan measure and the Γ-skinning size of P\mathcal{P} is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.     

$\mathcal{P}$-kernel normal systems for $\mathcal{P}$-inversive semigroups

As a generalization of Preston’s kernel normal systems, P\mathcal{P}-kernel normal systems for P\mathcal{P}-inversive semigroups are introduced, and strongly regular P\mathcal{P}-congruences on P\mathcal{P}-inversive semigroups in terms of their P\mathcal{P}-kernel normal systems are characterized. These results generalize the corresponding results for P\mathcal{P}-regular semigroups and P\mathcal{P}-inversive semigroups.