On a sheaf-theoretic version of the Witt’s decomposition theorem. A Lagrangian perspective

The approach to a counterpart, in Abstract Geometric Algebra, that is, Geometric Algebra via sheaves of modules, of the classical Witt’s decomposition theoremis based on the axiomatization of the classical context, which however leads to the formulation of a specific subcategory of the category of sheaves of modules: the full subcategory of convenient sheaves of modules. Convenient sheaves of modules turn out, by the very essence of the matter at hand, to be of further importance as far as the setting of results leading to the sheaf-theoretic aspect of several forms of the Witt’s theorem is concerned. Further versions of the Witt’s theorem are still to be treated elsewhere.      

The G-Fredholm Property of the \bar{\partial} -Neumann Problem

Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G→M→X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if G acts by holomorphic transformations in M, then the Laplacian on M has the following properties: The kernel of □ restricted to the forms Λ ^p,q with q>0 is a closed, G-invariant subspace in L ²(M,Λ ^p,q ) of finite G-dimension. Secondly, we show that if q>0, then the image of □ contains a closed, G-invariant subspace of finite G-codimension in L ²(M,Λ ^p,q ). These two properties taken together amount to saying that □ is a G-Fredholm operator. It is a corollary of the first property mentioned that the reduced L ²-Dolbeault cohomology spaces of M are finite G-dimensional for q>0. The boundary Laplacian □ _b has similar properties.      

Inverse Conductivity Problem on Riemann Surfaces

An electrical potential U on a bordered Riemann surface X with conductivity function σ>0 satisfies equation d(σ d ^c U)=0. The problem of effective reconstruction of σ from electrical currents measurements (Dirichlet-to-Neumann mapping) on the boundary: U| _bX ↦ σ d ^c U| _bX is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R. Novikov (Funkc. Anal. Ego Priloz. 22:11–22, 2008) for simply connected X. We apply for this new kernels for on the affine algebraic Riemann surfaces constructed in Henkin (, 2008).      

Lawvere Completeness in Topology

It is known since 1973 that Lawvere’s notion of Cauchy-complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper, we introduce the corresponding notion of Lawvere completeness for (\mathbbT,V)(\mathbb{T},\mathsf{V})-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones it means weak sobriety while for the latter it means Cauchy completeness. Further, we show that V\mathsf{V} has a canonical (\mathbbT,V)(\mathbb{T},\mathsf{V})-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; this structure permits us to define a Yoneda embedding in the realm of (\mathbbT,V)(\mathbb{T},\mathsf{V})-categories.     

The $\bar{\partial}$ -Approach to Monochromatic Inverse Scattering in Three Dimensions

We discuss a method for monochromatic inverse scattering in three dimensions of [Novikov in Int. Math. Res. Papers 2005(6):287–349, [2005]] and implemented numerically in [Alekseenko et al. in Acoust. J. 54(3), [2008]]. This method is obtained as a development of the -approach to inverse scattering at fixed energy in dimension d≥3 of [Beals and Coifman in Proc. Symp. Pure Math. 43:45–70, [1985]] and [Henkin and Novikov in Usp. Mat. Nauk 42(3):93–152, [1987]] and involves, in particular, some results of [Faddeev in Itogi Nauki Tech. Sovr. Prob. Math. 3:93–180, [1965], [1974]] and some ideas of the soliton theory (in particular, some ideas going back to [Manakov in Usp. Mat. Nauk 31(5):245–246, [1976]] and [Dubrovin et al. in Dokl. Akad. Nauk SSSR 229:15–18, [1976]]). Also, our studies go back, in particular, to [Regge in Nuovo Cimento 14:951–976, [1959]]. This article is an extended version of the talk given at International Conference in Mathematics in honor of G. Henkin at the occasion of his 65th birthday.      

Independence and convergence in non-additive settings

Some properties of convergence for archimedean t-conorms and t-norms are investigated and a definition of independence for events, evaluated by a decomposable measure, is introduced. This definition generalizes the concept of independence provided by Kruse and Qiang for λ-additive fuzzy measures. Finally, we derive the two Borel–Cantelli lemmas in the context of the general framework considered.     

The \bar \partial -Neumann operator and commutators of the Bergman projection and multiplication operators

We prove that compactness of the canonical solution operator to restricted to (0, 1)-forms with holomorphic coefficients is equivalent to compactness of the commutator defined on the whole L _(0,1)²(Ω), where is the multiplication by and is the orthogonal projection of L _(0,1)²(Ω) to the subspace of (0, 1) forms with holomorphic coefficients. Further we derive a formula for the -Neumann operator restricted to (0, 1) forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by z and . Partially supported by the FWF grant P19147-N13.     

Endpoint Properties of Localized Riesz Transforms and Fractional Integrals Associated to Schrödinger Operators

Let \({\mathcal L}\equiv-\Delta+V\) be the Schrödinger operator in \({{\mathbb R}^n}\), where V is a nonnegative function satisfying the reverse Hölder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined by V. In this paper, the authors characterize the localized Hardy spaces \(H^1_\rho({{\mathbb R}^n})\) in terms of localized Riesz transforms and establish the boundedness on the BMO-type space \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) of these operators as well as the boundedness from \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) to \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) via localized Riesz transforms. When ρ is the known auxiliary function determined by V, \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) is just the known space \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\), and \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) in this case is correspondingly denoted by \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\). As applications, when n?≥?3, the authors further obtain the boundedness on \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) of Riesz transforms \(\nabla{\mathcal L}^{-1/2}\) and their adjoint operators, as well as the boundedness from \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) to \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\) of their maximal operators. Also, some endpoint estimates of fractional integrals associated to \({\mathcal L}\) are presented.     

Boundedly UC Spaces and Topologies on Function Spaces

A new interesting topology on graphs of partial maps is introduced. This topology can be considered as a natural extension to a non locally compact setting of former topologies defined by P. Brandi, R. Ceppitelli and K. Back, having applications in mathematical economics, differential equations and in the convergence of dynamic programming models. New characterizations of boundedly Atsuji spaces are given by the coincidence of and the topology τ _ucb of uniform convergence on bounded sets on C(X,Y) and by topological properties of .      

The geometric invariants $${\Omega^{n}}$$ of a product of groups

We compute the geometric invariants of a product G × H of groups in terms of and . This gives a sufficient condition in terms of and for a normal subgroup of G × H with abelian quotient to be of type F _n . We give an example involving the direct product of the Baumslag–Solitar group BS_1,2 with itself.      

Analytic properties in the spectrum of certain Banach algebras

We show a sufficient condition for a domain in to be a H ^∞-domain of holomorphy. Furthermore if a domain has the Gleason property at a point and the projection of the n − 1th order generalized Shilov boundary does not coincide with Ω then is schlicht. We also give two examples of pseudoconvex domains in which the spectrum is non-schlicht and satisfy several other interesting properties.      

On unitary representability of topological groups

We prove that the additive group (E*, τ _k (E)) of an -Banach space E, with the topology τ _k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ _k (E)) is uniformly homeomorphic to a subset of for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces. Research partially supported by Spanish Ministry of Science, grant MTM2008-04599/MTM. The foundations of this paper were laid during the author’s stay at the University of Ottawa supported by a Generalitat Valenciana grant CTESPP/2004/086.     

An algorithm for matching point sets using the l_1 norm

The problem is considered of matching two sets of points in , by translation and rotation. There are many applications, for example in geodesy, computer vision and in the assessment of manufactured parts. When the matching criterion is least squares, there is a well known solution process based on the singular value decomposition of an matrix. Here we consider the use of the norm, which may be more appropriate than least squares in the context of wild points in the data. An algorithm is developed, and is illustrated by some examples for the case .     

Algorithms for finding clique-transversals of graphs

A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. It is NP-hard to determine the minimum cardinality τ _c of a clique-transversal of G. In this work, first we propose an algorithm for determining this parameter for a general graph, which runs in polynomial time, for fixed τ _c . This algorithm is employed for finding the minimum cardinality clique-transversal of [`(3K₂)]\overline{3K_{2}} -free circular-arc graphs in O(n ⁴) time. Further we describe an algorithm for determining τ _c of a Helly circular-arc graph in O(n) time. This represents an improvement over an existing algorithm by Guruswami and Pandu Rangan which requires O(n ²) time. Finally, the last proposed algorithm is modified, so as to solve the weighted version of the corresponding problem, in O(n ²) time.     

Quasi-regular Dirichlet Forms and L^p-resolvents on Measurable Spaces

We prove that for any semi-Dirichlet form on a measurable Lusin space E there exists a Lusin topology with the given -algebra as the Borel -algebra so that becomes quasi-regular. However one has to enlarge E by a zero set. More generally a corresponding result for arbitrary -resolvents is proven.     

Boundary Limits for Bounded Quasiregular Mappings

In this paper we establish results on the existence of nontangential limits for weighted -harmonic functions in the weighted Sobolev space , for some q>1 and w in the Muckenhoupt A _q class, where is the unit ball in . These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted -harmonic functions are weak solutions of the partial differential equation

Generalized Continuous Posets and a New Cartesian Closed Category

With every subset selection for posets, there is associated a certain ideal completion . As shown by Erné, such completions help to extend classical results on domains and similar structures in the absence of the required joins. Some results about –predistributive or –precontinuous posets and –continuous functions are summarized and supplemented. In particular, several central results on function spaces in domain theory are extended to the setting of productive closed subset selections. The category FSBP, in which objects are finitely separated and upper bounded posets and arrows are continuous functions between them, is shown to be cartesian closed. This research is supported by the National Natural Science Foundation of China, 10471035.     

Analysis and Geometry on Worm Domains

In this primarily expository article, we study the analysis of the Diederich-Fornæss worm domain in complex Euclidean space. We review its importance as a domain with nontrivial Nebenhülle, and as a counterexample to a number of basic questions in complex geometric analysis. Then we discuss its more recent significance in the theory of partial differential equations: the worm is the first smoothly bounded, pseudoconvex domain to exhibit global non-regularity for the \(\overline{\partial}\)-Neumann problem. We take this opportunity to prove a few new facts. Next, we turn to specific properties of the Bergman kernel for the worm domain. An asymptotic expansion for this kernel is considered, and applications to function theory and analysis on the worm are provided.     

CR-Admissible $$\mathbb{{Z}}_{2}$$-gradations and CR-symmetries

Complementing the results of (Lotta and Nacinovich, Adv. Math. 191(1): 114–146, 2005), we show that the minimal orbit M of a real form G of a semisimple complex Lie group in a flag manifold is CR-symmetric (see (Kaup and Zaitsev Adv. Math. 149(2):145–181, 2000)) if and only if the corresponding CR algebra admits a gradation compatible with the CR structure.