Universal Extension Research Articles

Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras

Let X be any rational surface. We construct a tilting bundle T on X. Moreover, we can choose T in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on X is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra A. The construction starts with a full exceptional sequence of line bundles on X and uses universal extensions. If X is any smooth projective variety with a full exceptional sequence of coherent sheaves (or vector bundles, or even complexes of coherent sheaves) with all groups Ext q for q≥2 vanishing, then X also admits a tilting sheaf (tilting bundle, or tilting complex, respectively) obtained as a universal extension of this exceptional sequence.

FRI Sampling With Arbitrary Kernels

This paper addresses the problem of sampling non-bandlimited signals within the Finite Rate of Innovation (FRI) setting. We had previously shown that, by using sampling kernels whose integer span contains specific exponentials (generalized Strang-Fix conditions), it is possible to devise non-iterative, fast reconstruction algorithms from very low-rate samples. Yet, the accuracy and sensitivity to noise of these algorithms is highly dependent on these exponential reproducing kernels - actually, on the exponentials that they reproduce. Hence, our first contribution here is to provide clear guidelines on how to choose the sampling kernels optimally, in such a way that the reconstruction quality is maximized in the presence of noise. The optimality of these kernels is validated by comparing with Cramér-Rao's lower bounds (CRB). Our second contribution is to relax the exact exponential reproduction requirement. Instead, we demonstrate that arbitrary sampling kernels can reproduce the “best” exponentials within quite a high accuracy in general, and that applying the exact FRI algorithms in this approximate context results in near-optimal reconstruction accuracy for practical noise levels. Essentially, we propose a universal extension of the FRI approach to arbitrary sampling kernels. Numerical results checked against the CRB validate the various contributions of the paper and in particular outline the ability of arbitrary sampling kernels to be used in FRI algorithms.

Cohomological finite generation for restricted Lie superalgebras and finite supergroup schemes

We prove that the cohomology ring of a finite-dimensional restricted Lie superalgebra over a field of characteristic p > 2 p > 2 is a finitely-generated algebra. Our proof makes essential use of the explicit projective resolution of the trivial module constructed by J. Peter May for any graded restricted Lie algebra. We then prove that the cohomological finite generation problem for finite supergroup schemes over fields of odd characteristic reduces to the existence of certain conjectured universal extension classes for the general linear supergroup G L ( m | n ) GL(m|n) that are similar to the universal extension classes for G L n GL_n exhibited by Friedlander and Suslin.

A general P300 brain–computer interface presentation paradigm based on performance guided constraints

An electroencephalographic-based brain–computer interface (BCI) can provide a non-muscular method of communication. A general model for P300-based BCI stimulus presentations is introduced – the “m choose n” or C(m (number of flashes per sequence), n (number of flashes per item)) paradigm, which is a universal extension of the previously reported checkerboard paradigm (CBP). C(m,n) captures all possible (unconstrained) ways to flash target items, and then applies constraints to enhance ERP's produced by attended matrix items. We explore a C(36,5) instance of C(m,n) called the “five flash paradigm” (FFP) and compare its performance to the CBP. Eight subjects were tested in each paradigm, counter-balanced. Twelve minutes of calibration data were used as input to a stepwise linear discriminant analysis to derive classification coefficients used for online classification. Accuracy was consistently high for FFP (88%) and CBP (90%); information transfer rate was significantly higher for the FFP (63bpm) than the CBP (48bpm). The C(m,n) is a novel and effective general strategy for organizing stimulus groups. Appropriate choices for “m,” “n,” and specific constraints can improve presentation paradigms by adjusting the parameters in a subject specific manner. This may be especially important for people with neuromuscular disabilities.

Universal extension for Sobolev spaces of differential forms and applications

This article is devoted to the construction of a family of universal extension operators for the Sobolev spaces Hk(d,Ω,Λl) of differential forms of degree l (0⩽l⩽d) in a Lipschitz domain Ω⊂Rd (d∈N, d⩾2) for any k∈N0. It generalizes the construction of the first universal extension operator for standard Sobolev spaces Hk(Ω), k∈N0, on Lipschitz domains, introduced by Stein [E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, NJ, 1970, Theorem 5, p. 181]. We adapt Steinʼs idea in the form of integral averaging over the pullback of a parametrized reflection mapping. The new theory covers extension operators for Hk(curl;Ω) and Hk(div;Ω) in R3 as special cases for l=1,2, respectively. Of considerable mathematical interest in its own right, the new theoretical results have many important applications: we elaborate existence proofs for generalized regular decompositions.

A universal crack extension criterion based on the equivalent stress gradient: I. theory and numerical verification

In this contribution, the microscopic fracture mechanism and extension criterion for mixed type crack in ductile material under plane mixed mode loading are investigated in details. A universal extension criterion for the mixed type crack, i.e. the crack propagates along the direction of the maximum gradient of equivalent stress, is suggested. This new criterion is used to predict the propagation direction of mixed type crack, showing a good agreement with other theories for different types of mode mixity. Moreover, the numerical verification is also carried out for the case of an edge crack with different mixed mode loadings. Finally, a potential application to three-dimensional fracture in the ductile material induced by holes is also discussed.

The Inverse Commutant Lifting Problem. I: Coordinate-Free Formalism

It is known that the set of all solutions of a commutant lifting and other interpolation problems admits a Redheffer linear-fractional parametrization. The method of unitary coupling identifies solutions of the lifting problem with minimal unitary extensions of a partially defined isometry constructed explicitly from the problem data. A special role is played by a particular unitary extension, called the central or universal unitary extension. The coefficient matrix for the Redheffer linear-fractional map has a simple expression in terms of the universal unitary extension. The universal unitary extension can be seen as a unitary coupling of four unitary operators (two bilateral shift operators together with two unitary operators coming from the problem data) which has special geometric structure. We use this special geometric structure to obtain an inverse theorem (Theorem 8.4) which characterizes the coefficient matrices for a Redheffer linear-fractional map arising in this way from a lifting problem. The main tool is the formalism of unitary scattering systems developed in Boiko et al. (Operator theory, system theory and related topics (Beer-Sheva/Rehovot 1997), pp. 89–138, 2001) and Kheifets (Interpolation theory, systems theory and related topics, pp. 287–317, 2002)

Universal extension crystals of 1-motives and applications

We use the crystalline nature of the universal extension of a 1-motive M to define a canonical Gauss–Manin connection on the de Rham realization of M . As an application we provide a construction of the so-called Manin map from a motivic point of view.

Toward an All-Around Semilocal Potential for Electronic Exchange

We test local and semilocal approximations of the exchange potential for a variety of systems including atoms, molecules, and atomic chains. In particular, we focus on a recent universal extension of the Becke−Johnson exchange potential [Rasanen, E.; Pittalis, S.; Proetto, C. R. J. Chem. Phys. 2010, 132, 044112]. It is shown that when this potential is used together with the Becke−Roussel approximation to the Slater potential [Becke, A. D.; Roussel, M. R. Phys. Rev. A 1989, 39, 3761−3767], a good overall agreement is obtained with experimental and numerically exact results for several systems, and with a moderate computational cost. Thus, this approximation is a very promising candidate in the quest for a simple and all-around semilocal potential.

Real interpolation of spaces of differential forms

In this paper, we study interpolation of Hilbert spaces of differential forms using the real method of interpolation. We show that the scale of fractional order Sobolev spaces of differential l-forms in H s with exterior derivative in H s can be obtained by real interpolation. Our proof heavily relies on the recent discovery of smoothed Poincare lifting for differential forms [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincare integral operators for de Rham complexes on Lipschitz domains, Math. Z. 265(2): 297–320, 2010]. They enable the construction of universal extension operators for Sobolev spaces of differential forms, which, in turns, pave the way for a Fourier transform based proof of equivalences of K-functionals.

Spectral theory for the fractal Laplacian in the context of h-sets

An h-set is a nonempty compact subset of the Euclidean n-space which supports a finite Radon measure for which the measure of balls centered on the subset is essentially given by the image of their radius by a suitable function h. In most cases of interest such a subset has Lebesgue measure zero and has a fractal structure. Let Ω be a bounded C∞ domain in with Γ ⊂ Ω. Let where (−Δ)−1 is the inverse of the Dirichlet Laplacian in Ω and trΓ is, say, trace type operator. The operator B, acting in convenient function spaces in Ω, is studied. Estimations for the eigenvalues of B are presented, and generally shown to be dependent on h, and the smoothness of the associated eigenfunctions is discussed. Some results on Besov spaces of generalised smoothness on and on domains which were obtained in the course of this work are also presented, namely pointwise multipliers, the existence of a universal extension operator, interpolation with function parameter and mapping properties of the Dirichlet Laplacian. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Canonical extensions of Néron models of Jacobians

Let A be the Neron model of an abelian variety AK over the fraction field K of a discrete valuation ring R. By work of Mazur and Messing, there is a functorial way to prolong the universal extension of AK by a vector group to a smooth and separated group scheme over R, called the canonical extension of A. Here we study the canonical extension when AK=JK is the Jacobian of a smooth, proper and geometrically connected curve XK over K. Assuming that XK admits a proper flat regular model X over R that has generically smooth closed fiber, our main result identifies the identity component of the canonical extension with a certain functor PicX∕R♮,0 classifying line bundles on X that have partial degree zero on all components of geometric fibers and are equipped with a regular connection. This result is a natural extension of a theorem of Raynaud, which identifies the identity component of the Neron model J of JK with the functor PicX∕R0. As an application of our result, we prove a comparison isomorphism between two canonical integral structures on the de Rham cohomology of XK.

The Absent-Minded Founder: Norway and the Establishment of the United Nations

This article examines Norwegian policy vis-à-vis the United Nations (UN) through the end of 1945. From here it will become clear that framing foreign policy orientations of the 1940s along conventional lines exaggerates the commitment of Norwegian politicians to two grand ideas. The novel idea of Atlantic alignment, developed by Norwegian circles in London exile, was more ambiguous than generally acknowledged and left room for universal extension. By contrast, the alleged turn in the mid-1940s toward support of the UN was in the form of lip-service as opposed to action that would have engaged actors from Norway. The government outsourced policy-making on the issue to a small circle of experts and made no attempt to exert leadership in regard to UN matters. Norway's indifference toward the UN in the 1940s stands in marked contrast to the country's later reputation as a faithful supporter of the world organization.

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” Ext ˆ X i ( F , G ) we introduce are extensions by groups of analytic types of the usual extension groups Ext X i ( F , G ) attached to O X -modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group Ext ˆ X 1 ( F , G ) – the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X – and we especially consider the case when X is an “arithmetic curve”, namely the spectrum Spec O K of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers. Namely, for any two hermitian vector bundles F ¯ and G ¯ over X : = Spec O K , we attach a logarithmic size s F ¯ , G ¯ ( α ) to any element α of Ext ˆ X 1 ( F , G ) , and we give an upper bound on s F ¯ , G ¯ ( α ) in terms of slope invariants of F ¯ and G ¯ . We further illustrate this notion by relating the sizes of restrictions to points in P 1 ( Z ) of the universal extension over P Z 1 to the geometry of PSL 2 ( Z ) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K ′ ): when the base field K is Q , we establish that the size, which cannot increase under base change, is actually invariant when the field K ′ is an abelian extension of K, or when F ¯ ∨ ⊗ G ¯ is a direct sum of root lattices and of lattices of Voronoi's first kind. The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.

Policy Issues in Enhancing the Output Of Agribusiness Small And Medium Scale Piggery Enterprises (AGRI-SMEs) in Abia State

The need for this study was rooted on the worrisome state of Agri-SMEs which has not responded positively despite government programmes, incentives and supportive services of research and extension. Bende agricultural zone of Abia state which has 12 local government areas in it was purposively chosen because of the presence of the State ADP, Ministry of Agriculture, National Root Crops Research Institute and University of Agriculture in it. Through random sampling, three Local Government Areas (L.G.As) comprising of Ikwuano, Umuahia North and South Local Government Areas were selected. Also, following a random sampling process, forty (40) agri-SMEs operators involved in piggery management were studied. They were selected from sampling frame of piggery operators provided by Local Government Area officials. The results presented in rank order of influence indicated that level of education and distance of operators to research and extension service centres were first and second factors hindering access to research and extension services in the area. Also, major factors affecting the output of operators included the education level, total cost of production and access to research and extension services. Therefore, policy considerations by the government should be directed towards achieving the universal basic education, bringing research and extension services sub-centres closer to the people and reducing total cost of production of agri-SME operators will immensely contribute to enhancement of output realization of piggery operators. Key-words: Policy, Enhancing, Output, Accessibility, Piggery, Enterprises

Tag-extension-based method for sensitive and specific genotyping of single nucleotide polymorphism on microarray

Background The obtainment of a large amount of single nucleotide polymorphism (SNP) information has emphasized a need for a sensitive, accurate and high-throughput strategy for SNP genotyping. We developed a reliable and potential microarray-based method to meet this demand. Methods A tag extension strategy is described to identify SNPs. The strategy is based on a fluorescent nucleotide extension from an extending primer that is hybridized to a bi-functional linker, which acts as an allele-specific primer that hybridizes with a PCR-amplified target DNA immobilized on a 3-dimensional (3-D) polyacrylamide gel microarray, as well as an extending template with a specific tag itself hybridized with a universal extension primer. Multiple fluorescence-dNTPs are simultaneously incorporated into the tagged linker. Results The method not only significantly enhanced the sensitivity, but also efficiently improved the specificity of SNP genotyping. 89 samples for 8025575C/G polymorphisms in γ-aminobutyric acid A receptor, β 3 (GABAB3) gene were accurately discriminated using this method. Sanger sequencing was performed to validate these results. Conclusion Our experiments successfully demonstrated that a tag-extension-based method on microarray could be used as a high-throughput and useful tool to obtain SNP information.

Milnor 𝐾-theory of local rings with finite residue fields

We propose a definition of improved Milnor K K -groups of local rings with finite residue fields, such that the improved Milnor K K -sheaf in the Zariski topology is a universal extension of the naive Milnor K K -sheaf with a certain transfer map for étale extensions of local rings. The main theorem states that the improved Milnor K K -ring is generated by elements of degree one.

Sharp de Rham realization

We introduce the sharp (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the sharp de Rham realization T ♯ by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. Thus we show how to provide one-dimensional sharp de Rham cohomology H ♯ - dR 1 of algebraic varieties.

EXTENDED BOTT–VIRASORO ALGEBRA, SEMIDIRECT PRODUCT, ⋆-LIE ALGEBRA OF DIFFEOMORPHISM AND NONCOMMUTATIVE INTEGRABLE SYSTEMS

We study noncommutative theory of a coadjoint representation of a universal extension of Vect (S1) ⋉ C∞(S1) algebra using the action of ⋆-deformed matrix Hill's operators Δ⋆ on the space of ⋆-deformed tensor densities. The centrally extended semidirect product algebra [Formula: see text] is a Lie algebra of extended semidirect product of the Bott–Virasoro group [Formula: see text]. The study of deformed diffeomorphisms, deformed semidirect product algebra and deformed Lie derivative action of Δ⋆ on ⋆ deformed tensor-densities on S1 allow us to construct noncommutative two component Korteweg–de Vries (KdV) equations, in particular, we derive the noncommutative Ito equation. This leads to a geometric formulation of ⋆-deformed quantization of the centrally extended semidirect product algebra [Formula: see text] and two component noncommutative KdV equations.

Evaluation of the 124-plex SNP typing microarray for forensic testing

Human identification systems such as criminal databases, forensic DNA testing and genetic genealogy require reliable and cost-effective genotyping of autosomal, mitochondrial and Y chromosome markers from different biological materials, including venous blood and saliva. Although many such assays are available, few systems are capable of simultaneously detecting all three targets in a single reaction. Employing the APEX-2 principle, we have characterized a novel 124-plex assay, using specific primer extension, universal primer amplification and single base extension on an oligonucleotide array. The assay has been designed for simultaneous genotyping of SNPs from the single copy loci (46 autosomal and 29 Y chromosomal markers) side by side with SNPs from the mitochondrial genome (49 markers) that appears in up to thousands of copies per cell in certain tissue types. All the autosomal SNPs (from the SNPforID Consortium) included in the multiplex assay are unlinked and are distributed widely across autosomes, enabling genetic fingerprints to be distinguished. Mitochondrial DNA and Y chromosome polymorphisms that define haplogroups common in European populations are included to allow for maternity and paternity testing and for the analysis of genetic genealogies. After assay optimization we estimated the accuracy (99.83%) and call rate (99.66%) of the protocol on 17 mother–father–child/children families and five internal control DNAs. In addition, 79 unrelated Estonian and Swedish DNA samples were genotyped and the accuracy of mtDNA and Y chromosome haplogroup inference by the multiplex method was assessed using conventional genotyping methods and direct sequencing.