Genome-wide scan of granular corneal dystrophy, type II: confirmation of chromosome 5q31 and identification of new co-segregated loci on chromosome 3q26.3

Granular corneal dystrophy, type II (CGD2; Avellino corneal dystrophy) is the most common corneal dystrophy among Koreans, but its pathophysiology is still poorly understood. Many reports showed that even though the causative mutation is the same TGFBI R124H mutation, there are severe and mild phenotypes of the corneal dystrophy. We also observed the phenotype differences in our samples. For this reason, we focused our effort on the identification of unknown genetic factor related to phenotype variation. A total 551 individuals from 59 families were genotyped with SNP chip and used in genome-wide linkage analysis. From single-point linkage analyses, we confirmed the known 5q31 region for TGFBI gene, and selected novel nine candidate loci for CGD2. In simulation analysis, the only 3q26.3 region including neuroligin 1 gene (NLGN1) was supported by empirical statistic significance. To investigate the effect of genetic heterogeneity in linkage analysis, we classified CGD2 families into two subgroups. Although we could not find a significant evidence for correlation between the 3q26.3 region and CGD2 phenotypes, this first genome-wide analysis with CGD2 families in Korea has a very important value for offering insights in genetics of CGD2. In addition, the co-segregating loci with CGD2 including 3q26.3 would be a good target for further study to understand the pathophysiology of CGD2.     

Derived Brauer–Thrall Type Theorem I for Artin Algebras

We define the global cohomological range for artin algebras, and define the derived bounded algebras to be the algebras with finite global cohomological range, then we prove the first Brauer–Thrall type theorem for bounded derived categories of artin algebras, i.e., derived bounded algebras are precisely the derived finite algebras. Moreover, the main theorem establishes that the derived bounded artin algebras are just piecewise hereditary algebras of Dynkin type, and can be also characterized as those artin algebras with derived dimension zero, which can be regarded as a generalization of the results of Han–Zhang [11 Han, Y., Zhang, C. Brauer-Thrall type theorems for derived category, arXiv:1310.2777. [Google Scholar], Theorem 1] and Chen–Ye–Zhang [4 Chen, X. W., Ye, Y., Zhang, P. (2008). Algebras of derived dimension zero. Comm. Algebra 36:1–10.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem] in the context of finite-dimensional algebras over algebraically closed fields, respectively.     

D3-Modules

A right R-module M is called a D3-module, if M ₁ and M ₂ are direct summands of M with M = M ₁ + M ₂, then M ₁ ∩ M ₂ is a direct summand of M. Following the work of Bass on projective covers, we introduce the notion of D3-covers and provide new characterizations of several well-known classes of rings in terms of D3-modules and D3-covers.     

Hereditary effects of exponentially damped oscillators with past histories

This paper presents hereditary effects of exponentially damped oscillators with past histories. Unlike the classical viscously damped oscillators, the nonviscously damped ones involve damping forces which depend on time-histories of vibrating motions via convolution integrals. As a result, equations of motion of such systems are a set of coupled second-order Volterra integro-differential equations. In this work, initial value problems for the integro-differential equations are revisited. The initial conditions should contain time-histories of vibrating motions. Then, initialization response of exponentially damped oscillators is obtained. It is used to characterize the hereditary effects on the dynamic response. At last, stability of initialization response is proved from the theoretical viewpoint and verified by numerical simulations. This reveals that the hereditary effects gradually recede with increasing of time.     

Jordan operator algebras: basic theory

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras.     

Non-commutative positive kernels and their matrix evaluations

We show that a formal power series in non-commuting indeterminates is a positive non-commutative kernel if and only if the kernel on -tuples of matrices of any size obtained from this series by matrix substitution is positive. We present two versions of this result related to different classes of matrix substitutions. In the general case we consider substitutions of jointly nilpotent -tuples of matrices, and thus the question of convergence does not arise. In the ``convergent' case we consider substitutions of -tuples of matrices from a neighborhood of zero where the series converges. Moreover, in the first case the result can be improved: the positivity of a non-commutative kernel is guaranteed by the positivity of its values on the diagonal, i.e., on pairs of coinciding jointly nilpotent -tuples of matrices. In particular this yields an analogue of a recent result of Helton on non-commutative sums-of-squares representations for the class of hereditary non-commutative polynomials. We show by an example that the improved formulation does not apply in the ``convergent' case.

     

Topological vector spaces problems in homology and homotopy theories

Three data are interesting here: domains of integration, integrands and integration itself. There is a lack of symmetry between polyhedral chains as domains of integration and differential forms as integrands. The non-symmetric situation disappears after considering the topological spaces of the de Rham differential forms and forms with compact supports and their strong duals, i.e., currents with compact supports and currents, respectively. This idea goes back to Schwartz distributions and Schwartz distributions with compact supports, in other terminology, generalized functions and generalized functions with compact supports.Some problems are raised, e.g., whether every quasi-complete barreled nuclear space E, whose strongly dual E^′ is nuclear, is strongly hereditary reflexive. This concerns the above mentioned de Rham spaces. Problems on R- and Q-homotopy, proper R- and Q-homotopy and proper R- and Q-homotopy at infinity are also considered as well as the coalgebra structure on currents and currents with compact supports.The classical theorem concerning derivation of additive functions with respect to volumes in points is generalized to a theorem on derivation of continuous m-forms with compact supports ωm of an oriented n-dimensional C¹-manifold Mn with respect to its m-dimensional oriented submanifolds Vm in compact regular oriented submanifolds Lk of Mn, 0?k<m?n.     

On classes of maps which preserve finitisticness

We shall prove the following: Let be a refinable map between paracompact spaces. Then is finitistic if and only if is finitistic. Let be a hereditary shape equivalence between metric spaces. Then if is finitistic, is finitistic.

     

Local dual spaces of Banach spaces of vector-valued functions

We show that is a local dual of , and is a local dual of , where is a Banach space. A local dual space of a Banach space is a subspace of so that we have a local representation of in satisfying the properties of the representation of in provided by the principle of local reflexivity.

     

Characteristic Modules of Dual Extensions and Groebner Bases

Let C be a finite dimensional directed algebra over an algebraically closed field k and A=A(C) the dual extension of C. The characteristic modules of A are constructed explicitly for a class of directed algebras, which generalizes the results of Xi. Furthermore, it is shown that the characteristic modules of dual extensions of a certain class of directed algebras admit the left Groebner basis theory in the sense of E. L. Green.