5,378 research outputs found
On homotopy categories of Gorenstein modules: compact generation and dimensions
Let be a virtually Gorenstein algebra of finite CM-type. We establish a
duality between the subcategory of compact objects in the homotopy category of
Gorenstein projective left -modules and the bounded Gorenstein derived
category of finitely generated right -modules. Let be a two-sided
noetherian ring such that the subcategory of Gorenstein flat modules
R\mbox{-}\mathcal{GF} is closed under direct products. We show that the
inclusion K(R\mbox{-}\mathcal{GF})\to K(R\mbox{-}{\rm Mod}) of homotopy
categories admits a right adjoint. We introduce the notion of Gorenstein
representation dimension for an algebra of finite CM-type, and establish
relations among the dimension of its relative Auslander algebra, Gorenstein
representation dimension, the dimension of the bounded Gorenstein derived
category, and the dimension of the bounded homotopy category of its Gorenstein
projective modules.Comment: arXiv admin note: text overlap with arXiv:0810.1401 by other author
Gorensteinness, homological invariants and Gorenstein derived categories
Relations between Gorenstein derived categories, Gorenstein defect categories
and Gorenstein stable categories are established. Using these, the
Gorensteinness of an algebra and invariants with respect to recollements of
the bounded Gorenstein derived category D^{b}_{gp}(A\mbox{-}{\rm mod}) of
are investigated. Specifically, the Gorensteinness of is characterized in
three ways: the existence of Auslander-Reiten triangles in
D^{b}_{gp}(A\mbox{-}{\rm mod}); recollements of D^{b}_{gp}(A\mbox{-}{\rm
mod}); and also Gorenstein derived equivalences. It is shown that the
finiteness of Cohen-Macaulay type and of finitistic dimension are invariant
with respect to the recollements of D^{b}_{gp}(A\mbox{-}{\rm mod}).Comment: arXiv admin note: text overlap with arXiv:1104.4006 by other author
ON THE 83-TH PROBLEM OF F. SMARANDACHE
Studying the properties of a Smarandache sequence, and giving an interesting asymptotic formula
Spherical Tiling by 12 Congruent Pentagons
The tilings of the 2-dimensional sphere by congruent triangles have been
extensively studied, and the edge-to-edge tilings have been completely
classified. However, not much is known about the tilings by other congruent
polygons. In this paper, we classify the simplest case, which is the
edge-to-edge tilings of the 2-dimensional sphere by 12 congruent pentagons. We
find one major class allowing two independent continuous parameters and four
classes of isolated examples. The classification is done by first separately
classifying the combinatorial, edge length, and angle aspects, and then
combining the respective classifications together.Comment: 53 pages, 40 figures, spherical geometr
Data-driven Efficient Solvers and Predictions of Conformational Transitions for Langevin Dynamics on Manifold in High Dimensions
We work on dynamic problems with collected data that
distributed on a manifold . Through the
diffusion map, we first learn the reaction coordinates where is a manifold isometrically embedded into an
Euclidean space for . The reaction coordinates
enable us to obtain an efficient approximation for the dynamics described by a
Fokker-Planck equation on the manifold . By using the reaction
coordinates, we propose an implementable, unconditionally stable, data-driven
upwind scheme which automatically incorporates the manifold structure of
. Furthermore, we provide a weighted convergence analysis of
the upwind scheme to the Fokker-Planck equation. The proposed upwind scheme
leads to a Markov chain with transition probability between the nearest
neighbor points. We can benefit from such property to directly conduct
manifold-related computations such as finding the optimal coarse-grained
network and the minimal energy path that represents chemical reactions or
conformational changes. To establish the Fokker-Planck equation, we need to
acquire information about the equilibrium potential of the physical system on
. Hence, we apply a Gaussian Process regression algorithm to
generate equilibrium potential for a new physical system with new parameters.
Combining with the proposed upwind scheme, we can calculate the trajectory of
the Fokker-Planck equation on based on the generated equilibrium
potential. Finally, we develop an algorithm to pullback the trajectory to the
original high dimensional space as a generative data for the new physical
system.Comment: 59 pages, 16 figure
Cosmology-Independent Distance Moduli of 42 Gamma-Ray Bursts between Redshift of 1.44 and 6.60
This report is an update and extension of our paper accepted for publication
in ApJ (arXiv:0802.4262). Since objects at the same redshift should have the
same luminosity distance and the distance moduli of type Ia supernovae (SNe Ia)
obtained directly from observations are completely cosmology independent, we
obtain the distance modulus of a gamma-ray burst (GRB) at a given redshift by
interpolating or iterating from the Hubble diagram of SNe Ia. Then we calibrate
five GRB relations without assuming a particular cosmological model, from
different regression methods, and construct the GRB Hubble diagram to constrain
cosmological parameters. Based upon these relations we list the
cosmology-independent distance moduli of 42 GRBs between redshift of 1.44 and
6.60, with the 1- uncertainties of 1-3%.Comment: 6 pages, 2 figures, 3 tables. To appear in the proceedings of "2008
Nanjing GRB conference", Nanjing, 23-27 June 200
- β¦