On homotopy categories of Gorenstein modules: compact generation and dimensions

Let $A$ 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 $A$-modules and the bounded Gorenstein derived category of finitely generated right $A$-modules. Let $R$ 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 $A$ and invariants with respect to recollements of the bounded Gorenstein derived category D^{b}_{gp}(A\mbox{-}{\rm mod}) of $A$ are investigated. Specifically, the Gorensteinness of $A$ 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 $\{\mathsf{x}_i\}$ that distributed on a manifold $\mathcal{M}\subset\mathbb{R}^p$. Through the diffusion map, we first learn the reaction coordinates $\{\mathsf{y}_i\}\subset \mathcal{N}$ where $\mathcal{N}$ is a manifold isometrically embedded into an Euclidean space $\mathbb{R}^\ell$ for $\ell \ll p$. The reaction coordinates enable us to obtain an efficient approximation for the dynamics described by a Fokker-Planck equation on the manifold $\mathcal{N}$. By using the reaction coordinates, we propose an implementable, unconditionally stable, data-driven upwind scheme which automatically incorporates the manifold structure of $\mathcal{N}$. Furthermore, we provide a weighted $L^2$ 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 $\mathcal{N}$. 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 $\mathcal{N}$ 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-$\sigma$ 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

Essays on product return management and closed loop-supply chain network design

This dissertation focuses on managerial and operational challenges associated with product return management and CLSC network design. The possibility of product return plays an important role in consumer\u27s purchase decisions. It also motivates firms to extend their forward-only supply chain network structures to a Closed-Loop Supply Chain (CLSC) network and handle both forward and reverse flows of products. While the configuration of the CLSC network is a complex problem comprised of the determination of the optimal locations and capacities of factories, warehouses and collection centers, this problem becomes even more complex under the potential regulations on carbon emissions. This dissertation follows a three-paper format. With a focus on product return management, the first paper studies the roles that pricing and return policy play in the product exchange process for refurbished products. We first apply netnography to study consumer attitudes, general opinions and experiences concerning refurbished electronics purchases, and then propose an analytical model that considers customers\u27 purchasing and return behavior as a result of the firm\u27s decisions regarding the pricing and return policy for refurbished products. The numerical results suggest that sellers should deliberately consider the market segmentation conditions, consumer valuation, and cost factors when choosing the appropriate price and return policy for refurbished products. The second and third paper focus on different aspects of CLSC network design. The second paper investigates a problem to design facility configurations that are robust to variations in possible carbon regulations and their cost and constraint implications. We establish a two-stage, multi-period stochastic programming model to include uncertain demand and return quantities and then extended it to incorporate the uncertainties in carbon regulation policy by the robust optimization method. We propose a hybrid model to account for either carbon tax or cap-and-trade regulatory policies and derive tractable robust counterparts under box and ellipsoidal uncertainty sets. Implications for network configuration, product allocation and transportation configuration are derived. We also present computational results that illustrate how the problem formulation under an ellipsoidal uncertainty set allows the decision maker to balance the trade-off between robustness and performance. The third paper formulates and solves an integrated model for product return management and CLSC network design considering uncertain carbon cost. We build a robust optimization model to address the carbon cost uncertainty, and develop a piecewise linear approximation for the nonlinear profit as a function of the refund. The results of the robust model are compared with those of deterministic models where no or only nominal carbon cost is considered. Extensive parametric analyses illustrate the impact of the cost, revenue and consumer profile parameters on the optimal refund, profit and network topology

Using Genetic Algorithm to solve Median Problem and Phylogenetic Inference

Genome rearrangement analysis has attracted a lot of attentions in phylogenetic com- putation and comparative genomics. Solving the median problems based on various distance definitions has been a focus as it provides the building blocks for maximum parsimony analysis of phylogeny and ancestral genomes. The Median Problem (MP) has been proved to be NP-hard and although there are several exact or heuristic al- gorithms available, these methods all are difficulty to compute distant three genomes containing high evolution events. Such as current approaches, MGR[1] and GRAPPA [2], are restricted on small collections of genomes and low-resolution gene order data of a few hundred rearrangement events. In my work, we focus on heuristic algorithms which will combine genomic sorting algorithm with genetic algorithm (GA) to pro- duce new methods and directions for whole-genome median solver, ancestor inference and phylogeny reconstruction. In equal median problem, we propose a DCJ sorting operation based genetic algorithms measurements, called GA-DCJ. Following classic genetic algorithm frame, we develop our algorithms for every procedure and substitute for each traditional genetic algorithm procedure. The final results of our GA-based algorithm are optimal median genome(s) and its median score. In limited time and space, especially in large scale and distant datasets, our algorithm get better results compared with GRAPPA and AsMedian. Extending the ideas of equal genome median solver, we develop another genetic algorithm based solver, GaDCJ-Indel, which can solve unequal genomes median prob- lem (without duplication). In DCJ-Indel model, one of the key steps is still sorting operation[3]. The difference with equal genomes median is there are two sorting di- rections: minimal DCJ operation path or minimal indel operation path. Following different sorting path, in each step scenario, we can get various genome structures to fulfill our population pool. Besides that, we adopt adaptive surcharge-triangle inequality instead of classic triangle inequality in our fitness function in order to fit unequal genome restrictions and get more efficient results. Our experiments results show that GaDCJ-Indel method not only can converge to accurate median score, but also can infer ancestors that are very close to the true ancestors. An important application of genome rearrangement analysis is to infer ancestral genomes, which is valuable for identifying patterns of evolution and for modeling the evolutionary processes. However, computing ancestral genomes is very difficult and we have to rely on heuristic methods that have various limitations. We propose a GA-Tree algorithm which adapts meta-population [4], co-evolution and repopulation pool methods In this paper, we describe and illuminate the first genetic algorithm for ancestor inference step by step, which uses fitness scores designed to consider co- evolution and uses sorting-based methods to initialize and evolve populations. Our extensive experiments show that compared with other existing tools, our method is accurate and can infer ancestors that are much closer to true ancestors