Graph Representation of Projective Resolutions

We generalize the concept -- dimension tree and the related results for monomial algebras to a more general case -- relations algebras A by bringing GrSbner basis into play. More precisely, we will describe the minimal projective resolution of a left A-module M as a rooted ＇weighted＇ diagraph to be called the minimal resolution graph for M. Algorithms for computing such diagraphs and applications as well will be presented.

Anick Resolution and the Minimal Projective Resolution of U_(q)+(C_(3))

In this paper,from the Anick’s resolution and Grobner-Shirshov basis for quantized enveloping algebra of type C_(3),we compute the minimal projective resolution of the trivial module U_(q)+(C_(3))and as an application,we obtain that the global dimension of U_(q)+(C_(3))is 9.

Grbner-Shirshov Basis and the Minimal Projective Resolution of U_q^+(G_2)

Abstract In this paper, by using the Anick＇s resolution and Grobner-Shirshov basis for quantized enveloping algebra of type G2, we compute the minimal projective resolution of the trivial module of Uq＋ （G2） and as an application we compute the global dimension of Uq＋（G2）.

Cotorsion Dimension of Weak Crossed Products

Let H be a finite-dimensional weak Hopf algebra over a field κ and A an associative algebra, and A#;H a weak crossed product. In this paper, a spectral sequence for Ext is constructed which yields an estimate for cotorsion dimension of A#;H in terms of the corresponding data for H and A.

Hochschild(Co)homology of Galois Coverings of Grassmann Algebras

Let ∧ be the Z2-Galois covering of the Grassmann algebra A over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of ∧ are calculated explicitly. And the cyclic homology of∧ can also be calculated when the underlying field is of characteristic zero. As a result, we prove that there is an isomorphism from i≥1 HH^i（∧） to i≥1 HH^i（∧）.

Derived 2-functors in(2-SGp)

In this paper, we construct the projective resolution of arbitrary symmetric 2-group, define thederived 2-functors in (2-SGp) and give some related properties of the derived 2-functors.

Global dimension and left derived functors of Hom

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom and the right projective resolutions of right R-modules. It is shown that rD(R) ? n (n ? 2) if and only if the gl right Proj-dim M R ? n?2 if and only if Ext n?1(N,M) = 0 for all right R-modules N and M if and only if every (n?2)th Projcosyzygy of a right R-module has a projective envelope with the unique mapping property. It is also proved that rD(R) ? n (n ? 1) if and only if every (n?1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Proj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R) ? 2 are characterized.

Vanishing of Tate Homology - An Application of Stable Homology for Complexes

It has been proved that the vanishing of Tare homology is a sufficient condition for the derived depth formula to hold in [J. Pure Appl. Algebra, 219, 464-481 （2015）]. In this paper, we inves- tigate when Tate homology vanishes by studying the stable homology theory for complexes. Properties such as the balancedness and vanishing of stable homology for complexes are studied. Our results show that the vanishing of this homology can detect finiteness of homological dimensions of complexes and regularness of rings.

LEVEL SET METHOD FOR NUMERICAL SIMULATION OF A CAVITATION BUBBLE COLLAPSING NEAR A RIGID WALL

The level set method, TVD scheme of second order upwind procedure coupled with flux limiter, ENO velocity extension procedure inside the huhble, and MAC projection algorithm were incorporated to simulate the whole collapse e volution of a cavitation bubble near a rigid wall with many complicated phenomena, such as topology distortion and shrinking, jet impact, bubble breaking into a toroidal form, and diminishing volume to zero, etc. The huhhle shape, evolution and distribution of velocity and pressure fields of the fluid during the bubble collapsing were investigated. It is found that the method is numerically stable and has good convergence property, and the results are in good agreements with those in previous work.

High-Throughput Chemotyping of Cannabis and Hemp Extracts Using an Ultraviolet Microplate Reader and Multivariate Classifiers

As the use of Cannabis products as natural medicines burgeons,it is also appearing as a food ingredient.It is important to screen Cannabis samples as ingredients by profiling their chemical compositions,which is referred to as chemotyping.Two sets of botanical extracts were studied.The first set is referred to as Cannabis contained plant materials from 15 samples of the sativa,indica,and hybrids of the two species.The second set contained 20 extracts from the variety of Cannabis sativa with low tetrahydrocannabinol(THC)concentrations,i.e.,below 0.3%,and,henceforth,will be referred to as hemp.An ultraviolet(UV)microplate reader provides a cost-effective and high-throughput method for identifying chemotypes of plant extracts by their spectra.The microplate reader affords rapid measurements of small volumes,e.g.,50μL,which demonstrates a potential to significantly reduce the analysis time and cost for Cannabis and hemp chemotyping or chemi-cal profiling.Replicate samples were measured on different days to demonstrate the robustness of the method.Projected difference resolution(PDR)maps were used to visualize the separations among the classes.Five multivariate classifiers,fuzzy rule-building expert system(FuRES),super partial least squares-discriminant analysis(sPLS-DA),support vector machine(SVM),and two tree-based support vector machines(SVMtreeG and SVMtreeH)were evaluated.The classifiers were validated with ten bootstrapped Latin partitions(BLPs).For the Cannabis extracts,the SVMtreeG yielded the best performance and the classification accuracy was 99.1±0.4%for spectra collected in the nonlinear absorbance range.For the hemp extracts,the SVM classifier performed the best with a 97.4±0.6%classification accuracy.These results demonstrate that the UV microplate reader coupled with multivariate classifiers can be used as a high-throughput and cost-effective approach for chemotyping Cannabis.