n-strongly Gorenstein Projective and Injective and Flat Modules

In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applications are given.

Notes on Strongly n-Gorenstein Projective, Injective and Flat Modules

In this paper, we mainly investigate some properties of strongly n-Gorenstein projective, injective and flat modules under the extension of rings, which mainly including excellent extensions, morita equivalences, polynomial extensions and localizations.

Strongly Ding projective modules with respect to a semidualizing module

This paper is a study of strongly Ding projective modules with respect to a semidualizing module. The class of strongly Ding flat modules with respect to a semidualizing module is also investigated, and the relationship between strongly Ding projective modules and strongly Ding flat modules with respect to a semidualizing module is characterized.Some well-known results on strongly Ding projective modules, n-strongly Ding projective modules and strongly D_C-projective modules are generalized and unified.

Relative Ding Projective Modules over Formal Triangular Matrix Rings

Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using the left global relative Ding projective dimensions of A and B, we estimate the relative Ding projective dimension of a left T-module.

STUDY OF FAILURE MODES OF MICROELECTRONIC PACKAGING MODULES BY HOLOGRAPHY QUASI PROJECTION MOIRE METHOD

In this paper, a novel interferometric method with a wide range of sensitivities, called holography quasi projection moire, is proposed. It combines the features of the variated double projection moire method and the holographic interferometry method. This technique is used to study the failure modes of microelectronic packaging modules.

Projective Hilbert modules and sequential approximations

We show that when A is a separable C^(*)-algebra,every countably generated Hilbert A-module is projective(with bounded module maps as morphisms).We also study the approximate extensions of bounded module maps.In the case where A is aσ-unital simple C^(*)-algebra with strict comparison,and every strictly positive lower semicontinuous affine function on quasitraces can be realized as the rank of an element in the Cuntz semigroup,we show that the Cuntz semigroup is equivalent to unitarily equivalent classes of countably generated Hilbert A-modules if and only if A has stable rank one.

F-perfect Rings and Modules

Let R be a ring, and let （F, C） be a cotorsion theory. In this article, the notion of F-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring R is said to be right dr-perfect if F is projective relative to R for any F ∈ F. We give some characterizations of F-perfect rings. For example, we show that a ring R is right F-perfect if and only if F-covers of finitely generated modules are projective. Moreover, we define F-perfect modules and investigate some properties of them.

Some Equivalences and Dualities via Static Modules

For a ring A, an extension ring B, a fixed right A-module M, the endomorphism ring D formed by M, the endomorphism ring E formed by , and the endomorphism ring F formed by HomA (B,M), we present equivalences and dualities between subcategories of B-modules which are finitely cogenerated injective as A-modules and E-modules and F-modules which are finitely generated projective as D-modules.

Eventually homological isomorphisms and Gorenstein projective modules

We prove that a certain eventually homological isomorphism between module categories induces triangle equivalences between their singularity categories,Gorenstein defect categories and stable categories of Gorenstein projective modules.Furthermore,we show that the Auslander-Reiten conjecture and the Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms.Applying these results to arrow removal and vertex removal,we describe the Gorenstein projective modules over some non-monomial algebras and verify the Auslander-Reiten conjecture for certain algebras.

Runoff simulation and hydropower resource prediction of the Kaidu River Basin in the Tianshan Mountains,China

The Tianshan Mountains of Central Asia,highly sensitive to climate change,has been comprehensively assessed for its ecosystem vulnerability across multiple aspects.However,studies on the region's main river systems and hydropower resources remain limited.Thus,examining the impact of climate change on the runoff and gross hydropower potential(GHP)of this region is essential for promoting sustainable development and effective management of water and hydropower resources.This study focused on the Kaidu River Basin that is situated above the Dashankou Hydropower Station on the southern slope of the Tianshan Mountains,China.By utilizing an ensemble of bias-corrected global climate models(GCMs)from Coupled Model Intercomparison Project Phase 6(CMIP6)and the Variable Infiltration Capacity(VIC)model coupled with a glacier module(VIC-Glacier),we examined the variations in future runoff and GHP during 2017-2070 under four shared socio-economic pathway(SSP)scenarios(SSP1-2.6,SSP2-4.5,SSP3-7.0,and SSP5-8.5)compared to the baseline period(1985-2016).The findings indicated that precipitation and temperature in the Kaidu River Basin exhibit a general upward trend under the four SSP scenarios,with the fastest rate of increase in precipitation under the SSP2-4.5 scenario and the most significant changes in mean,maximum,and minimum temperatures under the SSP5-8.5 scenario,compared to the baseline period(1980-2016).Future runoff in the basin is projected to decrease,with rates of decline under the SSP1-2.6,SSP2-4.5,SSP3-7.0,and SSP5-8.5 scenarios being 3.09,3.42,7.04,and 7.20 m^(3)/s per decade,respectively.The trends in GHP are consistent with runoff,with rates of decline in GHP under the SSP1-2.6,SSP2-4.5,SSP3-7.0,and SSP5-8.5 scenarios at 507.74,563.33,1158.44,and 1184.52 MW/10a,respectively.Compared to the baseline period(1985-2016),the rates of change in GHP under the SSP1-2.6,SSP2-4.5,SSP3-7.0,and SSP5-8.5 scenarios are-20.66%,-20.93%,-18.91%,and-17.49%,respectively.The Kaidu River Basin will face significant challenges in water and hydropower resources in the future,underscoring the need to adjust water resource management and hydropower planning within the basin.

平凡扩张上的投射余可解Gorenstein平坦模

设R是有单位元的结合环,M是R-R-双模,R■M是平凡扩张.刻画了平凡扩张上的投射余可解Gorenstein平坦模(简称PGF-模):设(X,α)是左(R■M)-模,M作为左R-模和右R-模的投射、平坦维数分别有限,且Ζ(R)=(R,0)是相容(R■M)-(R■M)-双模,则(X,α)是PGF-模当且仅当左R-模Coker(α)是PGF-模且左R-模的序列M■_(R)M■_(R)X→M■_(R)X→X是正合的.

Gorenstein projective modules and Frobenius extensions

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective,then its underlying module over the base ring is Gorenstein projective; the converse holds if the frobenius extension is either left-Gorenstein or separable(e.g., the integral group ring extension ZZG).Moreover, for the Frobenius extension RA = R[x]/(x^2), we show that: a graded A-module is Gorenstein projective in GrMod(A), if and only if its ungraded A-module is Gorenstein projective, if and only if its underlying R-module is Gorenstein projective. It immediately follows that an R-complex is Gorenstein projective if and only if all its items are Gorenstein projective R-modules.

Global lifting of projective modules modulo socles

For any ideal I of a ring R, let S=R/I. In this note, we consider the following global lifting problem: For any projective S-module Q, does there exist a projective R-module P such that Q and P/IP are isomorphic as S-modules? The concept of global lifting was introduced in Wu’s paper and one of our goals there is to study the computation of K0 groups. Thus in what follows, Q and P are sometimes assumed to be finitely generated in addition.

Singularity Categories with Respect to Ding Projective Modules

Abstract We introduce the singularity category with respect to Ding projective modules, Db dpsg（R）, as the Verdier quotient of Ding derived category Db DP（R） by triangulated subcategory Kb（DP）, and give some triangle equivalences. Assume DP is precovering. We show that Db DP（R） ≌K-,dpb（DP） and Dbpsg（R） ≌ DbDdefect（R）. We prove that each R-module is of finite Ding projective dimension if and only if Dbdpsg（R） = 0.

Derived representation type and Gorenstein projective modules of an algebra under crossed product

Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.

Macro mechanical parameters’ size effect of surrounding rock of Shuibuya project’s underground power station

Scale effect is one of the important aspects in the macro mechanical parame- ters’ research of rock mass, from a new point of view, by means of lab and field rock me- chanics test, establishment of E^Vp relation, classification of engineering rock mass, nu- merical simulation test and back analysis based on surrounding rock’s displacement monitoring results of Shuibuya Project’s underground power station, rock mass deforma- tion module’s size effect of surrounding rock of Shuibuya Project’s undegroud power sta- tion was studied. It’s shown that rock mass deformation module’s scale effect of sur- rounding rock of Shuibuya Project’s undeground power station is obvious, the rock mass deformation module to tranquilization is 20% of intact rock’s. Finally the relation between rock mass deformation modules and the scale of research was established.

f-Projective and f-Injective Modules

Let R be a ring. A fight R-module M is called f-projective if Ext^1 （M, N） = 0 for any f-injective right R-module N. We prove that （F-proj,F-inj） is a complete cotorsion theory, where （F-proj （F-inj） denotes the class of all f-projective （f-injective） right R-modules. Semihereditary rings, von Neumann regular rings and coherent rings are characterized in terms of f-projective modules and f-injective modules.

Robust Quantzation Index Modulation VIA Adaptive Watermarking

In this paper,we suggest an adaptive watermarkingmethod to improve both transparence and robustnessof quantization index modulation(QIM)scheme.Instead of a fixed quantization step-size,we apply astep-size adapted to image content in each 8×8block to make a balance of robust extraction andtransparent embedding.The modified step-size isdetermined by contrast masking thresholds ofWatson’s perceptual model.From a normalizedcrossed-correlation value between the original watermarkand the detected watermark,we could observethat our method is robust to attacks of additivewhite Gaussian noise(AWGN),Salt and Peppernoise and Joint Photographic Experts Group(JPEG)compression than the original QIM.By taking intoaccount the contrast insensitivity and visible thresholdsof human visual system,the suggested improvementachieves a maximum embedding strength andan appropriate quantization step-size which is consistentwith local values of a host signal.

MODIFIED QUANTIZATION INDEX MODULATION WATERMARKING ADAPTIVE TO CONTRAST MASKING THRESHOLDS

In this paper,we suggest an adaptive watermarking method to improve both transparence and robustness of quantization index modulation(QIM) scheme. Instead of a fixed quantization step-size,we apply a step-size adapted to image content in each 8×8 block to make a balance of robust extraction and transparent embedding.The modified step-size is determined by contrast masking thresholds of Watson’s perceptual model.From a normalized crossed-correlation value between the original watermark and the detected watermark,we could observe that our method is robust to attacks of additive white Gaussian noise(AWGN),Salt and Pepper noise and Joint Photographic Experts Group(JPEG) compression than the original QIM.By taking into account the contrast insensitivity and visible thresholds of human visual system,the suggested improvement achieves a maximum embedding strength and an appropriate quantization step-size which is consistent with local values of a host signal.

Homological Epimorphisms, Compactly Generated t-Structures and Gorenstein-Projective Modules

The aim of this paper is two-fold.Given a recollement(T′,T,T′′,i*,i_*,i~!,j!,j*,j*),where T′,T,T′′are triangulated categories with small coproducts and T is compactly generated.First,the authors show that the BBD-induction of compactly generated t-structures is compactly generated when i*preserves compact objects.As a consequence,given a ladder(T′,T,T′′,T,T′) of height 2,then the certain BBD-induction of compactly generated t-structures is compactly generated.The authors apply them to the recollements induced by homological ring epimorphisms.This is the first part of their work.Given a recollement(D(B-Mod),D(A-Mod),D(C-Mod),i*,i_*,i~!,j!,j*,j_*) induced by a homological ring epimorphism,the last aim of this work is to show that if A is Gorenstein,AB has finite projective dimension and j! restricts to D^b(C-mod),then this recollement induces an unbounded ladder(B-Gproj,A-Gproj,C-Gproj) of stable categories of finitely generated Gorenstein-projective modules.Some examples are described.