Duality theorem for L-R crossed coproducts

In this paper, the notion of L-R crossed coproduct is introduced as a unified approach for smash coproducts, crossed coproducts and L-R smash coproducts of Hopf algebras.A duality theorem for L-R crossed coproduct is proved.

Cylinder Coalgebras and Cylinder Coproducts for Quasitriangular Hopf Algebras

This paper introduces the concepts of cylinder coalgebras and cylinder coproducts for quasitriangular bialgebras, and points out that there exists an anti-coalgebra isomorphism （H,△^-）≌ （H,△^-）, where （H, △^-） is the cylinder coproduct, and （H,△^-） is the braided coproduct given by Kass. For any finite dimensional Hopf algebra H, the Drinfel＇d double （D（H）,△^-D（H）） is proved to be the cylinder coproduct. Let （H, H, R） be copaired Hopf algebras. If R ∈ Z（H×H） with inverse R-1 and skew inverse R, then the twisted coalgebra （H^R）^R-1 is constructed via twice twists, whose comultiplication is exactly the cylinder coproduct. Moreover, for any generalized Long dimodule, some solutions for Yang-Baxter equations, four braid pairs and Long equations are constructed via cylinder twists.

Nutrient profile and digestibility of tubers and agro-industrial coproducts determined using an in vitro model of swine

Exploring and evaluating alternative feed ingredients to be used in swine diet is essential due to highly variable cost and limited availability of conventional feed ingredients. Tubers and agro-industrial coproducts could provide the basis for producing affordable swine feed. However, information on the nutritional value of these potential alternative feedstuffs is necessary while considering their use in swine feeding program. Four tubers(purple sweet potato [PSP], okinawan sweet potato, taro and cassava) and 3 coproducts(okara, wheat millrun [WMR] and barley brewers grain [BBG]) were analyzed for their proximate nutrients, starch, fibers and gross energy(GE) content. Two independent in vitro studies were carried out for tubers and coproducts to determine their nutrients digestibility using a 3-step enzymatic assay(which mimics the digestion occurring in the gastrointestinal tract of swine) with 9 replicates of each sample digested in 3 batches equally. All replicate samples were used to determine in vitro dry matter digestibility(IVDDM) while 2 replicates from each batch were used to determine in vitro GE digestibility(IVDGE). Among tubers, CP content was the highest in taro(8.8%) and the lowest in cassava(3.7%), while CP content among coproducts was the highest in okara(22.7%) and the lowest in WMR(11.8%). Ether extract content among tubers ranged from 1.1% to 2.8%. The GE content among tubers, ranged from 4,134 to 4,334 kcal/kg whereas among coproducts it ranged from 4,270 to 4,794 kcal/kg. Among tubers, IVDDM for PSP was significantly higher(86.8%, P < 0.001) than taro(70.3%). Among coproducts, IVDDM of okara(74.1%) was significantly higher(P < 0.05) than BBG(61.3%). In conclusion,both tubers and coproducts can be used as a partial substitute of conventional energy feedstuffs in swine diets as these are rich in GE and other nutrients and are fairly digestible.

The Global Dimensions of Crossed Products and Crossed Coproducts

In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^＊ are semisimple, then gl.dim（A#σH）=gl.dim（A）, where a is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A^#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim（C α H）=gl.dim（C）.

PRODUCTS AND COPRODUCTS IN CATEGORY OF COMPLETELY DISTRIBUTIVE LATTICES AND THEIR STRUCTURES

Taking completely distributive lattices for objects and complete lattice homomorphisms for morphisms, we get a category Lat (see Refs. [1—3]). There is a primary question about Lat: Does every family of objects have product and coproduct in Lat? We prove that Lat has products and coproducts, and describe the intrinsic relation between

Cosemisimple Coextensions and Homological Dimension of Smash Coproducts

Let H be a finite dimensional cosemisimple Hopf algebra, C a left H-comodule coalgebra and let C = C/C（H^＊）^＋ be the quotient coalgebra and the smash coproduct of C and H. It is shown that if C/C is a eosemisimple coextension and C is an injective right C-comodule, then gl. dim（the smash coproduct of C and H） = gl. dim（C） = gl. dim（C）, where gl. dim（C） denotes the global dimension of coalgebra C.

Maschke-Type Theorems for Weak Smash Coproducts

In this paper we mainly give Maschke-type theorems for weak smash coproduct coalgebras which simultaneously show how to deal with the complicated computations of weak smash coproduct coalgebras.

Hopf Algebra of Labeled Simple Graphs

A lot of combinatorial objects have a natural bialgebra structure. In this paper, we prove that the vector space spanned by labeled simple graphs is a bialgebra with the conjunction product and the unshuffle coproduct. In fact, it is a Hopf algebra since it is graded connected. The main conclusions are that the vector space spanned by labeled simple graphs arising from the unshuffle coproduct is a Hopf algebra and that there is a Hopf homomorphism from permutations to label simple graphs.

Curating Spaces of Hope: Coproducing Shared Values in Uncertain Times

This paper sets out a new paradigm of faith based organisation(FBO)called Curating Spaces of Hope.The paper sets out the paradigm and the interdisciplinary literatures into which the paradigm is applied namely,the diversifying belief landscape in the UK,the postsecular,the redefinition of FBOs,and liminality as the new norm in policy.The paper then turns to ethnographic research to evidence the ability of the paradigm to map and coproduce shared values,before considering applications of Curating Spaces of Hope in post-pandemic contexts in the north west of England through case studies with ecumenical Christian,non-religious,and Turkish Muslim and interfaith contexts.

Super-Shuffle Product and Cut-Box Coproduct on (0,1)-Matrices

In 2014, Vargas first defined a super-shuffle product and a cut-box coproduct on permutations. In 2020, Aval, Bergeron and Machacek introduced the super-shuffle product and the cut-box coproduct on labeled simple graphs. In this paper, we generalize the super-shuffle product and the cut-box coproduct from labeled simple graphs to (0,1)-matrices. Then we prove that the vector space spanned by (0,1)-matrices with the super-shuffle product is a graded algebra and with the cut-box coproduct is a graded coalgebra.

Duality theorem for smash coproduct over quantum groupoids

The duality theorem of generalized weak smash coproducts of weak module coalgebras and comodule coalgebras over quantum groupoids is studied.Let H be a weak Hopf algebra,C a left weak H-comodule coalgebra and D a left weak H-module coalgebra.First,a weak generalized smash coproduct C×lH D over quantum groupoids is defined and the module and comodule structures on it are constructed.The weak generalized right smash coproduct C×rL D is similar.Then some isomorph-isms between them are obtained.Secondly,by introducing some concepts of a weak convolution invertible element,a weak co-inner coaction and a strongly relative co-inner coaction,a sufficient condition for C×rH D to be isomorphic to Cv D is obtained,where v∈WC（C,H）and the coaction of H on D is right strongly relative co-inner.Finally,the duality theorem for a generalized smash coproduct over quantum groupoids,（C×lH H）×lH H≌Cv（H×lH H）,is obtained.

Monoidal Hom-Hopf algebra on Hom-twisted product

Let （H, a） be a monoidal Hom-bialgebra and （B,p） be a left （H, a）-Hom-comodule coalgebra. The new monoidal Hom-algebra B#y H is constructed with a Hom-twisted product Ba[H] and a. B × H Hom-smash coproduct. Moreover, a sufficient and necessary condition for B#y / to be a monoidal Hom-bialgebra is given. In addition, let （H, a） be a Hom-σ- Hopf algebra with Hom-〇 --antipode SH, and a sufficient condition for this new monoidal Hom-bialgebra B#y H with the antipode S defined by S（b×h）=（1B×SH（a^-1）b（-1）））（SB（b（0））×1H to be a monoidal Hom-Hopf algebra is derived.

Reactive dividing-wall column for the co-production of ethyl acetate and n-butyl acetate

Reactive dividing-wall column(RDWC) technology plays a critical role in the energy saving and high efficiency of chemical process.In this article, the process of co-producing ethyl acetate(EA) and n-butyl acetate(BA) with RDWC was studied.BA was not only the product, but also acted as entrainer to remove the water generated by the two esterification reactions.Experiments and simulations of the co-production process were carried out.It was found that the experimental results were in good agreement with the simulation results.Two kinds of RDWC structures(RDWC-FC and RDWC-RS) were proposed, and the co-production process operating parameters of the two types of RDWC were optimized by Aspen Plus respectively.The optimal operating parameters of the RDWC-FC were determined as follows: 0.6 of the reflux ratio of aqueous phase(RR), 0.66 of the vapor split(R_V) and 0.51 of the liquid split(R_L).And the optimal operating parameters of the RDWC-RS were shown as follows: RR was 0.295 and R_V was 0.61.Furthermore, the energy saving analysis of the co-production process was based on the annual output of 10000 tons of EA, compared with the traditional reaction distillation(RD) to prepare EA and BA, the reboiler duty of the RDWC-FC column could save 20.4%, TAC saving 23.6%; RDWC-RS reboiler energy consumption could save 17.0%, TAC 22.2%.

Prevention of explosion in coal mine and management of coal mine gas

There are many problems in terms of safe coal production and the sound developmentof the coal industry.Accompanying the intensification and increasing efficiencyof coal production and the conducting of mining operations at deeper and more remoteareas of mines,the efficient recovery and utilization of Coal Mine Methane:(CMM) is animportant issue in improving and stabilizing the productivity in the coal mining industry withhigh levels of gas,where the incidence of gas outbursts is increasing.We plan to studyvarious aspects of the development of production technology and characteristics of themine site.This is to establish the technology for highly efficient coproduction coal and gasoperation rate.As a result,the productivity at the coal mine face will increase due to thereduction in gas emissions in the mining face.Effective use of recovered gas can be expectedto reduce global warming by reducing the amount of coal mine methane gas emissionin the air.

The quasi-triangular structures over Hom-ω-smash coproduct Hopf algebras

Let （C,α） and （H, β） be Hom-bialgebras and ω ： C × H → H × C a linear map. We introduce a Horn-ω-smash coproduct （Cω H, γ） and give necessary and sufficient conditions for （Cω H, γ） to be a Hom-bialgebra. We study the quasi-triangular structures over （Cω H, γ） and show the necessary and sufficient conditions for （Cω H, γ R） to be a quasi-triangular Hom-Hopf algebra. As applications of our results, we introduce the concept of D（H）＊ and construct quasi-triangular structures over D（H）＊.

The Structure Theorem for Module Coalgebras

This paper,mainly gives the structure theorem for module coalgebras by a kind of new method,and deletes the condition that the antipode S of the Hopf algebra H is bijective.

Constructing the Cocyclic Structures for Crossed Coproduct Coalgebras

In this paper, we construct a cocylindrical object associated to two coalgebras and a cotwisted map. It is shown that there exists an isomorphism between the cocyclic object of the crossed coproduet coalgebra induced from two coalgebras with a cotwisted map and the cocyclic object related to the diagonal of the cocylindrical object.

On the Structure of the B-cocleft Module Coalgebra

In this paper we develope the notions of crossed coproduct of Hopf algebras and study an equivalent theorem of generalized crossed coproduct of H weakly comodule coalgebras and H module coalgebras. The main result is to prove a structure theorem about B cocleft H module coalgebras.

A Generalized Biproduct Theorem

We develop the Radford's biproduct theorem which plays an important role in giving a negative answer to a conjecture of I Kaplansky. Let B, H be two Hopf algebras with H acting weakly on B and α, β : B → H H be two linear maps verifying suitable conditions. We consider in this paper a twisted Hopf crossed coproduct B ×βα H and derive a necessary and sufficient condition for B # ×βα H with a Hopf smash product structure to be a bialgebra which generalizes in [14, Theorem 1.1] and the well-known Radford biproduct theorem [10, Theorem 1] .

Social Welfare Transformation and Community Governance Model for the Aging Generation:Rural Community Care in Chiayi County as a Case Study

Over the past two decades,Taiwan’s welfare policies for the elderly have prepared for the arrival of an aged society with the aim of facilitating aging in place(AIP).The Long-Term Care Services Act passed by the Legislative Yuan in 2015 set regulations for long-term care personnel,thereby improving the welfare system for the elderly.The government’s AIP policies focus on providing welfare institutions or service units and do not sufficiently explore key factors such as the relationship between care services and community governance models.Since the basic focus of elderly welfare policies is AIP,it is important to actively empower grassroots community organizations and consider the elderly as important subjects of community participation.AIP policies emphasize providing long-term care and nurturing the elderly until death;such policies will be more proactive and innovative if coproduction is incorporated.This study briefly discusses theories related to coproduction using the case study of Chiayi County’s Dingcaiyian Community Development Association,which links six communities in remote rural villages and establishes a community care network based on the coproduction concept.Follow-up issues related to community care are considered and potential practices and ideas to guide social welfare transformation and community governance in the future are discussed.