On Direct Product of HX Groups

In the paper [1] the upgrade of the structure of groups has been considered, in which the concept of HX group has been raised. In this paper we will discuss the direct product of HX groups and AX groups on direct product group.

Continuous-Time Classical and Quantum Random Walk on Direct Product of Cayley Graphs

In this paper we define direct product of graphs and give a recipe for obtaining probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on direct product of graph is obtained by multiplication of probability on the corresponding to sub-graphs, where this method is useful to determining probability of walk on compficated graphs. Using this method, we calculate the probability of Continuous-time classical and quantum random walks on many of finite direct product Cayley graphs （complete cycle, complete Kn, charter and n-cube）. Also, we inquire that the classical state the stationary uniform distribution is reached as t→∞ but for quantum state is not always satisfied.

Geometric Origin of Staggered Fermion：Direct Product K—Cycle

Staggered formalism of lattice fermion can be cast into a form of direct product K-cycle in noncommutative geometry. We prove the correspondence between this staggered K-cycle and a canonically defined K-cycle for finitely generated Abelian groups where a lattice appears as a special case.

The Direct Product of Graded Finite Posets

A new concept Graded Finite Poset is proposed in this paper. Through discussing some basic properties of it, we come to that the direct product of graded finite posets is connected if and only if every graded finite poset is connected. The graded function of a graded finite poset is unique if and only if the graded finite poset is connected.

On total{k}-domatic number of Cartesian and direct product of graphs

For a positive integer k,the total{k}-dominating function(T{k}DF)of a graph G without isolated vertices is a function f from the vertex set V(G)to the set{0,1,2,…,k}such that for each vertex v∈V(G),the sum of the values of all its neighbors assigned by f is at least k.A set{f_(1),f_(2),…,f_(d)}of pairwise different T{k}DF s of G with the property that∑d i=1 f_(i)(v)≤k for each v∈V(G),is called a total{k}-dominating family(T{k}D family)of G.The total{k}-domatic number of a graph G,denoted by d^({k})_(t)(G),is the maximum number of functions in T{k}D family.In 2013,Aram et al.proposed a problem that whether or not d^({k})_(t)(C_(m)□C_(n))=3 when 4 nmk,and d^({k})_(t)(C m□C n)=4 when 4|nmk.It was shown that d^({k})_(t)(C_(m)□C_(n))=3 if 4 nmk and k≥2 or 4|nmk and 2 nk,which partially answered the above problem.In addition,the total{k}-domatic number of the direct product of a cycle and a path,two paths,and two cycles was studied,respectively.

Analysis of the System of Direct Subsidy for Grain Production in China

This paper introduces the status quo of the system structure of direct subsidy for grain production in China,and analyses the function and essence of direct subsidy for grain production in China as follows:the function of direct subsidy for grain production in China is to protect and promote the development of food industry,ensure food supply and national food security,protect grain growers' interests,and maintain the dominant position of national food trade;the essence of direct subsidy for grain production is the compensation for grain producers' reasonable interests,the compensation for positive externality of grain production,and social costs of adjusting equilibrium between food supply and demand.We discuss the problems existing in the system of direct subsidy for grain production in China and put forward corresponding countermeasures as follows:first,adhere to improving synergetic system of direct subsidy for grain production;second,adjust the direct subsidy method of grain production;third,establish long-term mechanism of subsidy for grain production;fourth,promote the information-based service level of the system of direct subsidy for grain production.

The Path-Positive Property on the Products of Graphs

The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.

Circular L(j,k)-labeling numbers of trees and products of graphs

Let j, k and m be three positive integers, a circular m-L（j, k）-labeling of a graph G is a mapping f： V（G）→{0, 1, …, m-1}such that f（u）-f（v）m≥j if u and v are adjacent, and f（u）-f（v）m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L（j, k）-labeling of G is called the circular L（j, k）-labeling number of G and is denoted by σj, k（G）. For any two positive integers j and k with j≤k,the circular L（j, k）-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined.

Study on regional material flow analysis of Chengyang District in Qingdao

Base on the framework of material flow analsis （MFA） proposed by Eurostat, this paper collected related data and conducted a regional MFA for the Chengyang Districgt in Qingdao. The indicators such as Direct Material Input（DMI）, Direct Material Consumption （DMC） and Exports are calculated or estimated for the period of year 1995 through 2004. Several derivable iudicators defining direct material productivity and material intenstiy are also evaluated and some valuable conclusions were drawn. DMI in absolute number increased from about 1645.9 kilotons in 1995to about 8052.5 kilotons in 2004. Imports contribute to about 47 to 69% of DMI, and the biggest component of imports is fossil fuel that approximatley accounts far 50% of imnports. DMI per capita of Chengyang District increased rapidly and reached 17. 4 tons in 2004. Compared with other five economies studied by WRI, material consumption intensity of Chengyang District is significantly lower than developed countries, which indicates that Chengyang District is still in a development stage The direct material productivity （actual GDP per DMI） increased 59.1% from 1995 to 2004 and has been higher than either of stadies in China, which indicates efficiency of resources utilization has improved obviously in Chengyang District. But there was significant increase in material input in the last decade, which shows that rapid increase of economy is highly dependent on requirement and consumption of natural resources. Therefore, in order to promote the sustainahility, it is essential to develop circular economy and improve utilization efficiency of resources.

Industrial Green Spatial Pattern Evolution of Yangtze River Economic Belt in China

We use the directional slacks-based measure of efficiency and inverse distance weighting method to analyze the spatial pattern evolution of the industrial green total factor productivity of 108 cities in the Yangtze River Economic Belt in 2003–2013.Results show that both the subprime mortgage crisis and ‘the new normal' had significant negative effects on productivity growth,leading to the different spatial patterns between 2003–2008 and 2009–2013.Before 2008,green poles had gathered around some capital cities and formed a tripartite pattern,which was a typical core-periphery pattern.Due to a combination of the polarization and the diffusion effects,capital cities became the growth poles and ‘core' regions,while surrounding areas became the ‘periphery'.This was mainly caused by the innate advantage of capital cities and ‘the rise of central China' strategy.After 2008,the tripartite pattern changed to a multi-poles pattern where green poles continuously and densely spread in the midstream and downstream areas.This is due to the regional difference in the leading effect of green poles.The leading effect of green poles in midstream and downstream areas has changed from polarization to diffusion,while the polarization effect still leads in the upstream area.

New Public Key Cryptosystems from Combinatorial Group Theory

External direct product of some low layer groups such as braid groups and general Artin groups, with a kind of special group action on it, provides a secure cryptographic computation platform, which can keep secure in the quantum computing epoch. Three hard problems on this new platform, Subgroup Root Problem, Multi-variant Subgroup Root Problem and Subgroup Action Problem are presented and well analyzed, which all have no relations with conjugacy. New secure public key encryption system and key agreement protocol are designed based on these hard problems. The new cryptosystems can be implemented in a general group environment other than in braid or Artin groups.

Bounds on Fractional Domination of Some Products of Graphs

Let γ f(G) and γ~t f(G) be the fractional domination number and fractional total domination number of a graph G respectively. Hare and Stewart gave some exact fractional domination number of P n×P m (grid graph) with small n and m . But for large n and m , it is difficult to decide the exact fractional domination number. Motivated by this, nearly sharp upper and lower bounds are given to the fractional domination number of grid graphs. Furthermore, upper and lower bounds on the fractional total domination number of strong direct product of graphs are given.

<i>Supereulerian Digraph</i>Strong Products

A vertex cycle cover of a digraph H is a collection C = {C1, C2, …, Ck} of directed cycles in H such that these directed cycles together cover all vertices in H and such that the arc sets of these directed cycles induce a connected subdigraph of H. A subdigraph F of a digraph D is a circulation if for every vertex in F, the indegree of v equals its out degree, and a spanning circulation if F is a cycle factor. Define f (D) to be the smallest cardinality of a vertex cycle cover of the digraph obtained from D by contracting all arcs in F, among all circulations F of D. Adigraph D is supereulerian if D has a spanning connected circulation. In [International Journal of Engineering Science Invention, 8 (2019) 12-19], it is proved that if D1 and D2 are nontrivial strong digraphs such that D1 is supereulerian and D2 has a cycle vertex cover C’ with |C’| ≤ |V (D1)|, then the Cartesian product D1 and D2 is also supereulerian. In this paper, we prove that for strong digraphs D1 and D2, if for some cycle factor F1 of D1, the digraph formed from D1 by contracting arcs in F1 is hamiltonian with f (D2) not bigger than |V (D1)|, then the strong product D1 and D2 is supereulerian.

On Some Embedment of Groups into Wreath Products

In this paper, we showed how groups are embedded into wreath products, we gave a simpler proof of the theorem by Audu (1991) (see [1]), also proved that a group can be embedded into the wreath product of a factor group by a normal subgroup and also proved that a factor group can be embedded inside a wreath product and the wreath product of a factor group by a factor group can be embedded into a group. We further showed that when the abstract group in the Universal Embedding Theorem is a p-group, cyclic and simple, the embedding becomes an isomorphism. Examples were given to justify the results.

Representations of Free Groups on Rational Line Q

A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of countable free groups on the rational line Q, which generalizes their results. As applications,we obtain the highly ordered transitive representation for the direct product of countable free groups,and the inverse limit of countable free groups must be an action on the set Q.

Fixed Subgroups are not Compressed in Direct Products of Surface Groups

By constructing counterexamples,the authors show that the fixed subgroups are not compressed in direct products of free and surface groups,and hence negate a conjecture in[Zhang,Q.,Ventura,E.and J.Wu,Fixed subgroups are compressed in surface groups,Internat.J.Algebra Comput.,25,2015,865–887].

Weak type(1, 1) of the Riesz transform on some direct product manifolds with exponential volume growth

In this paper, we are concerned with the Riesz transform on the direct product manifold H^(n)× M,where H^(n) is the n-dimensional real hyperbolic space, and M is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type(1, 1) property. In addition, we obtain the weak type(1, 1) of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type(1, 1) boundedness of some operators considered in the work of Li et al.(2016).

Regulatory framework of cosmetic in the European Union

Cosmetics that are personal or personalized are now becoming extremely prevalent.While compliance is mandated by European Union(EU)Cosmetics Directive 1223/2009,there seem to be no strict guidelines for maintaining obedience.Cosmetics must meet a number of conditions in order to be sold in the European Single Market;however,the focus of this article is on the Cosmetics Regulation 1223/2009.Regulations are examined for certain elements and several solutions are presented that allow for careful use of individually cloaked cosmetics that are available on the market.Metallic nanoparticles(NMs)have been proposed for usage as active ingredients/excipient in a number of cosmetics products.Due to fast-paced businesses in the cosmetic industry,cosmetology tends to focus on its distinctive characteristics to bring value to a diverse array of products,but due to the small size of nanometers,NMs may not always follow the very same handling guidelines as their conventional material.As a result,a nano-specific framework for regulating the use of nanomaterials&creation of nano-improved cosmetics is becoming increasingly prevalent.Scientific and industrial perspective into the NMs presently used for the m arketplace,with an emphasis on metallic NMs,and also an evaluation of the regulatory requirements and Scientific Commission on Consumer Safety(SCCS)Opinions.Considering the fact that the original Cosmetic products Directive(EU Legislation No 1223/2009)has precise restrictions on NMs,beauty materials containing unlawful NMs have already been supplied in the EU on numerous times.Researchers examine the risk evaluation indicated in Article 16 of the Cosmetology Code acts as a framework for the potential expansion to enhance nano-items,considering the long-term risks of nanomaterials if mistreated.The nation’s attention is on synchronizing efforts to integrate metallic NMs into cosmetic products but to the restricted fusion of metallic NMs with numerous non-metallic n anoparticles.Although Directive 76/768/EEC on the beauty items is an upright division of amendment that requires the European market for every cosmetic product placed to meet its exigencies would be irrational that it is for believings a stand-alone part of regulations is unaffected by other legal texts.In reality,D irective 76/768/EEC takes the form part of complicated legal action that began 40 years ago that ensure the free passage of goods throughout the EU while also European individuals’and their environment’s safety.The ongoing chapter outlines the most important aspects of the Directive Cosmetic Products along with the latest guidelines 2022 prepared by the COS law Team of what happened in the EU cosmetics regulatory framework between January and March,which serves as the book’s foundation.The trend of personal skincare seems to be high among clients.

On Direct Limits of Finite Products of Fields as Subrings of a Commutative Ring

Let R be a commutative ring. In this paper, we develop the existence of direct limits of finite products of fields as subrings of R.