Narrow Results

In this paper, the norm $L_i(G)$ of the lower central series in a finite group $G$ is introduced, which unifies the norm of derived subgroups and nilpotent residuals. Some propositions of $L_i(G)$ are obtained, and some related subgroups as well as their equivalent propositions can also be found.

For the given n numbers without any other prior information, how to obtain the minimum norm of them only by assigning their signs before them? Moreover, how to know one number is the multiplication of which ones in the given n numbers? In classical solutions, enumeration is the only way via trying one by one, whose complexity is about $\mathrm{\text{O}}(n{2}^{n-1})$ and this is a NP problem. In this paper, the parallel quantum algorithm is proposed to solve the two questions shown in above. Through the quantum design of linear expressions of angles in parallel circuits, only $\mathrm{\text{O}}(n^2)$ time’s quantum operations and about $\mathrm{\text{O}}(n^2)$ times’ quantum measurements in the average will give the correct answer in the successful probability of 0.97 instead of the traditional $\mathrm{\text{O}}(2^n)$ times. The example and theoretical analysis demonstrate the efficiency of the proposed method.

Let L≀X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the ℓ^p-norms and spectral radii of P, if one has an amenable (not necessarily discrete or unimodular) locally compact group of isometries that acts transitively on L. This applies, in particular, to wreath products K≀G of finitely-generated groups, where K is amenable. As a special case, this comprises a result of Żuk regarding the ℓ²-spectral radius of symmetric random walks on such groups.

Norms and institutions have been proposed to regulate multi-agent interactions. However, agents are intrinsically autonomous, and may thus decide whether to comply with norms. On the other hand, besides institutional norms, agents may adopt new norms by establishing commitments with other agents. In this paper, we address these issues by considering an electronic institution that monitors the compliance to norms in an evolving normative framework: norms are used both to regulate an existing environment and to define contracts that make agents' commitments explicit. In particular, we consider the creation of virtual organizations in which agents commit to certain cooperation efforts regulated by appropriate norms. The supervision of norm fulfillment is based on the notion of institutional reality, which is constructed by assigning powers to agents enacting institutional roles. Constitutive rules make a connection between the illocutions of those agents and institutional facts, certifying the occurrence of associated external transactions. Contract specification is based on conditional prescription of obligations. Contract monitoring relies on rules for detecting the fulfillment and violation of those obligations. The implementation of our normative institutional environment is supported by a rule-based inference engine.

The norm $N(G)$ of a group $G$ is the intersection of the normalizers of all subgroups in $G$. In this paper, the norm is generalized by studying on Sylow subgroups and ${\mathscr{H}}$-subgroups in finite groups which is denoted by $C(G)$ and $A(G)$, respectively. It is proved that the generalized norms $A(G)$ and $C(G)$ are all equal to the hypercenter of $G$.

Agent-based modeling is being increasingly used to simulate socio-techno-ecosystems that involve social dynamics. Humans face constraints that they sometimes wish to challenge, and when they do so, they often trigger changes at the scale of the social group too. Including such adaptation dynamics explicitly in our models would allow simulation of the endogenous emergence of rule changes. This paper discusses such an approach in an institutional framework and develops a sequence that allows modeling of endogenous rule changes. Parts of this sequence are implemented in a NetLogo KISS model to provide some illustrative results.

Social issues are generally discussed by highly-involved and less-involved people to build social norms defining what has to be thought and done about them. As self-involved agents share different attitude dynamics to other agents [Wood, W., Pool, G., Leck, K. and Purvis, D., Self-definition, defensive processing, and influence: The normative impact of majority and minority groups, J. Pers. Soc. Psychol. (1996) 1181–1193], we study the emergence and evolution of norms through an individual-based model involving these two types of agents. The dynamics of self-involved agents is drawn from [Huet, S. and Deffuant, G., Openness leads to opinion stability and narrowness to volatility, Adv. Complex Syst.13 (2010) 405–423], and the dynamics of others, from [Deffuant, G., Neau, D., Amblard, F. and Weisbuch, G., Mixing beliefs among interacting agents, Adv. Complex Syst.3 (2001) 87–98]. The attitude of an agent is represented as a segment on a continuous attitudinal space. Two agents are close if their attitude segments share sufficient overlap. Our agents discuss two different issues, one of which, called main issue, is more important for the self-involved agents than the other, called secondary issue. Self-involved agents are attracted to both issues if they are close to the main issue, but shift away from their peer’s opinion if they are only close on the secondary issue. Differently, non-self-involved agents are attracted by other agents when they are close on both the main and secondary issues. We observe the emergence of various types of extreme minor clusters. In one or different groups of attitudes, they can lead to an already-built moderate norm or a norm polarized on secondary and/or main issues. They can also push disagreeing agents gathered in different groups to a global moderate consensus.

For x ∈ [0, 1]ⁿ with ‖x‖₁ = 1 and y ∈ [1, ∞)ⁿ, we prove that

The norm N(G) of a group G is the intersection of normalizers of all the subgroups of G. Let G be a finite group, p a prime dividing the order of G, and P a Sylow p-subgroup of G. In this paper, it is proved that G is p-nilpotent if Ω₁(P) ≤ N(N_G(P)), and when p=2, . Some applications of this result are given. Finally, a class of finite p-groups in which the index of the norm is exactly p is described.

The norm N(G) of a group G is the intersection of the normalizes of all the subgroups of G. A group is called capable if it is a central factor group. In this paper, we give a necessary and sufficient condition for a capable group to satisfy N(G)=ζ(G), and then some sufficient conditions for a capable group with N(G)=ζ(G) are obtained. Furthermore, we discuss the norm of a nilpotent group with cyclic derived subgroup.

Let $H_d[x_1,\dots ,x_m]$ be the complex vector space of homogeneous polynomials of degree $d$ with the independent variables $x_1,\dots ,x_m$. Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_1,\dots ,x_m$. For any linear operator $T$ acting on $V$, there is a (unique) induced operator $P(T)$ acting on $H_d[x_1,\dots ,x_m]$ satisfying

We present some recent results on variable exponent Lebesgue spaces, which include: A variant of the definition of the norm in the variable exponent Lebesgue space; The Amemiya norm equals the Orlicz norm in the variable exponent Lebesgue space; An exact inequality involving the Luxemburg norm and the conjugate-Orlicz norm in the variable exponent Lebesgue space. We also present some results and open problems on the solutions of the p(x)–Laplacian equations.

An automorphism of a group G is said to be a power automorphism if it leaves every subgroup of G invariant. In this paper, we collect our recently work about power automorphisms and induced automorphisms.

In this paper, addition and number multiplication are defined in the classical logic metric space, and the definition of norm is introduced based on the degree of formula. It also is proved that the classical logic metric space builds a Z(2)- normable linear logic space. And the concept of a Z(2)- normable linear logic sub-space is introduced in. According to the definition, it is obtained that the set of n-symmetric logical formula constitutes a Z(2)- normable linear logic sub-space.