A centre of a differential system in the plane $ {\mathbb {R}}^2$ is an equilibrium point $p$ having a neighbourhood $U$ such that $U\setminus \{p\}$ is filled with periodic orbits. A centre $p$ is global when $ {\mathbb {R}}^2\setminus \{p\}$ is filled with periodic orbits. In general, it is a difficult problem to distinguish the centres from the foci for a given class of differential systems, and also it is difficult to distinguish the global centres inside the centres. The goal of this paper is to classify the centres and the global centres of the following class of quintic polynomial differential systems

\begin{align*} \dot{x}= y,\quad \dot{y}={-}x+a_{05}\,y^5+a_{14}\,x\,y^4+a_{23}\,x^2\,y^3+a_{32}\,x^3\,y^2+a_{41}\,x^4\,y+a_{50}\,x^5, \end{align*}

$ {\mathbb {R}}^2$

Let $G$ be a compact Abelian group and $E$ a subset of the group $\widehat {G}$ of continuous characters of $G$. We study Arens regularity-related properties of the ideals $L_E^1(G)$ of $L^1(G)$ that are made of functions whose Fourier transform is supported on $E\subseteq \widehat {G}$. Arens regularity of $L_E^1(G)$, the centre of $L_E^1(G)^{\ast \ast }$ and the size of $L_E^1(G)^\ast /\mathcal {WAP}(L_E^1(G))$ are studied. We establish general conditions for the regularity of $L_E^1(G)$ and deduce from them that $L_E^1(G)$ is not strongly Arens irregular if $E$ is a small-2 set (i.e. $\mu \ast \mu \in L^1(G)$ for every $\mu \in M_E^1(G)$), which is not a $\Lambda (1)$-set, and it is extremely non-Arens regular if $E$ is not a small-2 set. We deduce also that $L_E^1(G)$ is not Arens regular when $\widehat {G}\setminus E$ is a Lust-Piquard set.

Suppose that R is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra $L_R(E)$ is simple if and only if R is simple, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.

We prove that for a Banach algebra A having a bounded $\mathcal {Z}(A)$ -approximate identity and for every $\mathbf {[IN]}$ group G with a weight w which is either constant on conjugacy classes or satisfies $w \geq 1$ , $\mathcal {Z}(L^{1}_{w}(G) \otimes ^{\gamma } A) \cong \mathcal {Z}(L^{1}_{w}(G)) \otimes ^{\gamma } \mathcal {Z}(A)$ . As an application, we discuss the conditions under which $\mathcal {Z}(L^{1}_{\omega }(G,A))$ enjoys certain Banach algebraic properties, such as weak amenability or semisimplicity.

An object located in the centre position is believed to be the most attended and well remembered, which increases its likelihood of being chosen (i.e., centrality preference). However, the literature has yielded inconsistent evidence. With the support of an eye-tracking technique, this study tried to provide another means of examining the relationship between preference and attention. Thirty undergraduates were asked to choose one of five similar items presented on a horizontal line. The findings on eye fixation points and looking duration positively related to the probability of an item being chosen as the preferred item. Yet performance in a recall test revealed an independence between preference and remembering. Furthermore, an unexpectedly large proportion of the participants also preferred the items on the leftmost side of the array. The mental number line and social norms, together with centrality preference, were used to provide an explanation of our implicit preference in decision making.

If the centre of a group $G$ is trivial, then so is the centre of its automorphism group. We study the structure of the centre of the automorphism group of a group $G$ when the centre of $G$ is a cyclic group. In particular, it is shown that the exponent of $Z(\text{Aut}(G))$ is less than or equal to the exponent of $Z(G)$ in this case.

In this paper, we determine when $\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $, the complement of the zero divisor graph ${\Gamma }_{I} (L)$ with respect to a semiprime ideal $I$ of a lattice $L$, is connected and also determine its diameter, radius, centre and girth. Further, a form of Beck’s conjecture is proved for ${\Gamma }_{I} (L)$ when $\omega (\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} )\lt \infty $.

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systems where h, K, Ψ and ξ are analytic functions defined in the neighbourhood of O with K(O) ≠ 0 or Ψ(O) ≠ 0 and n ≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.

Let R be an integral domain and A a symmetric cellular algebra over R with a cellular basis {CλS,T∣λ∈Λ,S,T∈M(λ)}. We construct an ideal L(A) of the centre of A and prove that L(A) contains the so-called Higman ideal. When R is a field, we prove that the dimension of L(A) is not less than the number of nonisomorphic simple A-modules.

We extend the group theoretic notions of transfer and stable elements to graded centres of triangulated categories. When applied to the centre Z*(Db(B) of the derived bounded category of a block algebra B we show that the block cohomology H*(B) is isomorphic to a quotient of a certain subalgebra of stable elements of Z*(Db(B)) by some nilpotent ideal, and that a quotient of Z*(Db(B)) by some nilpotent ideal is Noetherian over H*(B).

This article analyses the competition strategies adopted by the Christian Democratic (DC) and post-Christian Democratic (post-DC) parties after the electoral reforms of 1993 and 2005. Four main aspects are considered: the significance of cultural (ideological and cognitive) factors in the DC's exit from the political stage; the need to adapt post-DC strategy to bipolarism and the nostalgia for an autonomous centre; the nature and geography of the post-DC vote; and the attitudes of the middle-level elites of the three principal post-DC parties (the DL, the UDEUR and the UDC) vis-à-vis the competitive and strategic decisions they had to make. The article reaches the following conclusions: the post-DC parties share the same cultural orientations and have similar politico-electoral characteristics (a confessional background, the relevance of patronage networks and personalistic vote mobilisation); all three parties adapted, in different ways and with different degrees of success, to the new structure of coalitional bipolarism in the decade 1996–2006; and both research data on the main post-DC parties’ national congress delegates and the evolution of their electoral strategies on the eve of the 2008 elections show that the post-Christian Democrats felt more at home in the centre-left alignment, together with the post-communist PDS-DS, than in the centre-right alignment led by Berlusconi.

The general question underlying this article is how much have parties changed in Italy in the transition from the First to the Second Republic? The disintegration of the Christian Democrats (DC) in the early 1990s created a large opening on the centre-right of the political spectrum, which was promptly occupied by Forza Italia (FI). A totally new party organisationally speaking, FI, with its centralisation, its personalisation and its charismatic leadership, marked a clear break with all that the DC had represented in terms of its unrestricted pluralism based on strong factions. However, the franchise model adopted by Berlusconi's party, characterised by the co-existence of unity of command at the top, and differentiated and quasi-autonomous bodies locally, may facilitate the convergence of the parties representing the various interests and outlooks on the centre-right in the new rassemblement party that is the PdL. Through the latter, and given the new systemic conditions, Italy may achieve a transformation in the nature of the centre from Christian Democratic to conservative, enabling it to integrate most of the political right as well.

Let $A$ be a non-commutative normed algebra with centre $Z(A)$. A simple application of the triangle inequality shows that if $a \in A$ is close to $Z(A)$ then $a$ almosts commutes with elements of the unit ball of $A$. The extent to which the converse holds is a fundamental question in the study of non-commutativity and has been previously investigated for various classes of Banach algebras, including von Neumann algebras and other C*-algebras. In this context, $K(A)$ is defined to be the smallest number in $[0, \infty]$ such that $$ {\rm distance}(a, Z(A)) \leq K(A) \Vert {\rm ad}(a) \Vert $$ for all $a \in A$, where ${\rm ad}(a)$ is the inner derivation $x \to xa-ax$ for $(x \in A)$. For a C*-algebra $A$, the constant $K_s(A)$ is defined similarly, restricting to self-adjoint $a \in A$.

For a unital C*-algebra $A$, it is already known that the only possible values for $K_s(A)$ are $ \frac{1}{2}, 1, \frac{3}{2}, \ldots, \infty $, that $K(A) \in \{\frac{1}{2}, \frac{1}{\sqrt{3}}\} \cup [1, \infty]$ and that these cases are distinguished by purely topological conditions on the primitive ideal space ${\rm Prim}(A)$. In this paper, further progress is made on the question of possible values for $K(A)$ in $[1, \infty]$. It is shown that $ K(A) \leq \frac{2}{\sqrt{3}}K_s(A)$ and that either $ K(A) \leq \frac{4}{\sqrt{15}} \sim 1.033$ or $ K(A) \geq \frac{1}{2} + \frac{1}{\sqrt{3}} \sim 1.077$. The values $\frac{4}{\sqrt{15}}$ and $\frac{1}{2} + \frac{1}{\sqrt{3}}$ are attained and each of them arises as the maximal bounding radius of a constrained subset of the plane which is obtained from the structure of ${\rm Prim}(A)$.

It is also shown that if $A$ has trivial centre then $K(A)$ is further restricted: $$ K(A) \in \{\frac{1}{2}, 1, \frac{1}{2} + \frac{1}{\sqrt{3}}\} \cup [\frac{3}{2}, \infty].$$ Again, the various cases are distinguished by purely topological conditions on ${\rm Prim}(A)$, some of which are expressed in terms of primal ideals.

Xiong proved that if f: I → I is any map of the unit interval I, then the depth of the centre of f is at most 2, and Ye proved that for any map f: T → T of a finite tree T, the depth of the centre of f is at most 3. It is natural to ask whether the result can be dendrites. In this note, we show that there is dendrite D such that for any countable ordinal number λ there is a map f: D →D such that the depth of centre of f is λ. As a corollary, we show that for any countable ordinal number λ there is a map (respectively a homeomorphism) f of a 2-dimensional ball B2 (respectively a 3-dimensional ball B3) such that the depth of centre of f is λ.