In this paper, we present two proofs of the reciprocal butterfly theorem. The statement of the butterfly theorem is: Let us consider a chord PQ of midpoint M in the circle Ω(O). Through M, two other chords AB and CD are drawn, such that A and C are on the same side of PQ. We denote by X and U the intersection of AD respectively CB with PQ. Consequently, XM = YM. For the proof of this theorem, see [1]. The (...) reciprocal of the butterfly theorem has the following statement: In the circle Ω(O), let us consider the chords PQ, AB and CD which are concurrent in the point M≠O, such as the points A and C are on the same side of the line PQ. Let X and Y respectively be the intersections of the chord PQ with AD and BC respectively. If XM = YM, then M is the middle of the chord PQ. (shrink)

The second of these lines seems to be the result of an interpolation designed to spell out the implicit sense of the first. In 241–6 Ion has expressed amazement that Creusa should be weeping at the sight of Apollo's sanctuary, a sight which brings other visitors joy. She prefaces her explanation of this with an assurance which in its transmitted form is elegantly translated by Grégoire: ‘Il n'est point discourtois de ta part, étranger, de marquer ta surprise au sujet de (...) mes pleurs.’ But there are reasons for doubting I the authenticity of line 248: Line 247 is self-sufficient, as is shown by IA 1402: τ μν σόν, νενι, γενναίως χει. This also gives the closest parallel for the use of τ σόν in reference to the attitude which a previous speaker's words have just expressed. Other instances of τ σόν, tout court, are helpfully grouped in Allen and Italie's Concordance to Euripides, s.v. σός. τ σόν…[adverbial phrase] χει occurs also in Hek. 1195 and Med. 312; cf. HF 165, Hel. 893, Pho. 995 with τομόν. These make it unlikely that τ χόν in our passage is to be taken adverbially, as perhaps Grégoire intended with ‘de ta part’, rather than as subject of χει. Tro. 82 might be adduced for the alternative , but there is no reason for Creusa to be saying emphatically ‘as far as you are concerned’. (shrink)

Relativistic mean field theory with mesons σ, ω, π and ρ mediating interactions and nucleons as basic fermions has been very successful in describing nuclear matter and finite nuclei. However, in heavy-ion collisions, where the c. m. energy of two colliding nucleons will be in the hundreds of GeV region, nucleons are not expected to behave as point-like particles. Analyses of elastic pp and ¯pp scattering data in the relevant c. m. energy range show that the nucleon is a composite (...) object—a topological soliton or Skyrmion embedded in a condensed quark-antiquark ground state. Against this backdrop, we formulate an effective field theory model of nuclear matter based on the gauged linear σ-model where quarks are the basic fermions, but the mesons still mediate the interactions. The model describes the nucleon as a Skyrmion and produces a q¯q ground state analogous to a superconducting ground state. Quarks are quasi-particles in this ground state. When the temperature exceeds a critical value, the scalar field in the ground state vanishes, quarks become massless, and a chiral phase transition occurs leading to chiral symmetry restoration. We explore the possibility of a first order phase transition in this model by introducing suitable self-interactions of the scalar field. Internal structures of the Skyrmions are ignored, and they are treated as point-like fermions. (shrink)

Ion channels are transport proteins present in the lipid bilayers of biological membranes. They are involved in many physiological processes, such as the generation of nerve impulses, hormonal secretion, and heartbeat. Conformational changes in the ion channel-forming protein allow the opening or closing of pores to control the ionic flux through the cell membranes. The opening and closing of the ion channel have been classically treated as a random kinetic process, known as a Markov process. Here the time the channel (...) remains in a given state is assumed to be independent of the condition it had in the previous state. More recently, however, several studies have shown that this process is not random but a deterministic one, where both the open and closed dwell-times and the ionic current flowing through the channel are history-dependent. This property is called long memory or long-range correlation. However, there is still much controversy regarding how this memory originates, which region of the channel is responsible for this property, and which models could best reproduce the memory effect. In this article, we provide a review of what is, where it is, its possible origin, and the mathematical methods used to analyze the long-term memory present in the kinetic process of ion channels. (shrink)

ῥαΨῳδ⋯σαι λ⋯γεται καí τó φλυαρ⋯σαι, τò áπλ⋯ς †λαβεíν κπ´ παγγεîλαι χωρíς †ἒργου τινός. This is how W. C. Greene prints the last sentence of the Schol. ad Ion 530a ῥαψδν, which is repeated ad Rep. 373b and in Photius, Suda, Et. Magn., and Lex. Bekk. s.v.αφῳδοί.