In this paper, the idea of an objective scale axis enriching Nature with a new dimension is explicated and illustrated on
the problem of heat conduction. Physical description of Nature thus has to be formulated on individual scale levels parametrized
by points of the scale axis), space and time are intrinsic parameters of each level. A ‘global’space-time then becomes a useful
construction expressing a possibility of a scale-independent description. Consequently, the cases in which such a description
is not possible (e.g. the problem of thermal waves) might lead to a contradiction with the concept of a global space-time,
which may manifest itself by a presence of some fractal structures.
After three decades of our personal, publicly conducted discussion with Ernő Lendvai, in 1999 at a conference organized in memory of Bence Szabolcsi, I raised again my objections related his theories. Since my lecture was given in Hungarian, and its printed version was published in Hungarian language (Muzsika 2000, Bartók-analitika 2003), I feel necessary to present some of my objetions on an international forum as well, with particular aspect to the fact that in the Bartók literature - in spite of serious criticism (Petersen, Gillies) - several analysts employ up to now Lendvai's theories in a servile way. My objections are focussed upon four points. 1. The extension of Riemann's three-function theory to the twelvetone system is a theoretical arbitrariness and an impasse. 2. The axis interpretation of the tonalities - by identification of polar keys - is in flat contradiction with Bartók's tonal thinking. 3. The pentatony interpreted as a golden section system is very much doubtful according basic experiences of the ethnomusicology. 4. The typical Bartókian chord structures - named by Lendvai α, β etc. - are phenomenologically correct, but their interpretation by Fibonacci figures is arbitrary, because the actual intervals represent another ratios.
Let t be an infinite graph, let p be a double ray in t, and letd anddp denote the distance functions in t and in p, respectively. One calls p anaxis ifd(x,y)=dp(x,y) and aquasi-axis if lim infd(x,y)/dp(x,y)>0 asx, y range over the vertex set of p anddp(x,y)?8. The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if a is a translation whose directionD(a) is a thin end, then there exists an axis inD(a) andD(a-1) invariant under ar for somer not exceeding the maximum number of disjoint rays inD(a).The thinness ofD(a) is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.
diseases such as amyotrophic lateral sclerosis (ALS), multiple sclerosis (MS), AD, and PD, since the gut microbiome is facile and changed via diet and exercise [ 27 – 30 ]. A microbiota–gut–brain axis is a cross-talk between gut and microbiota, in which