Qualitative changes in sound intensities frequently occur in musical signals and Hopf bifurcation control is the most natural mathematical approach to study them. We propose a natural formulation for ...investigating and generating varieties of pitch simultaneity and sound intensity using Eulerian flows with n-tuple Hopf singularity for a sufficiently large n. Chord leaves and chord limit sets are defined as two critical flow-invariant manifolds. The reduction of flows on chord leaves facilitate the study and bifurcation control through a generic codimension-three (a three-parametric family) scalar bifurcation problem. There is a one-to-one correspondence between bifurcations of positive equilibria in the scalar equation and bifurcations of invariant chord limit sets. An ordered set of double saddle-node and pitchfork bifurcation control of subcritical and supercritical types can describe and generate the sound intensity bifurcations of a sheet music. These scalar bifurcations are interpreted in terms of appearances and disappearances of various chord leaf-stable and unstable flow-invariant chord limit sets. We apply our proposed method to the first three bars of a sheet music. This is implemented in Matlab so that one can hear the smooth dynamic effect of bifurcations using a computer through a classical sheet music. Then, one can hear how pitchfork (subcritical and supercritical) and double fold-bifurcations of invariant hypertori sound in music.
Chord measures are newly discovered translation-invariant geometric measures of convex bodies in
, in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals ...of convex bodies and random lines. Prescribing the
chord measures is called the
chord Minkowski problem in the
Brunn-Minkowski theory, which includes the
Minkowski problem as a special case. This article solves the
chord Minkowski problem when
and the symmetric case of
This paper will evaluate chord mu and alpha length in patients with Fuchs endothelial corneal dystrophy (FECD) and its changes following Descemet membrane endothelial keratoplasty (DMEK). Patients ...with FECD that underwent DMEK surgery were included in this retrospective study. Scheimpflug Tomography was carried out in order to calculate chord mu and chord alpha lengths prior to surgery and at 3 and 12 months postoperative. This study included 27 eyes from 27 patients. Significant changes in chord mu were observed within the first three months (from 0.47 ± 0.32 to 0.29 ± 0.21 mm, p < 0.01) and remained stable 12 months postoperative (0.30 ± 0.21 mm, p > 0.05). However, chord alpha remained stable throughout the 12 months post surgery (from 0.53 ± 0.19 to 0.49 ± 0.14 mm, p > 0.05). In addition to the pupillary center distance from the corneal center (from 0.35 ± 0.25 to 0.34 ± 0.20 mm, p > 0.05) also remain stable. In FECD patients undergoing DMEK surgery, chord mu length decreased, and chord alpha length remained stable after 12 months of follow-up.
The two volumes presented here (A. E. Colla, Trattato di armonia moderna e contemporanea and M. Quagliarini, Costruire la musica) are part of a current trend of renewal in harmony and composition ...treatises in Italy. Colla addresses post-tonal harmony in an innovative way by showing harmonic-compositional techniques developed from the 20th century to the present day; specifically, he devotes his two volumes to "harmonic partials" (secondary frequencies emitted together with a main frequency) and various forms of symmetry. In his book, Quagliarini introduces a compositional method based on the construction of chords and how to tie them together to build coherent structures of increasing size. Both volumes relate concepts that are pecualir to musical syntax to extra-musical elements.
A chord diagram refers to a set of chords with distinct endpoints on a circle. The intersection graph of a chord diagram C is defined by substituting the chords of C with vertices and by adding edges ...between two vertices whenever the corresponding two chords cross each other. Let Cn and Gn denote the chord diagram chosen uniformly at random from all chord diagrams with n chords and the corresponding intersection graph, respectively. We analyze Cn and Gn as n tends to infinity. In particular, we study the degree of a random vertex in Gn, the k-core of Gn, and the number of strong components of the directed graph obtained from Gn by orienting edges by flipping a fair coin for each edge. We also give two equivalent evolutions of a random chord diagram and show that, with probability approaching 1, a chord diagram produced after m steps of these evolutions becomes monolithic as m tends to infinity and stays monolithic afterward forever.
For a fixed convex body K in \mathbb{R}^n, the chord distribution on the affine Grassmann manifold of lines is introduced and then the entropy of chord distribution is defined. Using the integral ...geometric method, we establish that the entropy of chord distribution of a convex body K attains its minimum if and only if K is a ball.
Akkord Kühn, Clemens
MGG Online,
11/2016
Reference
I. Begriff
Im musiktheoretischen Sinne meint der Begriff Akkord, elementar verstanden, das Zusammenklingen verschiedener Töne. Daß dazu drei oder mehr Töne gehören, gilt für die Dur-Moll-tonale Mus...
This study focuses on the cold-formed stainless steel square hollow section (SHS) tubular T-joints prone to failure by chord sidewall yielding, crushing or instability under axial compression in ...brace member. An experimental investigation was conducted to understand the joint behaviour subjected to chord axial loading during the brace compression. The dimensions of the tubular sections were 80x80x2mm and 40x40x2mm, corresponding to 2γ values of 20 and 40, respectively. Based on the obtained experimental results, it was observed that the T-joint resistances decrease when subjected to chord compressive load, especially in the cases with elevated values for the chord sidewall slenderness (2γ). On the other hand, if a chord tension force is considered, an increase in the T-joint resistance is observed. With the results obtained from the tests, the design standards EN 1993-1-8 in association with EN 1993-1-4, NBR 16239, CIDECT, ISO 14346 and AISC 360-22 were assessed, as well as the predictions proposed by other authors from the literature. Finally, it was found that design codes are very conservative, except for AISC 360-22 26, which provided unsafe predictions. On the other hand, equations proposed by other authors from the literature presented better results.
•Experimental investigation of cold-formed stainless steel tubular T-joints with loaded chord;•Ten cold-formed stainless steel tubular T-joints with β=1 and chord axially loaded under tension and compression;•Design standards EN1993-1-8, EN1993-1-4, NBR16239, ISO14346, AISC360-22 were assessed to predict the joint's strength;•Other design procedures were also investigated if they can predict the stainless steel tubular T-joints strength;
This study aimed to investigate the effect of polymer type on solidification rate of PLGA polymeric microparticles and particle size/distribution of the emulsion droplets/hardened PLGA polymeric ...microparticles during solvent evaporation process using FBRM (Focussed Beam Reflectance Measurement).
PLGA polymeric microparticles were prepared by an O/W solvent evaporation method using various PLGA polymers, including PLGA Resomer® RG503H, RG502H and RG752H. The particle size mean, chord length distribution (CLD), and chord count of the emulsion droplets/hardened microparticles were monitored by FBRM. The morphology of polymeric microparticles were characterised by optical microscopy and scanning electron microscopy (SEM).
The transformation of the emulsion droplets into solid microparticles occurred within the first 30 (± 1.04), 34 (± 1.15) and 37 (± 0.82) min and square weighted mean chord lengths are 64.08 (± 3.18), 52.36 (± 5.27) and 42.18 (± 4.61) µm when PLGA Resomer® RG503H, RG502H and RG752H were used respectively. Larger square weighted mean chord length of PLGA polymeric microparticles gave lower chord counts. PLGA RG752H microparticles gave smallest square weighted mean chord length and the chord counts was the highest. The CLDs measured by FBRM showed that a larger particle size mean gave longer CLD and a lower peak of particle number. SEM data revealed that the morphology of microparticles was influenced by type and physical properties of polymer.
FBRM can be employed for online monitoring of the shift in the microparticle CLD and detect transformation of the emulsion droplets into solid microparticles during the solvent evaporation process. The microparticle CLD and transformation process were strongly influenced by polymer type.
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