A graph-theoretic approach to quasigroup cycle numbers

Norton and Stein associated a number with each idempotent quasigroup or diagonalized Latin square of given finite order n, showing that it is congruent mod 2 to the triangular number T(n). In this paper, we use a graph-theoretic approach to extend their invariant to an arbitrary finite quasigroup. We call it the cycle number, and identify it as the number of connected components in a certain graph, the cycle graph. The congruence obtained by Norton and Stein extends to the general case, giving a simplified proof (with topology replacing case analysis) of the well-known congruence restriction on the possible orders of general Schroeder quasigroups. Cycle numbers correlate nicely with algebraic properties of quasigroups. Certain well-known classes of quasigroups, such as Schroeder quasigroups and commutative Moufang loops, are shown to maximize the cycle number among all quasigroups belonging to a more general class.     

室内声学中散射研究进展

近20年,界面散射研究在声场预测、定量测量和产品研发等方面取得了重要进展,加深了我 们对散射及其作用的认识。人们可以定量地评估、设计和优化散射产品,用更加丰富的手法进行室内 音质设计。     

Acoustic design of small rectangular rooms: Normal frequency statistics

The subject of discrete modes in small rectangular rooms has been considered. A new procedure for selecting optimum geometric proportions of rooms has been proposed, taking into account the eigenfrequencies up to the Schroeder frequency and considering also the surface averaged sound absorption coefficient (α) of a given room. This new procedure leads to a series of plots describing the geometric proportions of small rectangular rooms corresponding to the smoothest frequency response for different absorption conditions. When taking α into account, the range of the acceptable dimension ratios X:Y has proved relatively wide, so the standard deviation calculated for the distances between subsequent modes does not exceed 1.5 (as in Bolt’s work). However, the range of the acceptable dimension ratios decreases with decreasing α and for mean absorption coefficient lower or equal 0.3 there are only a few of the ratios for which a uniform distribution of eigenmodes is obtained.     

Characterization and improvement of absorption and scattering by profiled architectural surfaces without specialized test facilities

A turntable apparatus and measurement procedure, based on ISO 17497-1, were developed for measuring the random-incidence absorption and scattering of surfaces, without specialized facilities. While not highly accurate, they proved adequate for determining the general characteristics of surfaces. Two original profiled wooden architectural surfaces, and a hard, flat reference surface, were characterized, and found to have low absorption and scattering. They were then modified for increased absorption by creating Helmholtz-resonator configurations, obtaining good performance at low- and mid-frequencies. They were also modified for increased scattering, in one case by creating a Schroeder diffuser. In the other case, the surface was integrated into a novel frame-array configuration, which scattered sound very well above 500 Hz.     

Theoretical analysis on sound absorption by pseudostochastic diffusors

I.IntroductionIn1979,M.R.S.h.o.der[1]pro-posedanewdesignofsounddiffusorwhoseperiodcomprisesNelements(slotsorwells,referedtoaschannelsinfol1owing)ofequalwidths.ThedepthsofchanneIsvaryaccordingtoapseu-dostochasticsequencewithinonepe-riod.Atypicalstructureofq1ladraticresiduediffusor(N=11)isi1IustratedinFig.1.BychoosingthedePtl1ofchannels,thescatteringcharacteristicsofthestructurecanbeoptirnizeds11chastodistributethescattereden(}rgyequalIyoveralloweddirections.In1992,K.Fuiwaraandothers['repo…