Approximate Generalized Conditional Symmetries for Perturbed Evolution Equations

The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via their A GCSs is illustrated with examples.

Artificial boundary conditions for Euler–Bernoulli beam equation

In a semi-discretized Euler-Bernoulli beam equa- tion, the non-nearest neighboring interaction and large span of temporal scales for wave propagations pose challenges to the effectiveness and stability for artificial boundary treat- ments. With the discrete equation regarded as an atomic lattice with a three-atom potential, two accurate artificial boundary conditions are first derived here. Reflection co- efficient and numerical tests illustrate the capability of the proposed methods. In particular, the time history treatment gives an exact boundary condition, yet with sensitivity to nu- merical implementations. The ALEX （almost EXact） bound- ary condition is numerically more effective.

Effects of metronomic sounds on a self-paced tapping task in budgerigars and humans

The origin of rhythmic synchronization or entrainment to a musical beat in animals has been widely discussed. Parrots are suitable animals to exami ne the relati on ship betwee n the capability of vocal learning and spontaneous rhythmic synchronization. In this study, budgerigars Melopsittacus undulatus learned to tap (peck) 2 keys alternately at a self-paced rate. Then, the metronomic sounds were played in the background during test sessions while the birds were performing the key pecking task, although they were not required to synchronize tap timing with the metronome. We found modest but significant effects of the metronome rhythms on the tap timing in some subjects. We also tested humans Homo sapiens using almost the same method. In contrast to the birds, a number of huma n subjects synchr on ized tap timing to the on set of the metronome without verbal or documented instructions. However, we failed to find an effect of the metronome on self-paced tap timing in some human subjects, although they were capable of rhythmic synchronization. This is the first report describing the effects of metronomic sounds on self-paced tapping in nonhuman vocal learners. This study introduces a new method that can be used in future research comparing birds that differ in vocal learning capacities, social structure, age, sex, hormonal status, and so on as part of exami nations of the evolutionary foun dati ons of beat processing.

Approximate Self-Similarity of a Class of Cookie-Cutter-Like Sets

According to the bounded variation theory and the bounded distortion property of cookie-cutter-like （CCL） sets, the approximate self-similarity of cookie-cutter-like sets satisfying certain conditions is studied. Based on the mean value theorem, it is proved that a class of special cookie-cutter-like sets is approximately self-similar. The results obtained in this paper extend the corresponding results that have already existed. K