Exact Lyapunov exponents of the generalized Boole transformations

The generalized Boole transformations have rich behavior ranging from the \textit{mixing} phase with the Cauchy invariant measure to the \textit{dissipative} phase through the \textit{infinite ergodic} phase with the Lebesgue measure. In this Letter, by giving the proof of mixing property for $0<\alpha<1$ we show an \textit{analytic} formula of the Lyapunov exponents $\lambda$ which are explicitly parameterized in terms of the parameter $\alpha$ of the generalized Boole transformations for the whole region $\alpha>0$ and bridge those three phase \textit{continuously}. We found the different scale behavior of the Lyapunov exponent near $\alpha=1$ using analytic formula with the parameter $\alpha$. In particular, for $0<\alpha<1$, we then prove an existence of extremely sensitive dependency of Lyapunov exponents, where the absolute values of the derivative of Lyapunov exponents with respect to the parameter $\alpha$ diverge to infinity in the limit of $\alpha\to 0$, and $\alpha \to 1$. This result shows the computational complexity on the numerical simulations of the Lyapunov exponents near $\alpha \simeq$ 0, 1.Comment: 11 pages, 5 figures, 2 table

Proof of Irreversibility for Systems with Time Reversal Symmetry

The positivity of entropy change rate was rigorously proven in a certain microscopic system associated with a Hamiltonian using only information about the microscopic system. This microscopic system obeys an equation with time-reversal symmetry. We proved that a symplectic map with time-reversal symmetry has the same ergodic properties as an Anosov diffeomorphism. This result guarantees that any initial density function defined except for a zero volume set converges to the unique invariant density (uniform distribution) in the sense of mixing. Moreover, we proved that the Kolmogorov--Sinai entropy, which is the entropy change rate, is \textit{positive} using the property of Anosov diffeomorphisms. In addition, we discovered a mathematical structure connecting the convergence of initial density functions to the uniform distribution and positivity of the entropy change rate (Anosov diffeomorphism)

Representing Difficult People

[Excerpt] Few stewards would argue that most of their union work flows directly from problems with management. Contract misinterpretations and outright violations, thoughtless supervision, paperwork foulups and a million other things go wrong all the time, adding up to a real handful for stewards. That’s why it can be such a frustration when some of your most difficult problems come not from management, but from your own ranks

PENGARUH PROBIOTIK PADA DIARE AKUT : PENELITIAN DENGAN 3 PREPARAT PROBIOTIK

ABSTRAK Latar belakang. Probiotik diketahui memiliki efek yang menguntungkan dalam pengobatan diare akut pada anak. Probiotik mempengaruhi frekuensi dan durasi diare dengan meningkatkan respon imun, produksi substansi antimikroba dan menghambat pertumbuhan kuman patogen penyebab diare. Probiotik dengan strain spesifik efektif menurunkan frekuensi dan durasi diare. Tujuan. Mengetahui efektifitas suplementasi probiotik tunggal maupun kombinasi pada anak dengan diare akut. Metode. Uji klinis acak tersamar buta ganda terhadap pasien diare akut usia 6-24 bulan dengan diare akut di RS Dr. Kariadi Semarang periode Juli 2010 - Februari 2011. Subyek dibagi dalam 3 kelompok perlakuan (L.reuteri; L.acidophilus dan LGG; L.acidophilus, S.faecium and B.longum) dan kontrol. Probiotik diberikan selama 5 hari. Setiap kelompok mendapat terapi standar berupa rehidrasi dan dietetik. Hasil. Dari 84 anak yang masuk dalam penelitian, rerata durasi diare lebih pendek pada kelompok L.reuteri (37,4 ± 14,4 jam) dan L.acidophilus-LGG (38,6 ± 19,6 jam) dibanding kelompok 3 strain probiotik dan kontrol. Didapatkan perbedaan yang bermakna pada penurunan durasi diare antar kelompok (p=0,002). Rerata frekuensi diare menurun pada kelompok L.reuteri (5,6 ± 2,9 kali) dan L.acidophilus-LGG (6,9 ± 8,4 kali) dibanding dengan kelompok 3 strain probiotik dan kontrol. Didapatkan perbedaan yang bermakna pada penurunan frekuensi diare antar kelompok (p=0,02). Kesimpulan. Probiotik L. reuteri dan L.acidophilus-LGG efektif menurunkan durasi dan frekuensi diare dan dapat digunakan sebagai terapi tambahan pada anak dengan diare akut. Kata kunci: probiotik, diare akut, durasi, frekuensi