Harnack Inequalities on Manifolds with Boundary and Applications

On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$ the Neumann heat kernel w.r.t. a volume type measure $\mu$ and for $K$ a constant, the curvature condition \Ric-\nn Z\ge K together with the convexity of the boundary is equivalent to the heat kernel entropy inequality \int_M p_t(x,z)\log \ff{p_t(x,z)}{p_t(y,z)} \mu(\d z)\le \ff{K\rr(x,y)^2}{2(\e^{2Kt}-1)}, t>0, x,y\in M, where \rr is the Riemannian distance. The main result is partly extended to manifolds with non-convex boundary and applied to derive the HWI inequality.Comment: 24 page

Pengaruh Servant Leadership terhadap Nilai Kinerja Guru

Abstract: Servant leadership is a form of leadership which is oriented to serve the needs of subordinates in order to improve their pay scale. The term of servant leadership was first introduced by Robert K. Greenleaf in 1970. Servant leadership can be implemented in educational institution. School as one of those intuition that runs teaching and learning process is highly determined by the practices of servant leadership performed by the principal. This research aims at investigating the effect of servant leadership on teachers and employees at Sekolah Islam Athirah Makassar. This is a quantitative research using regression analysis in analyzing the data. The variables of this research are servant leadership (X) and teachers and employees grade performance (Y). The total participant is 90 which was taken randomly from each unit including primary school (SD), junior high school (SMP), and senior high school (SMA). The result of regression analysis showed that there was significant effect by0,02 1.991) which means Ho is rejected and there is effect of servant leadership on teachers and employees’ performance grade. The regression equation obtained is Y = 76,058 + 0,187 X.  Abstrak: Servant leadership atau kepemimpinan yang melayani adalah konsep kepemimpinan yang pertama kali diperkenalkan oleh Greenleaf (1970) yang dimaksudkan sebagai bentuk kepemimpinan yang berorientasi pada sifat melayani kepentingan bawahan agar mereka menjadi lebih sejahtera. Praktik kepemimpinan yang melayani (servant leadership) juga dapat dilaksanakan pada institusi pendidikan. Sekolah sebagai institusi pendidikan yang menjalankan proses pendidikan dan pembelajaran bagi peserta didik sangat ditentukan oleh praktik kepemimpinan yang dijalankan oleh kepala sekolah. Penelitian ini bertujuan menganalisis pengaruh kepemimpinan yang melayani (servant leadership) terhadap kinerja guru dan karyawan di Sekolah Islam Athirah Makassar. Metode penelitian kuantitatif dengan menggunakan analisis regresi, variabel penelitian adalah servant leadership (X) dan nilai kinerja guru dan karyawan (Y), jumlah sampel sebanyak 90 orang guru dari unit SD, SMP dan SMA. Bedasarakan analisis regresi diperoleh nilai signifikan 0,02 ttabel (2,297 > 1,991) sehingga H0 ditolak artinya ada pengaruh servant leadership terhadap nilai kinerja guru dan karyawan. Dengan Persamaan Regresi yang terbentuk adalah Y = 76,058 + 0,187 X.

Rate of Converrgence for ergodic continuous Markov processes : Lyapunov versus Poincare

We study the relationship between two classical approaches for quantitative ergodic properties : the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincar\'e type). We show that they can be linked through new inequalities (Lyapunov-Poincar\'e inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker-Planck equation recently studied by H\'erau-Nier, Helffer-Nier and Villani is in particular discussed in the final section

Subelliptic Li-Yau estimates on three dimensional model spaces

We describe three elementary models in three dimensional subelliptic geometry which correspond to the three models of the Riemannian geometry (spheres, Euclidean spaces and Hyperbolic spaces) which are respectively the SU(2), Heisenberg and SL(2) groups. On those models, we prove parabolic Li-Yau inequalities on positive solutions of the heat equation. We use for that the $\Gamma_{2}$ techniques that we adapt to those elementary model spaces. The important feature developed here is that although the usual notion of Ricci curvature is meaningless (or more precisely leads to bounds of the form $-\infty$ for the Ricci curvature), we describe a parameter $\rho$ which plays the same role as the lower bound on the Ricci curvature, and from which one deduces the same kind of results as one does in Riemannian geometry, like heat kernel upper bounds, Sobolev inequalities and diameter estimates

Weighted Nash Inequalities

Nash or Sobolev inequalities are known to be equivalent to ultracontractive properties of Markov semigroups, hence to uniform bounds on their kernel densities. In this work we present a simple and extremely general method, based on weighted Nash inequalities, to obtain non-uniform bounds on the kernel densities. Such bounds imply a control on the trace or the Hilbert-Schmidt norm of the heat kernels. We illustrate the method on the heat kernel on \dR naturally associated with the measure with density $C_a\exp(-|x|^a)$, with $