we shall give a lower estimate of harmonic‎ ‎functions of order greater than one in a half space‎, ‎which‎ ‎generalize the result obtained by b‎. ‎ya‎. ‎levin in a half plane‎.

We give estimates on the number ALH(x) of arithmetic lattices Γ of covolume at most x in a simple Lie group H . In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most x. Our main result is for the classical case H = PSL(2,R) where we show that lim x→∞ log ALH(x) x log x = 1 2π . The proofs use several different techniques: geometric (boundin...

We provide a sharp generalization to the nonautonomous case of the well-known Kobayashi estimate for proximal iterates associated with maximal monotone operators. We then derive a bound for the distance between a continuous-in-time trajectory, namely the solution to the differential inclusion ẋ+A(t)x 3 0, and the corresponding proximal iterations. We also establish continuity properties with re...

We show that for every n ≥ 4, 0 ≤ k ≤ n − 3, p ∈ (0, 3] and every origin-symmetric convex body K in Rn, the function ‖x‖−k 2 ‖x‖ −n+k+p K represents a positive definite distribution on Rn, where ‖·‖2 is the Euclidean norm and ‖·‖K is the Minkowski functional of K. We apply this fact to prove a result of Busemann-Petty type that the inequalities for the derivatives of order (n−4) at zero of X-ra...

* Correspondence: ksrim@sogang. ac.kr Department of Mathematics, Sogang University, Seoul 121-742, Korea Abstract In this article, first, we prove that weighted-norm inequalities for the M-harmonic conjugate operator on the unit sphere whenever the pair (u, v) of weights satisfies the Ap-condition, and uds, vds are doubling measures, where ds is the rotationinvariant positive Borel measure on t...

Current experimental data indicate that two unitarity triangles of the CKM quark mixing matrix V are almost the right triangles with α ≈ 90. We highlight a very suggestive parametrization of V and show that its CP-violating phase φ is nearly equal to α (i.e., φ − α ≈ 1.1). Both φ and α are stable against the renormalizaton-group evolution from the electroweak scale MZ to a superhigh energy scal...

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct ...

We study positive solutions to the heat equation on graphs. prove variants of Li-Yau gradient estimate and differential Harnack inequality. For some graphs, we can show estimates be sharp. establish new com- putation rules for operators discrete spaces introduce a re- laxation function that governs time dependency in estimate

We compute an asymptotic formula for the supremum of resolvent norm ||(ζ−T)−1|| over |ζ|≥1 and contractions T acting on n-dimensional Hilbert space, whose spectral radius does not exceed a given r∈(0,1). prove that this is achieved unit circle by analytic Toeplitz matrix.