In this paper we study the injectivity radius based at a fixed point along Weil–Petersson geodesics. We show that square root of is 0.3884-Lipschitz on Teichmüller space endowed with metric. As an application reprove systole function uniformly Lipschitz metric, where constant can be chosen to 0.5492. Applications asymptotic geometry moduli Riemann surfaces for large genus will also discussed.

In this paper, a stability result on an inverse source scattering problem is shown for Helmholtz equation in multi-layered medium. The estimate consists of Lipschitz term low frequencies and logarithmically unstable high frequencies, based the Green's function problem. Moreover, logarithmic decreases as frequency grows, which indicates almost Lipschitz. This demonstrates that increasing problem...

We prove the following new characterization of C (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a C smooth (Lipschitz) bump function if and only if it has another C smooth (Lipschitz) bump function f such that f ′(x) 6= 0 for every point x in the interior of the support of f (that is, f does not satisfy Rolle’s theorem). Moreover, the support of this bump ca...

This paper provides novel analytic expressions for the incomplete Toronto function, TB(m,n, r), and the incomplete Lipschitz-Hankel Integrals of the modified Bessel function of the first kind, Iem,n(a, z). These expressions are expressed in closed-form and are valid for the case that n is an odd multiple of 1/2, i.e. n±0.5 ∈ N. Capitalizing on these, tight upper and lower bounds are subsequentl...

The pool adjacent violators (PAV) algorithm is an efficient technique for the class of isotonic regression problems with complete ordering. The algorithm yields a stepwise isotonic estimate which approximates the function and assigns maximum likelihood to the data. However, if one has reasons to believe that the data were generated by a continuous function, a smoother estimate may provide a bet...

This paper presents a Riemannian trust region algorithm for unconstrained optimization problems with locally Lipschitz objective functions defined on complete Riemannian manifolds. To this end we define a function Φ : TM → R on the tangent bundle TM , and at k-th iteration, using the restricted function Φ|TxkM where TxkM is the tangent space at xk, a local model function Qk that carries both fi...

The construction of a Lyapunov function characterizing the pullback attraction of a cocycle attractor of a nonautonomous discrete time dynamical system involving Lipschitz continuous mappings is presented.

Consider the problem of approximate function maximization: Given a continuous real function with domain [0, 1], what is its approximate maximum of absolute error 2−n? More precisely, we are interested in the complexity of approximating the real functional MAX : C[0,1] 3 f 7→max0≤x≤1 ∈ R up to a prescribed absolute error 2−n. Regarding the complexity, we examine quantitative and parameterized bo...

The existence of a locally Lipschitz continuous homogeneous Lyapunov function is proven for class asymptotically stable infinite dimensional systems with unbounded nonlinear operators.