appear (the uj run over an orthonormal basis of automorphic Laplaceeigenforms), so our formula (Theorem 1) is a duality between such integrals and certain geodesic integrals of u. New integral transformations are involved depending on the Laplace-eigenvalue of u. We invert these integral transformations in Section 5, Theorem 2. We develop the formula for finite volume Fuchsian groups, so (as in...

The main goal of this paper is to present an alternative, real variable proof of the T(1)-Theorem for the Cauchy Integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of the T(1)-Theorem. An example shows that the L 1-BMO estimate for the Cauchy Integral does not follow from L 2 boundedness when the underlying measur...

We prove analytically a connection between the generalized Molchan-Golosov integral transform (see [4], Theorem 5.1) and the generalized Mandelbrot-VanNess integral transform (see [8], Theorem 1.1) of fractional Brownian motion (fBm). The former changes fBm of arbitrary Hurst index K into fBm of index H by integrating over [0, t], whereas the latter requires integration over (−∞, t] for t > 0. ...

This paper deals with a nonlinear fractional differential equation with integral boundary condition of the following form: { D t x(t) = f(t, x(t), D β t x(t)), t ∈ (0, 1), x(0) = 0, x(1) = ∫ 1 0 g(s)x(s)ds, where 1 < α ≤ 2, 0 < β < 1. Our results are based on the Schauder fixed point theorem and the Banach contraction principle. Keywords—Fractional differential equation; Integral boundary condi...

The most general form of a linear continuous functional operation on a function f(x) continuous on (a, b) has been expressed by F. Rieszf as a Stieltjes integral of f(x) with respect to a function of bounded variation. By virtue of the extension theorem of Hahn,$ this result can be extended to the case where f(x) has discontinuities of the first kind on (a, b). In establishing this more general...

This paper studies Lebesgue integral of a fuzzy closed set-valued stochastic process with respect to the time t. Firstly, a progressively measurable fuzzy closed set-valued stochastic process is discussed and an almost everywhere problem in the former Aumann type Lebesgue integral of the level-set process is pointed out. Secondly, a new definition of the Lebesgue integral by decomposable closur...

In a series of papers, we proved theorems characterizing the value function in exit time optimal control as the unique viscosity solution of the corresponding Bellman equation that satisfies appropriate side conditions. The results applied to problems which satisfy a positivity condition on the integral of the Lagrangian. This positive integral condition assigned a positive cost to remaining ou...

Following Ohtsuki, Garoufalidis, and Habegger we provide an elementary introduction to nite type invariants of integral homology spheres, culminating with a proof of the upper bound for the magnitude of the space I of such invariants in terms of the space A(;) of oriented trivalent graphs modulo the so-called AS and IHX relations. We raise the issue of \the fundamental theorem" for nite type in...

1. Historical survey 2. Convergence of Sequences 2.1 Definition of Convergence 2.2. The Basic Property of Real Numbers. 2.3. Real Line 3. Continuous Functions 3.1 Continuous Functions and Their Limits 3.2 Properties of Continuous Functions. 3.2.1 The Intermediate Value Theorem 3.2.2. Maxima and Minima of Continuous Functions 3.3. The graph of a function 4. Differential Calculus 4.1 Derivative 4...

The purpose of this paper is to extend and clarify the methods of the papers [G1] and [G2] by using the methods of [G3], incorporating ideas from Carnot-Caratheodory geometry such as those of the fundamental paper [NSW]. We will thereby prove an analogue for singular Radon transforms to the T (1) theorem of David and Journe [DJ]. As in [G1] and [G2], we will associate a singular integral operat...