1. We consider differential topology to be the study of differentiable manifolds and differentiable maps. Then, naturally, manifolds are considered equivalent if they are diffeomorphic, i.e., there exists a differentiable map from one to the other with a differentiable inverse. For convenience, differentiable means C; in the problems we consider, C' would serve as well. The notions of different...

In this paper, within the scope of local fractional derivative theory, bidirectional propagation system equations are researched. Compared with unidirectional nonlinear waves in a pipeline, studied paper can better describe channel. This study is groundbreaking and offers more possibilities for simulation pipeline. The exact traveling wave solutions non-differentiable type defined on Cantor set...

Complex computer simulators are increasingly used across fields of science as generative models tying parameters of an underlying theory to experimental observations. Inference in this setup is often difficult, as simulators rarely admit a tractable density or likelihood function. We introduce Adversarial Variational Optimization (AVO), a likelihood-free inference algorithm for fitting a non-di...

In the present use of fuzzy logic systems tuning of their parameters has become an important issue. Many techniques, mainly based on the application of gradient descent, have been applied to this task generally in order to minimize a quadratic error function. The class of fuzzy logic systems using piecewise linear membership functions (e.g., triangular or trapezoidal) and/or minimum or maximum ...

The theory of scale relativity consists of developing the consequences of the withdrawal of the hypothesis of space-time differentiability. Spacetime acquires a fractal geometry, namely, it becomes explicitly dependent on the observation scale. The space-time resolutions are redefined as characterizing the state of scale of the reference system, then we set the principle of scale relativity, ac...

The theory of derivative of noninteger order goes back to Leibniz, Liouville, Riemann, Grunwald and Letnikov. Derivatives of fractional order have found many applications in recent studies in mechanics, physics, economics, medicine.Classes of fractional differentiable systems have studied in [10], [4]. In the first section the fractional tangent bundle to a differentiable manifold is defined, u...

In the recent paper Communications in Nonlinear Science and Numerical Simulation. Vol.18. No.11. (2013) 2945-2948, it was demonstrated that a violation of the Leibniz rule is a characteristic property of derivatives of non-integer orders. It was proved that all fractional derivatives Dα, which satisfy the Leibniz rule Dα(fg) = (Dαf) g+ f (Dαg), should have the integer order α = 1, i.e. fraction...

‎In this paper‎, ‎we shall establish some extended Simpson-type inequalities‎ ‎for differentiable convex functions and differentiable concave functions‎ ‎which are connected with Hermite-Hadamard inequality‎. ‎Some error estimates‎ ‎for the midpoint‎, ‎trapezoidal and Simpson formula are also given‎.