Stochastic integration,trace and the skeleton of wiener functionals

It is shown that the notion of trace induced by a given complete orthonormal system relates the Skorohod integral with a corresponding Ogawa‐type integral evaluated with respect to the same orthonormal systems. Similarly the multiple Wiener‐Ito integral is shown to be related to a multiple Ogawa‐type integral induced by a complete orthonormal system via the Hu‐Meyer formula with suitably defined multiple traces. The notion of skeleton of a Wiener functional relative to a given orthonormal system is defined and yields what seems to be a “natural” extension of Wiener functionals to the Cameron Martin space and the Wiener processes with a different scale.     

Areas and the Fundamental Theorem of Calculus

We present a concise, yet self-contained module for teaching the notion of area and the Fundamental Theorem of Calculus for different groups of students. This module contains two different levels of rigour, depending on the class it used for. It also incorporates a technological component.     

杰出人物论微积分

摘编14位杰出人物对微积分的精辟论述。     

Transient transition behaviors of fractional-order simplest chaotic circuit with bi-stable locally-active memristor and its ARM-based implementation

This paper proposes a fractional-order simplest chaotic system using a bi-stable locally-active memristor. The characteristics of the memristor and transient transition behaviors of the proposed system are analyzed, and this circuit is implemented digitally using ARM-based MCU. Firstly, the mathematical model of the memristor is designed, which is nonvolatile, locally-active and bi-stable. Secondly, the asymptotical stability of the fractional-order memristive chaotic system is investigated and some sufficient conditions of the stability are obtained. Thirdly, complex dynamics of the novel system are analyzed using phase diagram, Lyapunov exponential spectrum, bifurcation diagram, basin of attractor, and coexisting bifurcation, coexisting attractors are observed. All of these results indicate that this simple system contains the abundant dynamic characteristics. Moreover, transient transition behaviors of the system are analyzed, and it is found that the behaviors of transient chaotic and transient period transition alternately occur. Finally, the hardware implementation of the fractional-order bi-stable locally-active memristive chaotic system using ARM-based STM32F750 is carried out to verify the numerical simulation results.     

Modeling and character analyzing of multiple fractional-order memcapacitors in parallel connection

Recently,the memory elements-based circuits have been addressed frequently in the nonlinear circuit theory due to their unique behaviors.Thus,the modeling and characterizing of the mem-elements become essential.In this paper,the analysis of the multiple fractional-order voltage-controlled memcapacitors model in parallel connection is studied.Firstly,two fractional-order memcapacitors are connected in parallel,the equivalent model is derived,and the characteristic of the equivalent memcapacitor is analyzed in positive or negative connection.Then a new understanding manner according to different rate factor K and fractional orderαis derived to explain the equivalent modeling structure conveniently.Additionally,the negative order appears,which is a consequence of the combination of memcapacitors in different directions.Meanwhile,the equivalent parallel memcapacitance has been drawn to determine that multiple fractional-order memcapacitors could be calculated as one composite memcapacitor.Thus,an arbitrary fractional-order equivalent memcapacitor could be constructed by multiple fractional-order memcapacitors.     

Local well-posedness for quasi-linear NLS with large Cauchy data on the circle

We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schrödinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of coordinates in order to transform the system into a paradifferential one with symbols which, at the positive order, are constant and purely imaginary. This allows to obtain a priori energy estimates on the Sobolev norms of the solutions.     

Malliavin and Dirichlet structures for independent random variables

On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron–Stein inequality can be interpreted as a Poincaré inequality or that the Hoeffding decomposition of $U$-statistics can be interpreted as an avatar of the Clark representation formula. Thanks to our framework, we obtain a bound for the distance between the distribution of any functional of independent variables and the Gaussian and Gamma distributions.