Whitney's type theorems for infinite dimensional spaces

It is proved that for any $f$ is an element of $C^k(L,R)$, where k is a natural number and L is a closed linear subspace of a nuclear Frechet space $X$, the function $f$ can be extended to a function of class $C^{k-1}$ defined on the entire space $X$. It is also proved that for any $f$ is an element of $C^k(L, R)$, where $k$ is a natural number of infinity and L is a closed linear subspace of a dual $X$ of a nuclear Frechet space, the function $f$ can be extended to a function of class $C^k$ defined on the entire space $X$. In addition, it is proved that under these conditions, the existence of a linear extension operator is equivalent to the complementability of the subspace.

An infinite-dimensional separable pre-Hilbert space non-homeomorphic to any of its closed hyperplanes

An example is constructed of an infinite-dimensional separable pre-Hilbert space non-homeomorphic to any of its closed hyperplanes.

Counterexample to Peano's theorem in infinite-dimensional F '-spaces

Let $E$ be a nonnormable Frechet space, and let $E'$ be the space of all continuous linear functionals on $E$ in the strong topology. A continuous mapping $f : E' \to E'$ such that for any $t_0\in R$ and $x_0\in E'$, the Cauchy problem $\dot x= f(x)$, x(t_0) = x_0$ has no solutions is constructed.

Simple heuristics: From one infinite regress to another?

Gigerenzer, Todd, and the ABC Research Group argue that optimisation under constraints leads to an infinite regress due to decisions about how much information to consider when deciding. In certain cases, however, their fast and frugal heuristics lead instead to an endless series of decisions about how best to decide.

Jordan isomorphism of purely infinite C*-algebras

We prove that every unital bounded linear mapping from a unital purely infinite C*-algebra of real rank zero into a unital Banach algebra which preserves elements of square zero is a Jordan homomorphism. This entails a description of unital surjective spectral isometries as the Jordan isomorphisms in this setting.

Robust Dynamic Pricing Over Infinite Horizon in The Presence of Model Uncertainty( 2* Journal)

This paper studies a problem of dynamic pricing faced by a retailer with limited inventory, uncertain about the demand rate model, aiming to maximize expected discounted revenue over an infinite time horizon. The retailer doubts his demand model which is generated by historical data and views it as an approximation. Uncertainty in the demand rate model is represented by a notion of generalized relative entropy process, and the robust pricing problem is formulated as a two-player zero-sum stochastic differential game. The pricing policy is obtained through the Hamilton-Jacobi-Isaacs (HJI) equation. The existence and uniqueness of the solution of the HJI equation is shown and a verification theorem is proved to show that the solution of the HJI equation is indeed the value function of the pricing problem. The results are illustrated by an example with exponential nominal demand rate.

Infinite Dimensional Banach spaces of functions with nonlinear properties

The aim of this paper is to show that there exist infinite dimensional Banach spaces of functions that, except for 0, satisfy properties that apparently should be destroyed by the linear combination of two of them. Three of these spaces are: a Banach space of differentiable functions on Rn failing the Denjoy-Clarkson property; a Banach space of non Riemann integrable bounded functions, but with antiderivative at each point of an interval; a Banach space of infinitely differentiable functions that vanish at infinity and are not the Fourier transform of any Lebesgue integrable function.

Organometallic polymers: An infinite organoplatinum chain in the solid state formed by (C CH center dot center dot center dot ClPt)hydrogen bonds

The new platinum complex [PtCl[C6H2(CH(2)NMe(2))(2) -2,6-(C=CH)-4)] exhibits a polymeric linear -C=CH ... ClPt-hydrogen-bonded structure in the solid state.

Optimal proportional-integral-derivative control of shape memory alloy actuators using genetic algorithm and the Preisach model

Shape memory alloy (SMA) actuators, which have the ability to return to a predetermined shape when heated, have many potential applications in aeronautics, surgical tools, robotics, and so on. Although the number of applications is increasing, there has been limited success in precise motion control owing to the hysteresis effect of these smart actuators. The present paper proposes an optimization of the proportional-integral-derivative (PID) control method for SMA actuators by using genetic algorithm and the Preisach hysteresis model.

A comparison of stress intensity factors through crack closure integral and other approaches using eXtended element-free Galerkin method

In this paper, a new approach for extracting stress intensity factors (SIFs) by the extended element-free Galerkin method, through a crack closure integral (CCI) scheme, is proposed. The CCI calculation is used in conjunction with a local smoothing technique to improve the accuracy of the computed SIFs in a number of case studies of linear elastic fracture mechanics. The cases involve problems of mixed-mode, curved crack and thermo-mechanical loading. The SIFs by CCI, displacement and stress methods are compared with those based on the M-integral technique reported in the literature. The proposed CCI method involves very simple relations, and still gives good accuracy. The convergence of the results is also examined.