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.

On stratifiable locally convex spaces

In the present paper we prove several results on the stratifiability of locally convex spaces. In particular, we show that a free locally convex sum of an arbitrary set of stratifiable LCS is a stratifiable LCS, and that all locally convex F'-spaces whose bounded subsets are metrizable are stratifiable. Moreover, we prove that a strict inductive limit of metrizable LCS is stratifiable and establish the stratifiability of many important general and specific spaces used in functional analysis. We also construct some examples that clarify the relationship between the stratifiability and other properties.

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.

Some results on solvability of ordinary linear differential equations in locally convex spaces

Let $\Gamma$ be the class of sequentially complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem $\dot x = Ax$, $x(0) = x_0$ with respect to functions $x: R\to E$. It is proved that if $E\in \Gamma$, then $E\times R^A$ is-an-element-of $\Gamma$ for an arbitrary set $A$. It is also proved that a topological product of infinitely many infinite-dimensional Frechet spaces, each not isomorphic to $\omega$, does not belong to $\Gamma$.

Even order harmonic Series-L/Parallel-Tuned Class-E frequency multiplier

In this brief, we propose a new Class-E frequency multiplier based on the recently introduced Series-L/Parallel-Tuned Class-E amplifier. The proposed circuit produces even-order output harmonics. Unlike previously reported solutions the proposed circuit can operate under 50% duty ratio which minimizes the conduction losses. The circuit also offers the possibility for increased maximum operating frequency, reduced peak switch voltage, higher load resistance and inherent bond wire absorption; all potentially useful in monolithic microwave integrated circuit implementations. In addition, the circuit topology suggested large transistors with high output capacitances can be deployed. Theoretical design equations are given and the predictions made using these are shown to agree with harmonic balance circuit simulation results.

Inverse Class-E Amplifier With Transmission-Line Harmonic Suppression

This paper reports on the design methodology and experimental characterization of the inverse Class-E power amplifier. A demonstration amplifier with excellent second and third harmonic-suppression levels has been designed, constructed, and measured. The circuit fabricated using a 1.2-min gate-width GaAs MESFET is shown to be able to deliver 22-dBm output power at 2.3 GHz. The amplifier achieves a peak power-added efficiency of 64 % and drain efficiency of 69 %, and exhibits 11.6 dB power gain when operated from a 3-V supply voltage. Comparisons of simulated and measured results are given with good agreement between them being obtained. Experimental results are presented for the amplifier's response to Gaussian minimum shift keying modulation, where a peak error vector modulation value of 0.6% is measured.

2.4 GHz Class-E power amplifier with transmissionline harmonic terminations

The design procedure, fabrication and measurement of a Class-E power amplifier with excellent second- and third-harmonic suppression levels are presented. A simplified design technique offering compact physical layout is proposed. With a 1.2 mm gate-width GaAs MESFET as a switching device, the amplifier is capable of delivering 19.2 dBm output power at 2.41 GHz, achieves peak PAE of 60% and drain efficiency of 69%, and exhibits 9 dB power gain when operated from a 3 V DC supply voltage. When compared to the classical Class-E two-harmonic termination amplifier, the Class-E amplifier employing three-harmonic terminations has more than 10% higher drain efficiency and 23 dB better third-harmonic suppression level. Experimental results are presented and good agreement with simulation is obtained. Further, to verify the practical implementation in communication systems, the Bluetooth-standard GFSK modulated signal is applied to both two- and three-harmonic amplifiers. The measured RMS FSK deviation error and RMS magnitude error were, for the three-harmonic case, 1.01 kHz and 0.122%, respectively, and, for the two-harmonic case, 1.09 kHz and 0.133%. © 2007 The Institution of Engineering and Technology.

Protest practice and (tree) cultures of conflict: understanding the spaces of ‘tree maiming’ in eighteenth- and early nineteenth-century England

Developing understandings of protest and cultures of resistance has been a central theme of the 'new' cultural geography of the 1990s and 2000s. But whilst geographers of the here and now have been highly sensitive to the importance of acts of protest which occur outside of the context of broader social movements, geographers concerned with past protests have tended to focus overwhelmingly upon either understanding the development of social movements or highly specific place-based studies. Through a focus upon the hitherto ignored practice of 'tree maiming', this paper demonstrates not only the value of examining specific protest practices in helping to better understand the complexity of conflict, but also how in periods of acute socio-economic change the evolving relationship between humans and the non-human – in this case trees – is a central discourse to the protest practices of the poor. Such attacks often involved complex cultural understandings about the ways in which trees should – and should not – be socially enrolled.