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.

Hypercyclic operators on countably dimensional spaces

According to Grivaux, the group GL(X) of invertible linear operators on a separable infinite dimensional Banach space X acts transitively on the set s (X) of countable dense linearly independent subsets of X. As a consequence, each A? s (X) is an orbit of a hypercyclic operator on X. Furthermore, every countably dimensional normed space supports a hypercyclic operator. Recently Albanese extended this result to Fréchet spaces supporting a continuous norm. We show that for a separable infinite dimensional Fréchet space X, GL(X) acts transitively on s (X) if and only if X possesses a continuous norm. We also prove that every countably dimensional metrizable locally convex space supports a hypercyclic operator.

Temporal extension of Laplacian Eigenmaps for unsupervised dimensionality reduction of time series

A novel non-linear dimensionality reduction method, called Temporal Laplacian Eigenmaps, is introduced to process efficiently time series data. In this embedded-based approach, temporal information is intrinsic to the objective function, which produces description of low dimensional spaces with time coherence between data points. Since the proposed scheme also includes bidirectional mapping between data and embedded spaces and automatic tuning of key parameters, it offers the same benefits as mapping-based approaches. Experiments on a couple of computer vision applications demonstrate the superiority of the new approach to other dimensionality reduction method in term of accuracy. Moreover, its lower computational cost and generalisation abilities suggest it is scalable to larger datasets. © 2010 IEEE.

Hypercyclic tuples of operators on $C^n$ and $R^n$

A tuple $(T_1,\dots,T_n)$ of continuous linear operators on a topological vector space $X$ is called hypercyclic if there is $x\in X$ such that the the orbit of $x$ under the action of the semigroup generated by $T_1,\dots,T_n$ is dense in $X$. This concept was introduced by N.~Feldman, who have raised 7 questions on hypercyclic tuples. We answer those 4 of them, which can be dealt with on the level of operators on finite dimensional spaces. In
particular, we prove that the minimal cardinality of a hypercyclic tuple of operators on $\C^n$ (respectively, on $\R^n$) is $n+1$ (respectively, $\frac n2+\frac{5+(-1)^n}{4}$), that there are non-diagonalizable tuples of operators on $\R^2$ which possess an orbit being neither dense nor nowhere dense and construct a hypercyclic 6-tuple of operators on $\C^3$ such that every operator commuting with each member of the tuple is non-cyclic.

Interacting kinetics of neutral and ionic species in an atmospheric-pressure helium-oxygen plasma with humid air impurities

We unravel the complex chemistry in both the neutral and ionic systems of a radio-frequency-driven atmospheric-pressure plasma in a helium-oxygen mixture (He-0.5% O) with air impurity levels from 0 to 500 ppm of relative humidity from 0% to 100% using a zero-dimensional, time-dependent global model. Effects of humid air impurity on absolute densities and the dominant production and destruction pathways of biologically relevant reactive neutral species are clarified. A few hundred ppm of air impurity crucially changes the plasma from a simple oxygen-dependent plasma to a complex oxygen-nitrogen-hydrogen plasma. The density of reactive oxygen species decreases from 10 to 10 cm, which in turn results in a decrease in the overall chemical reactivity. Reactive nitrogen species (10 cm ), atomic hydrogen and hydroxyl radicals (10-10 cm) are generated in the plasma. With 500 ppm of humid air impurity, the densities of positively charged ions and negatively charged ions slightly increase and the electron density slightly decreases (to the order of 10 cm). The electronegativity increases up to 2.3 compared with 1.5 without air admixture. Atomic hydrogen, hydroxyl radicals and oxygen ions significantly contribute to the production and destruction of reactive oxygen and reactive nitrogen species. © 2013 IOP Publishing Ltd.

Chemical kinetics and reactive species in atmospheric pressure helium-oxygen plasmas with humid-air impurities

In most applications helium-based plasma jets operate in an open-air environment. The presence of humid air in the plasma jet will influence the plasma chemistry and can lead to the production of a broader range of reactive species. We explore the influence of humid air on the reactive species in radio frequency (rf)-driven atmospheric-pressure helium-oxygen mixture plasmas (He-O, helium with 5000 ppm admixture of oxygen) for wide air impurity levels of 0-500 ppm with relative humidities of from 0% to 100% using a zero-dimensional, time-dependent global model. Comparisons are made with experimental measurements in an rf-driven micro-scale atmospheric pressure plasma jet and with one-dimensional semi-kinetic simulations of the same plasma jet. These suggest that the plausible air impurity level is not more than hundreds of ppm in such systems. The evolution of species concentration is described for reactive oxygen species, metastable species, radical species and positively and negatively charged ions (and their clusters). Effects of the air impurity containing water humidity on electronegativity and overall plasma reactivity are clarified with particular emphasis on reactive oxygen species. © 2013 IOP Publishing Ltd.

Efficient tracking of human poses using a manifold hierarchy

In this paper a 3D human pose tracking framework is presented. A new dimensionality reduction method (Hierarchical Temporal Laplacian Eigenmaps) is introduced to represent activities in hierarchies of low dimensional spaces. Such a hierarchy provides increasing independence between limbs, allowing higher flexibility and adaptability that result in improved accuracy. Moreover, a novel deterministic optimisation method (Hierarchical Manifold Search) is applied to estimate efficiently the position of the corresponding body parts. Finally, evaluation on public datasets such as HumanEva demonstrates that our approach achieves a 62.5mm-65mm average joint error for the walking activity and outperforms state-of-the-art methods in terms of accuracy and computational cost.

Investigating the dynamics of laser induced sparks in atmospheric helium using Rayleigh and Thomson scattering

We have used optical Rayleigh and Thomson scattering to investigate the expansion dynamics of laser induced plasma in atmospheric helium and to map its electron parameters both in time and space. The plasma is created using 9 ns duration, 140 mJ pulses from a Nd:YAG laser operating at 1064 nm, focused with a 10 cm focal length lens, and probed with 7 ns, 80 mJ, and 532 nm Nd:YAG laser pulses. Between 0.4 μs and 22.5 μs after breakdown, the electron density decreases from 3.3 × 1017 cm−3 to 9 × 1013 cm−3, while the temperature drops from 3.2 eV to 0.1 eV. Spatially resolved Thomson scattering data recorded up to 17.5 μs reveal that during this time the laser induced plasma expands at a rate given by R ∼ t0.4 consistent with a non-radiative spherical blast wave. This data also indicate the development of a toroidal structure in the lateral profile of both electron temperature and density. Rayleigh scattering data show that the gas density decreases in the center of the expanding plasma with a central scattering peak reemerging after about 12 μs. We have utilized a zero dimensional kinetic global model to identify the dominant particle species versus delay time and this indicates that metastable helium and the He2 + molecular ion play an important role.

Two-dimensional Banach spaces with polynomial numerical index zero

We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.

On the solvability of Hamilton's equations in Hilbert spaces

We construct a bounded function $H : l_2\times l_2 \to R$ with continuous Frechet derivative such that for any $q_0\in l_2$ the Cauchy problem $\dot p= - {\partial H\over\partial q}$, $\dot q={\partial H\over\partial p}$, $p(0) = 0$, q(0) = q_0$ has no solutions in any neighborhood of zero in R.

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.