Decompositions of spaces of measures

Let M be the Banach space of sigma-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of M is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon-Nikodym theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen-Wintner purity theorem for our decompositions.

Sequentially continuous non-linear fundamental systems of solutions of affine equations in locally convex spaces

According to the Mickael's selection theorem any surjective continuous linear operator from one Fr\'echet space onto another has a continuous (not necessarily linear) right inverse. Using this theorem Herzog and Lemmert proved that if $E$ is a Fr\'echet space and $T:E\to E$ is a continuous linear operator such that the Cauchy problem $\dot x=Tx$, $x(0)=x_0$ is solvable in $[0,1]$ for any $x_0\in E$, then for any $f\in C([0,1],E)$, there exists a continuos map $S:[0,1]\times E\to E$, $(t,x)\mapsto S_tx$ such that for any $x_0\in E$, the function $x(t)=S_tx_0$ is a solution of the Cauchy problem $\dot x(t)=Tx(t)+f(t)$, $x(0)=x_0$ (they call $S$ a fundamental system of solutions of the equation $\dot x=Tx+f$). We prove the same theorem, replacing "continuous" by "sequentially continuous" for locally convex spaces from a class which contains strict inductive limits of Fr\'echet spaces and strong duals of Fr\'echet--Schwarz spaces and is closed with respect to finite products and sequentially closed subspaces. The key-point of the proof is an extension of the theorem on existence of a sequentially continuous right inverse of any surjective sequentially continuous linear operator to some class of non-metrizable locally convex spaces.

On weak and strong Peano theorems

We say that the Peano theorem holds for a topological vector space $E$ if, for any continuous mapping $f : {\Bbb R}\times E \to E$ and any $(t(0), x(0))$ is an element of ${\Bbb R}\times E$, the Cauchy problem $\dot x(t) = f(t,x(t))$, $x(t(0)) = x(0)$, has a solution in some neighborhood of $t(0)$. We say that the weak version of Peano theorem holds for $E$ if, for any continuous map $f : {\Bbb R}\times E \to E$, the equation $\dot x(t) = f (t, x(t))$ has a solution on some interval. We construct an example (answering a question posed by S. G. Lobanov) of a Hausdorff locally convex topological vector space E for which the weak version of Peano theorem holds and the Peano theorem fails to hold. We also construct a Hausdorff locally convex topological vector space E for which the Peano theorem holds and any barrel in E is neither compact nor sequentially compact.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

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 $\Gamma$.

High-density and low electron temperature direct current reflex plasma source

A new type of direct current, high-density, and low electron temperature reflex plasma source, obtained as a hybrid between a modified hollow-cathode discharge and a Penning ionization gauge discharge is presented. The plasma source was tested in argon, nitrogen, and oxygen over a range pressure of 1.0-10(-3) mbar, discharge currents 20-200 mA, and magnetic field 0-120 Gauss. Both external parameters, such as breakdown potential and the discharge voltage-current characteristic, and its internal parameters, like the electron energy distribution function, electron and ion densities, and electron temperature, were measured. Due to the enhanced hollow-cathode effect by the magnetic trapping of electrons, the density of the bulk plasma is as high as 10(18) m(-3), and the electron temperature is as low as a few tenths of electron volts. The plasma density scales with the dissipated power. Another important feature of this reflex plasma source is its high degree of uniformity, while the discharge bulk region is free of an electric field. (C) 2004 American Institute of Physics.

Bimodules over nest algebras and Deddens'

We generalise Dedden's Theorem for nest algebras to nest algebra bimodules. We define an object which extends the Jacobson radical of a nest algebra, and characterose it generalising a theorem of Erdos.

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.

The Ectocarpus genome and the independent evolution of multicellularity in brown algae

Brown algae (Phaeophyceae) are complex photosynthetic organisms with a very different evolutionary history to green plants, to which they are only distantly related(1). These seaweeds are the dominant species in rocky coastal ecosystems and they exhibit many interesting adaptations to these, often harsh, environments. Brown algae are also one of only a small number of eukaryotic lineages that have evolved complex multicellularity (Fig. 1). We report the 214 million base pair (Mbp) genome sequence of the filamentous seaweed Ectocarpus siliculosus (Dillwyn) Lyngbye, a model organism for brown algae(2-5), closely related to the kelps(6,7) (Fig. 1). Genome features such as the presence of an extended set of light-harvesting and pigment biosynthesis genes and new metabolic processes such as halide metabolism help explain the ability of this organism to cope with the highly variable tidal environment. The evolution of multicellularity in this lineage is correlated with the presence of a rich array of signal transduction genes. Of particular interest is the presence of a family of receptor kinases, as the independent evolution of related molecules has been linked with the emergence of multicellularity in both the animal and green plant lineages. The Ectocarpus genome sequence represents an important step towards developing this organism as a model species, providing the possibility to combine genomic and genetic(2) approaches to explore these and other(4,5) aspects of brown algal biology further.

SK1 of graded division algebras

The reduced Whitehead group $\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that $\SK$ of a tame valued division algebra over a henselian field coincides with $\SK$ of its associated graded division algebra. Furthermore, it is shown that $\SK$ of a graded division algebra is isomorphic to $\SK$ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes $\SK$ for generic abelian crossed products.

Reducing Uncertainty in Aeroelastic Flutter Boundaries Using Experimental Data

Flutter prediction as currently practiced is usually deterministic, with a single structural model used to represent an aircraft. By using interval analysis to take into account structural variability, recent work has demonstrated that small changes in the structure can lead to very large changes in the altitude at which
utter occurs (Marques, Badcock, et al., J. Aircraft, 2010). In this follow-up work we examine the same phenomenon using probabilistic collocation (PC), an uncertainty quantification technique which can eficiently propagate multivariate stochastic input through a simulation code,
in this case an eigenvalue-based fluid-structure stability code. The resulting analysis predicts the consequences of an uncertain structure on incidence of
utter in probabilistic terms { information that could be useful in planning
flight-tests and assessing the risk of structural failure. The uncertainty in
utter altitude is confirmed to be substantial. Assuming that the structural uncertainty represents a epistemic uncertainty regarding the
structure, it may be reduced with the availability of additional information { for example aeroelastic response data from a flight-test. Such data is used to update the structural uncertainty using Bayes' theorem. The consequent
utter uncertainty is significantly reduced across the entire Mach number range.

Optical source model for the 23.2-23.6 nm radiation from the multielement germanium soft X-ray laser

Distributions of source intensity in two dimensions (designated the source model), averaged over a single laser pulse, based on experimental measurements of spatial coherence, are considered for radiation from the unresolved 23.2/23.6 nm spectral lines from the germanium collisional X-ray laser. The model derives from measurements of the visibility of Young slit interference fringes determined by a method based on the Wiener-Khinchin theorem. Output from amplifiers comprising three and four target elements have similar coherence properties in directions within the horizontal plane corresponding to strong plasma refraction effects and fitting the coherence data shows source dimensions (FWHM) are similar to 26 mu m (horizontal), significantly smaller than expected by direct imaging, and similar to 125 mu m (vertical: equivalent to the height of the driver excitation). (C) 1999 Elsevier Science B.V. All rights reserved.

Spectroscopic characterization of prepulsed x-ray laser plasmas

Through the use of time-integrated space-resolved keV spectroscopy, we investigate line plasmas showing gain in Ne-like nickel, copper, and zinc for irradiation using the prepulse technique. The experiments were conducted at 1.06 mu m with the prepulse to main pulse intensity contrast ranging from 10(-6) to 10(-2). The effect of the prepulses on the plasma conditions is inferred through spectroscopic line ratio diagnostics for the electron temperature, the Ne-like ground-state density, and the lateral size of the Ne-like region. It is observed that neither the value of the electronic temperature nor its spatially resolved profile along the linear focus axis varies significantly with the prepulse level, contrary to the lateral width and the density of the Ne-like region in the plasma, which are seen to increase. These results explain, at least in part, why prepulsed x-ray lasers show such high gain and brightness.

Coherence and bandwidth measurements of harmonics generated from solid surfaces irradiated by intense picosecond laser pulses

We present measurements of the transverse and longitudinal coherence lengths of the fourth harmonic of a 1053-nm, 2.5-ps laser generated during high-intensity (up to 10(19) W cm(-2)) interactions with a solid target. Coherence lengths were measured by use of a Young's double-slit interferometer. The effective source size, as defined by the Van Cittert-Zernicke theorem, was found to be 10-12 mu m, and the coherence time was observed to be in the range 0.02-0.4 ps.