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Inverse problems aimed at locating cracks and voids inside a dielectric medium by means of electromagnetic waves involve knowledge of transmission eigenvalues. The subject has been actively investigated by many Authors for some years and results have been presented in articles and books. Unfortunately, the frequency dependence of dielectric permittivity and magnetic permeability of the material has not been taken into account. A theorem which provides existence and asymptotic properties of real transmission eigenvalues is shown to be physically inconsistent, because it ignores the Kramers - Kronig relations.

Let ${\mit\Omega}\subset I\hspace{-3pt}R^2$ be bounded by two heteroclinic orbits, ${\mit\Gamma}_1, {\mit\Gamma}_2$ of the $\nabla u$-flow. Then $\nabla\cdot (c \nabla u)=0$ in ${\mit\Omega}$ implies $c\equiv 0$ in $\bar{\mit\Omega}$ [\textsc{Chicone} and \textsc{Gerlach}, 1987]. Let $u\in {\cal C}^2 ({\mit\Omega})\cap{\cal C}^0 (\bar{\mit\Omega})$ be known. The (unique) conductivity $\hat a$, which complies with $\nabla\cdot(\hat a\nabla u)=f$, can be identified by minimising with respect to $b$ the norm of $\nabla\tilde a [b] \times\nabla u - \nabla b\times\nabla p$ under constraints, where $\nabla\cdot(b\nabla p)=f$ and $\tilde a[b]\partial_j u := b\partial_j p$, $j=$ 1 \textbf{or} 2. This is an attempt at justifying the ``comparison model'' algorithm [\textsc{Scarascia} and \textsc{Ponzini}, 1972], which has seen successful practical applications to inverse hydrogeology ever since.

Inverse problems aimed at locating cracks and voids inside a dielectric medium by means of electromagnetic waves involve knowledge of transmission eigenvalues. The subject has been actively investigated by many Authors for some years and results have been presented in articles and books. Unfortunately, the frequency dependence of dielectric permittivity and magnetic permeability of the material has not been taken into account. A theorem which provides existence and asymptotic properties of real transmission eigenvalues is shown to be physically inconsistent, because it ignores the Kramers-Kronig relations.

The doubly-fed induction generator (DFIG ) is a key constituent of energy conversion plants. The control of a DFIG is a challenge, whenever the primary energy supply (e.g., the wind velocity field) is characterised by intermittency. The mathematical model and control of a DFIG rely on the Blondel-Park transformation, which is known to simplify the governing equations. The distinctive feature of this contribution consists of showing how the Blondel-Park transformation derives from a set of conditions to be met by a group. Such a group is shown to exist and to continuously depend on one parameter. The uniqueness of its infinitesimal generator is also shown. As an application, the well-known electric torque theorem is proved in a simple way, which relies on a “product of matrices” formula. The latter, in turn, is a by-product of the axiomatic deduction of the Blondel-Park transformation.

The elastic scattering of the mass-less (${\mit\omega}^2 = c^2 k^2$) complex scalar field by a potential, $q[.]$, of bounded support is a prototype model which finds applications to acoustics and electromagnetics in a classical (i.e., non quantum mechanical) setting. Of particular interest are the symmetries of the scattered wave, $u[.]$ and of the quantities derived therefrom ($A[.]$, $|A[.]|^2$) caused by symmetry operations on $q[.]$, such as translation, rotation, reflection, and scaling. This investigation is motivated by the analysis of scattering patterns of environmental interest.

TAOS (two-dimensional angle-resolved optical scattering) is an experimental method to detect single, micrometer-sized airborne material particles, illuminate them by a single pulse of laser (${\mit\lambda} = 532$nm, pulse duration = 30ns) and record their scattered light intensity patterns over the angular sector $\{75^\circ\le{\mit\theta}\le 135^\circ\}\times\{0^\circ\le{\mit\varphi}\le 360^\circ\}$ at high resolution (one pattern $\doteq1024^2$ pixels). Particles of reference materials and from outdoor environmental sampling have been analyzed and thousands of scattering patterns (TAOS patterns, hereinafter) have been stored . One of the goals of classification is the discrimination of bacterial spore patterns ($Bq$). A typical result is: a set of 957 $K5$ (outdoor dust) patterns is analysed; 98 patterns (10\%) are falsely recognised as $Bq$ (lower halfplane), whereas the remainder is assigned to the other two training classes, $Fq$ (top) and $Pq$ (middle).

Cauchy partial data imply the uniqueness of the potential. Complex geometrical optics solutions are needed to prove the claim.

The fluid dynamic model under investigation stems from the tropical climate model [D. M. W. Frierson, A. J. Majda and O. M. Pauluis, Commun. Math. Sci. 2 (2004), no. 4, 591--626; MR2119930 (2006a:86015)] written, under suitable approximations, for the three observables: baroclinic velocity u⃗ , temperature deviation T, and moisture q, all made dimensionless. The interested reader should also look at the references, [D. M. W. Frierson, A. J. Majda and O. M. Pauluis, op. cit.] in particular, in order to fully appreciate the breadth of this paper.

The boundary value problems addressed in the paper physically correspond to electromagnetic scattering by a smooth, open surface S in R3∖S− and include the magnetic-magnetic crack type, the electric-electric screen type, the magnetic-electric and the mixed-mixed types.

Experimental: TAOS (two-angle optical scattering) instruments record the intensity patterns of laser light scattered by single airborne particles over a wide range of the scattering angles {θ,ϕ} [1]. Large data sets are available, which consist of scattering patterns from a variety of known materials and from environmental sampling. Data analysis: Due to the lack of methods which exactly solve the inverse obstacle scattering problem, artificial intelligence techniques can be used. A classifier has been developed that is based on the spectrum enhancement algorithm [2]. This classifier extracts vectors of morphological descriptors from TAOS patterns and submits them to principal components analysis. Supervised training of the classifier occurs by processing hundreds of patterns from known materials and maximizing a suitable figure of merit. The trained and validated classifier is applied to patterns from new materials for the purpose of recognition. A typical recent classification result is shown by Figure 1 below. The purpose of ongoing work is to design classification experiments with patterns from new materials (NaCl crystals, soot, outdoor dust, . . .) and apply the above outlined method to particle recognition and scoring

Motivation: Fractal models apply to radiation patterns from the so-called fractal antennas and to the description of wave propagation through a complicated environment. In antenna design the prototype antenna array factor has been provided by Weierstraß functions and their band- limited approximations. In antenna characterization one may want to assign a given (synthesized or measured) radiation pattern to a fractal class. Fractal Pattern Synthesis: Array antenna factors are synthesized by means of Weierstraß functions [1]. Two- dimensional fractal radiation patterns can be synthesized by separation of variables. Classification of Synthesized Fractal Patterns: The spectrum enhancement (ση) algorithm, introduced a few years ago [2 and references quoted therein] operates on the Fourier transform of a function of two variables. It evaluates derivatives of integer [3] or of fractional order [4], followed by some non-linear operations. By applying ση to a synthesized fractal pattern one obtains a vector of morphological descriptors, which are submitted to a trainable classifier. A typical classification result is displayed by Figure 1 where the center panel shows the fractal pattern class centroids on the plane of the first two principal components z1 and z2 . Each of the five classes (labelled 1, 3, 5, 6, 9) is composed of patterns such that b = 14 and 16. Moreover, in class 1, D = 1.1; in class 3, D = 1.3 (sample pattern on the right panel); in class 5, D = 1.5; in class 6, D = 1.6 and in class 9, D = 1.9 (sample on the left panel). Problem to be Addressed: There are critical values of the wavenumber [5] at which the nu- merical estimation of dimB [graph[f ]] is affected by relevant errors. The goal of this investigation is to determine whether or not the same wavenumber values are critical for the ση algorithm as well and, in case they are, provide an explanation.

The parabolic integro-differential equation under investigation, $$ {\beta}\partial_t u = Au - m * Au + {\chi}\tag1 $$(1) subject to suitable initial and boundary conditions (BC), represents linear, non-homogeneous diffusion with memory kernel $m = m[t]$, in the space-time domain $Q$.

Diagnostic images at all biological scales benefit from quantitative morphological analysis which can lead to their classification and recognition. With reference to an image from an optical microscope, quantitative morphology consists of extracting from the raw image, by a suitable algorithm, some numerical indicators or "descriptors". The latter replace the image and represent it for subsequent analysis by algorithms capable of: classification, recognition, feature extraction and other higher level functions. Herewith the extraction of morphological descriptors relies on the "spectrum enhancement" algorithm: the discrete Fourier transform of the image undergoes some nonlinear operatons and comparison with a model power spectral density. As a result the graph of a function of wavenumber, the enhanced spectrum, is obtained, which carries information about the structure and texture of the image. Enhanced spectra are submitted to a classifier based on multivariate statistics, which can be trained, validated and applied to recognize new images. The whole procedure will be illustrated by applications at different scales: a) (sub-cell) cytoskeletal structures altered by xenobiotics; b) (colony) cell colonies ("foci") from in vitro neoplastic transformation assays; c) (tissue) cross sections of reconstituted human corneal epithelia. The results of diagnostic and pathological interest, which will be presented, include the quantitative estimation of morphological alterations, dose-response relationships and recovery from damage. The same paradigm can apply to images acquired by different (optical) instruments and is intended for integration with other methods, the eventual goal being the high content analysis of biological data.

The steady state scattering and absorption of light intensity within a two- or three-dimensional bounded domain $B$ with piacewise smooth boundary $\partial B$ are described by the equation $$-\nabla \cdot (a \nabla u) + cu =0$$.

An apparently "innocuous'' problem derived from the heat equation, namely the determination of the boundary values $u(1,t)$ of the function $u(\cdot,\cdot)$ which satisfies $$ \partial_t u(x,t) =\partial^2_{xx}u(x,t)+a(x)u(x,t),\quad x\in (0,1),\ t\ge 0,\tag1 $$,(1) $$ \gathered u(x,0) = 0,\ x\in [0,1],\quad u(0,t) = 0,\ t\ge 0,\\ u(x_0,t) = f(t),\ 0< x_0 < 1,\ t\ge 0,\endgathered \tag2$$,(2) from knowledge of $a(\cdot)\in {\scr C}^2 ([0,1])$ and $f(\cdot)$, gives rise to a logarithmic estimate for an "approximate solution'' $u_{\delta}(1,t)$, $t\ge 0$.

A numerical method is proposed to simultaneously reconstruct a piecewise constant conductivity ${\mit\sigma}$ and a permittivity ${\mit\epsilon}$ in a one-dimensional layered medium. In addition to the surface current density, the layer stack geometry has to be known.

Spectrum Enhancement: Let g denote a scalar function which is at least continuous on the surface T of a torus. Let u ≡ {u1 , u2 } denote the spatial frequency vector and G[u] the (distribution-valued) Fourier transform of g. The spectrum enhancement (SE ) algorithm, which was introduced a few years ago [e.g., [1]] consists of suitable transformations of the function H (p) [u] := |u| 2β |G[u]| 2 |a0,0 | 2 + δ[u], where δ is the Dirac measure, a0,0 is the Fourier amplitude at the origin such that F (g)[0] = a0,0 δ[u], β ∈ R + is the enhancement order and β = 2p. It has been recently shown that SE is related to Fourier transforms of derivatives of g[.], which are of integer order whenever β ∈ N [2] and of fractional order otherwise (β / ∈ N) [3]. Herewith, the role played by the enhancement order β in the fractal analysis of g[.] will be described. Fractal Radiation Patterns: Array antennas exhibiting fractal radiation patterns are synthe- sized by means of generalized Weierstraß functions [4]. For example, in one spatial dimension (x1 ) one has f [x1 ] := ∞ m=1 η(D−2)mh[ηmx1 ] where 1 < D < 2, η > 1 and h[.] is a suitably chosen bounded, periodic function of one variable. It can be shown that the box-counting (B) fractal dimension (dimB ) of the graph of f [.] (graph[f ]) satisfies dimB [graph[f ]] = D. Two-dimensional fractal radiation patterns can be synthesized in a similar way e.g., by separation of variables. Recognition of Fractal Radiation Patterns: Let simulated or measured radiation patterns be available on the whole surface of the unit sphere or part thereof. Let the pattern dimB be unknown or assume the pattern has to be classified by fractal dimension. The SE algorithm can be used to extract features from said patterns and estimate their dimB . Moreover, patterns can be classified by a procedure which includes supervised training. Some preliminary classification and recognition results will be provided, based on numerically synthesized fractal radiation patterns.

TAOS (two-angle optical scattering) is a well-known experimental technique which records the intensity patterns of laser light scattered by single aerosol particles over an extended range of the scattering angles {θ, φ} [1]. In the absence of a method which solves the inverse obstacle problem from intensity data, patterns have to be classified by artificial intelligence tech- niques. The classifier described herewith extracts features from TAOS patterns by the spectrum enhancement algorithm [2], which is controlled by a few parameters (the n-tuple ψ), not described herewith. Training corresponds to finding a ψ which maximizes a suitable figure of merit (F ). The newly developed training-validation scheme is illustrated by the following example. 1) Out of 100 TAOS patterns produced by clusters of polystyrene spheres of controlled size (material a6, class 1) and 100 patterns from single spores of Bacil lus globigii (material bg, class 2), form e.g., 10 training (T ) sets of 50 patterns from each class, which differ by at least 5 + 5 patterns from one another. Two sample patterns are shown by Figures 1 and 2 below

Let $\Omega$ be a an open, bounded sobset of $\bbd R ^2$ with smooth boundary and $D \subset\Omega$ an open subset such that $\Omega - \bar D$ is connected. Let the background conductivity be a symmetric, positive definite matrix and $u$ solve the conductivity equation in $\Omega$ plus suitable boundary conditions.

The paper focuses on the asymptotic formula for Dirichlet-to-Neumann maps.

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The coefficients which appear in the representation of the far-field scattered by a twodimensional, perfectly electrically conducting obstacle, depend on the normal derivative of the scattered field on the obstacle boundary, $\partial_N v |_{\Gamma}$. A family of functions, W, deduced from the minimization of a boundary defect, is shown to be linearly independent and complete. As a consequence the approximation of @Nv|gamma results from a well posed algebraic problem. The approximation error is estimated from an a priori bound on the equation error in terms of the eigenvalues of a boundary integral operator. Rayleigh's hypothesis is nowhere required of the obstacle. These results justify some obstacle inversion methods which originated from heuristic arguments

A two- or three-dimensional bounded domain $\Omega$ exhibits an unknown conductivity $\sigma $ such that $\sigma=1+\kappa$ $ (\ge c>0)$ in $\overline D$ and $\sigma=1$ in $\Omega \sbs\overline D$, where $\overline D$ $(\subset\Omega)$ is also unknown.

The paper provides a uniqueness result for a three-dimensional inverse domain problem in linear elasticity. In some respects it can be regarded as an extension of [Đạ̌ng Điñh Áng, Dang Duc Trong and M. Yamamoto, J. Elasticity 56 (1999), no. 3, 199--212 (2000); MR1772729 (2001e:74032)].

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The inverse problem consists of finding {ε[x]−ε0,σ[x]} from the knowledge of the wave fronts τk[x] on ∂B and of the tangential components {FH[x,t,v(k),FE[x,t,v(k)]}≡{F(k)H,F(k)E} of the four electromagnetic fields {H(k),E(k)} on S[v(k)], k=1,2,3,4. Reviewer's remarks: 1. To the extent to which both the dependence of {ε,σ} on frequency and the Kramers-Kronig relations can be neglected, these results can be applied to inverse problems for penetrable obstacles in the time domain. 2. The idea of "identifying coefficients'' (in quite general terms) by means of multiple data sets and by solving algebraic systems derived therefrom is another basic joint contribution by the author [M. M. Lavrentʹev, V. G. Romanov and S. P. Šišatskiĭ, Ill-posed problems of mathematical physics and analysis (Russian), "Nauka'', Moscow, 1980; MR0591674 (82g:65003) (Chapter VII, Section 2)]: this paper further demonstrates how fruitful such a scheme still is, even in complex situations.

A non-conductive, magneto-dielectric medium with constant parameters is confined within a plane layer. A qualitative argument is provided on how one could obtain two-sided boundary conditions when the medium moves at constant velocity within a thin shell of arbitrary shape.