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We generalize the classical Riesz–Dunford holomorphic functional calculus with weakly locally nilpotent operators in a Frechet space to the case of the algebra of all formal power series. We obtain a counterpart of the spectral mapping theorem and formulas similar to the Taylor and Riesz–Dunford formulas. We show that the weak local nilpotency of an operator is necessary for constructing the formal functional calculus. Bibliography: 12 titles.

Detailed investigation on texture and mineralogy of beach sediments helps to understand their nature of deposition and potential targets of strategic mineral deposits. The beach sediments from the ...

In this article, we have processed Landsat-8 spectral bands, ground gravity, and magnetic data for delineating the mafic cumulates within the anorthositic-layered complex in Sittampundi, Tamilnadu,...

In this paper, we consider the asymptotic behavior of solutions to the linear spatially homogeneous Boltzmann equation for hard potentials without angular cutoff. We obtain an optimal rate of exponential convergence towards equilibrium in a L1-space with a polynomial weight. Our strategy is taking advantage of a spectral gap estimate in the Hilbert space L 2 ( μ − 1 2 ) and a quantitative spectral mapping theorem developed by Gualdani et al. (2017).

Herein we report the detection and differentiation of plasmonic and non-plasmonic nanoparticles simultaneously administered to A549 lung epithelium cells using dark-field microscopy and hyperspectral imaging. Reflectance spectra-based hyperspectral mapping and image analysis allows for the effective quasi-quantitative identification of nanomaterials in cultured human cells. © 2018 The Chemical Society of Japan. The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University and funded by Russian presidential grant MD-6655.2018.4 and by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (6.7743.2017/6.7).

The behavior of the domain, the range, the kernel and the multi-valued part of a rational function in a linear relation is analyzed, respectively. We give some basic properties of such linear relations, and we prove that the rational form of the spectral mapping theorem holds in terms of ascent, essential ascent, descent and essential descent.

Although it is known that the Raman spectroscopic signature of single-walled carbon nanotubes (SWCNTs) is highly chirality dependent, using Raman spectroscopy with several laser excitations as a tool for quantifying fraction of either metallic or semiconducting nanotubes in a sample has become a widely used analytical method. In this work, using the electron diffraction technique as a basis, we have examined the validity of Raman spectroscopy for quantitative evaluation of metallic fractions (M%) in single-walled carbon nanotube samples. Our results show that quantitative Raman spectroscopic evaluations of M% by using several discrete laser lines, either by using integrated intensities of chirality-associated radial breathing modes (RBMs) or, as has been more commonly utilized in recent studies, by statistically counting the numbers of RBMs can be misrepresentative. Specifically, we have found that the occurrence numbers of certain types of RBMs in Raman spectral mapping depend critically on the diameter distribution, resonant coupling between transition energies and excitation laser energy, and the chirality-dependent Raman scattering cross sections rather than simply on the metallic and semiconducting SWCNT fractions. These dependencies are similar to those observed in the integrated intensities of RBMs. Our findings substantially advance the understanding of the proper use of Raman spectroscopy for carbon nanotube quantification, which is important for carbon nanotube characterization and crucial to guide research in SWCNT growth and their applications.

We extend and deepen the theory of functional calculus for semigroup generators, based on the algebra $\mathcal B$ of analytic Besov functions, which we initiated in a previous paper. In particular, we show that our construction of the calculus is optimal in several natural senses. Moreover, we clarify the structure of $\mathcal B$ and identify several important subspaces in practical terms. This leads to new spectral mapping theorems for operator semigroups and to wide generalisations of a number of basic results from semigroup theory., 51 pages. This is a version of the paper to appear in Journal of Functional Analysis

In this paper, we consider the quantum walk on $\mathbb{Z}$ with attachment of one-length path periodically. This small modification to $\mathbb{Z}$ provides localization of the quantum walk. The eigenspace causing this localization is generated by finite length round trip paths. We find that the localization is due to the eigenvalues of an underlying random walk. Moreover we find that the transience of the underlying random walk provides a slow down of the pseudo velocity of the induced quantum walk and a different limit distribution from the Konno distribution.

We describe a closed operator functional calculus in Banach modules over the group algebra L 1 ( R ) and illustrate its usefulness with a few applications. In particular, we deduce a spectral mapping theorem for operators in the functional calculus, which generalizes some of the known results. We also obtain an estimate for the spectrum of a perturbed differential operator in a certain class.

We discuss the description of eigenspace of a quantum walk model $U$ with an associating linear operator $T$ in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of $U$ without the spectral decomposition of $T$. Arguments in this direction reveal the eigenspaces of $U$ characterized by the generalized kernels of linear operators given by $T$., Comment: 17 pages

Firstly, we establish spectral mapping theorems for normal eigenvalues (due to Browder) of a -semigroup and its generator. Secondly, we discuss relationships between normal eigenvalues of the compact monodromy operator and the generator of the evolution semigroup on associated with the -periodic evolutionary process on a Banach space X, where stands for the space of all -periodic continuous functions mapping to X.

Originally developed to classify multispectral and hyperspectral images, spectral mapping methods were used to classify Light Detection and Ranging (LiDAR) data to estimate the vertical structure of vegetation for Fuel Type (FT) mapping. Three spectral mapping methods generated spatially comprehensive FT maps for Cabaneros National Park (Spain): (1) Spectral Mixture Analysis (SMA), (2) Spectral Angle Mapper (SAM), and (3) Multiple Endmember Spectral Mixture Analysis (MESMA). The Vegetation Vertical Profiles (VVPs) describe the vertical distribution of the vegetation and are used to define each FT endmember in a LiDAR signature library. Two different approaches were used to define the endmembers, one based on the field data collected in 1998 and 1999 (Approach 1) and the other on exploring spatial patterns of the singular FT discriminating factors (Approach 2). The overall accuracy is higher for Approach 2 and with best results when considering a five-FT model rather than a seven-FT model. The agreement with field data of 44% for MESMA and SMA and 40% for SAM is higher than the 38% of the official Cabaneros National Park FTs map. The principal spatial patterns for the different FTs were well captured, demonstrating the value of this novel approach using spectral mapping methods applied to LiDAR data. The error sources included the time gap between field data and LiDAR acquisition, the steep topography in parts of the study site, and the low LiDAR point density among others.

Band parameters can effectively be used as a rapid precursor processing step to enhance and augment the use of imaging spectrometer data by scientists that are not spectroscopy experts. Simple algorithms can be applied to large regional areas with consistent interpretation of surface composition and mineral exposures. Where these methods have been tested, they provide good agreement with results from more time consuming spectral feature fitting and detailed spectral mapping methods. We have tested band depths determined using three channels and derivative methods that have been shown to work well for CRISM.

We report the analysis of Hubble Space Telescope (HST) UV–optical spectroscopy of the nova-like variable V348 Puppis with eclipse mapping techniques. We measured the eclipse width at disc centre and determined a lower limit on the mass ratio of q > 0.3, with a tight relation between q and the binary inclination i. For 0.31 ≤ q ≤ 0.6, we have 79$_{.}^{\circ}$5 ≤ i ≤ 88$_{.}^{\circ}$2. Simulations with 3D eclipse mapping give no support to the suggestion of self-shielding of the accretion disc of V348 Pup, indicating a geometrically thin disc. Eclipse maps reveal two structures in the accretion disc interpreted as tidally induced spiral arms, which account for 50–60 per cent of the disc flux in the continuum. The uneclipsed component accounts for 30–50 per cent of the light, indicating that a significant fraction of the light comes from a vertically extended disc chromosphere + wind. Its spectrum can be fitted by an isothermal slab of gas with temperature T = 9600 K and column density Σ ∼1010 g cm−2, corresponding to optically thick thermal emission. Superhumps had faded by the epoch of the HST observations; the disc shrank in size while the disc wind flux increased with respect to observations when superhumps were present. This may be explained if the disc wind outflow is the dominant source of angular momentum loss from the accretion disc. Previously observed accretion disc radii of V348 Pup and of two other binaries with large discs are used to derive an empirical expression for the maximum possible disc radius, which implies that superhumps may be found in systems with up to q ≃ 0.7.

The spectrum and point spectrum of the Cesaro averaging operator \(\mathsf {C}\) acting on the Frechet space \(C^\infty ({\mathbb R}_+)\) of all \(C^\infty \) functions on the interval \([0,\infty )\) are determined. We employ an approach via \(C_0\)-semigroup theory for linear operators. A spectral mapping theorem for the resolvent of a closed operator acting on a locally convex space is established; it constitutes a useful tool needed to establish the main result. The dynamical behaviour of \(\mathsf {C}\) is also investigated.

We study truncated Toeplitz operators in model spaces K θ p for 1 p ∞ , with essentially bounded symbols in a class including the algebra C ( R ∞ ) + H ∞ + , as well as sums of analytic and anti-analytic functions satisfying a θ-separation condition, using their equivalence after extension to Toeplitz operators with 2 × 2 matrix symbols. We establish Fredholmness and invertibility criteria for truncated Toeplitz operators with θ-separated symbols and, in particular, we identify a class of operators for which semi-Fredholmness is equivalent to invertibility. For symbols in C ( R ∞ ) + H ∞ + , we extend to all p ∈ ( 1 , ∞ ) the spectral mapping theorem for the essential spectrum. Stronger results are obtained in the case of operators with rational symbols, or if the underlying model space is finite-dimensional.

For large fully connected neuron networks, we study the dynamics of homogenous assemblies of interacting neurons described by time elapsed models. Under general assumptions on the firing rate which include the ones made in previous works (Pakdaman et al. in Nonlinearity 23(1):55–75, 2010; SIAM J Appl Math 73(3):1260–1279, 2013, Mischler and Weng in Acta Appl Math, 2015), we establish accurate estimate on the long time behavior of the solutions in the weak and the strong connectivity regime both in the case with and without delay. Our results improve (Pakdaman et al. 2010, 2013) where a less accurate estimate was established and Mischler and Weng (2015) where only smooth firing rates were considered. Our approach combines several arguments introduced in the above previous works as well as a slightly refined version of the Weyl’s and spectral mapping theorems presented in Voigt (Monatsh Math 90(2):153–161, 1980) and Mischler and Scher (Ann Inst H Poincare Anal Non Lineaire 33(3):849–898, 2016).

Visual and thermal cameras have large modality gap while providing different information of the same scene. The spectral mapping of thermal imagery into color imagery is a challenging task due to inherit nonlinear relationship. This paper deals with the automatic colorization of thermal imagery into color images using deep encoder-decoder convolutional neural network architecture. The presented approach is trained and evaluated on an online thermal/color dataset, where the network is trained on the spectral mapping of thermal to color images. The training is automatic without requiring user intervention. Extensive experimentation is carried out on the online available dataset, demonstrating consistent and harmonious appearance of generated colorized images from input thermal images.

We study Lipschitz, positively homogeneous and finite suprema preserving mappings defined on a max-cone of positive elements in a normed vector lattice. We prove that the lower spectral radius of such a mapping is always a minimum value of its approximate point spectrum. We apply this result to show that the spectral mapping theorem holds for the approximate point spectrum of such a mapping. By applying this spectral mapping theorem we obtain new inequalites for the Bonsall cone spectral radius of max type kernel operators., arXiv admin note: text overlap with arXiv:1612.01755

Fusion of the high-spatial-resolution hyperspectral (HHS) image using low-spatial- resolution hyperspectral (LHS) and high-spatial-resolution multispectral (HMS) image is usually formulated as a spatial super-resolution problem of LHS image with the help of an HMS image, and that may result in the loss of detailed structural information. Facing the above problem, the fusion of HMS with LHS image is formulated as a nonlinear spectral mapping from an HMS to HHS image with the help of an LHS image, and a novel cluster-based fusion method using multi-branch BP neural networks (named CF-BPNNs) is proposed, to ensure a more reasonable spectral mapping for each cluster. In the training stage, considering the intrinsic characteristics that the spectra are more similar within each cluster than that between clusters and so do the corresponding spectral mapping, an unsupervised clustering is used to divide the spectra of the down-sampled HMS image (marked as LMS) into several clusters according to spectral correlation. Then, the spectrum-pairs from the clustered LMS image and the corresponding LHS image are used to train multi-branch BP neural networks (BPNNs), to establish the nonlinear spectral mapping for each cluster. In the fusion stage, a supervised clustering is used to group the spectra of HMS image into the clusters determined during the training stage, and the final HHS image is reconstructed from the clustered HMS image using the trained multi-branch BPNNs accordingly. Comparison results with the related state-of-the-art methods demonstrate that our proposed method achieves a better fusion quality both in spatial and spectral domains.

The paper presents a novel artificial bee colony clustering (ABCC) algorithm with a self-adaptive multidimensional search mechanism based on difference bias for insula functional parcellation, called as DABCC. In the new algorithm, the preprocessed functional magnetic resonance imaging (fMRI) data was mapped into a low-dimension space by spectral mapping to reduce its dimension in the initialization. Then, clustering centers in the space were searched by the search procedure composed of employed bee search, onlooker bee search and scout bee search, where a self-adaptive multidimensional search mechanism based on difference bias for employed bee search was developed to improve search capability of ABCC. Finally, the experiments on fMRI data demonstrate that DABCC not only has stronger search ability, but can produce better parcellation structures in terms of functional consistency and regional continuity.

SMI (SPICA Mid-infrared Instrument) is one of the two focal-plane scientific instruments planned for new SPICA, and the Japanese instrument proposed and managed by a university consortium in Japan. SMI covers the wavelength range of 12 to 36 μm, using the following three spectroscopic channels with unprecedentedly high sensitivities: low-resolution spectroscopy (LRS; R = 50 - 120, 17 - 36 μm), mid-resolution spectroscopy (MRS; R = 1300 - 2300, 18 - 36 μm), and high-resolution spectroscopy (HRS; R = 28000, 12 - 18 μm). The key functions of these channels are high-speed dustband mapping with LRS, high-sensitivity multi-purpose spectral mapping with MRS, and high-resolution molecular-gas spectroscopy with HRS. This paper describes the technical concept and scientific capabilities of SMI.

Hyperspectral imaging is a powerful tool for mineral mapping and increasingly used in poorly-accessible areas. It only requires a limited amount of validation sample points, but can fail to discriminate spectrally-similar features. In this manuscript, we show that we improve the identification of interesting targets by including geomorphological data in the spectral mapping scheme. We jointly use geomorphic and spectral features to locate gossanous ironstone ridges as an indicator for possible Pb-Zn-Ag-mineralization and provide an application around Mount Isa and George Fisher/Hilton mine, Queensland, Australia. We combine hyperspectral HyMap data using mixture tuned matched filtering with topographical indices, such as maximum curvature and the topographical position index. As it is often the case with structurally-controlled mineralization, the amount of training sites is limited, and supervised classification methods cannot be implemented. Therefore, we implement expert knowledge in a decision tree to take advantage of the relationship between mineralization, alteration and structure. Optimized rock sampling and spectral measurements provided data for validation. We are able to map sets of gossanous ridges with a minimum of validation points, not only within the Mount Isa mining area itself, but also outside the commonly-accepted host rocks. The ridges are parallel to north-south trending geomorphological features and probably associated with the Paroo fault zone. Similarities between the ridges were confirmed by field observations, spectral measurements and a qualitative rock sample analysis. We identified new mineralized ridges that we could subsequently attribute to a poorly-known and sub-economic deposit known as the Mount Novit Pb-Zn-deposit.

The potential for Raman spectroscopy to provide early and improved diagnosis on a wide range of tissue and biopsy samples in situ is well documented. The standard histopathology diagnostic methods of reviewing H&E and/or immunohistochemical (IHC) stained tissue sections provides valuable clinical information, but requires both logistics (review, analysis and interpretation by an expert) and costly processing and reagents. Vibrational spectroscopy offers a complimentary diagnostic tool providing specific and multiplexed information relating to molecular structure and composition, but is not yet used to a significant extent in a clinical setting. One of the challenges for clinical implementation is that each Raman spectrometer system will have different characteristics and therefore spectra are not readily compatible between systems. This is essential for clinical implementation where classification models are used to compare measured biochemical or tissue spectra against a library training dataset. In this study, we demonstrate the development and validation of a classification model to discriminate between adenocarcinoma (AC) and non-cancerous intraepithelial metaplasia (IM) oesophageal tissue samples, measured on three different Raman instruments across three different locations. Spectra were corrected using system transfer spectral correction algorithms including wavenumber shift (offset) correction, instrument response correction and baseline removal. The results from this study indicate that the combined correction methods do minimize the instrument and sample quality variations within and between the instrument sites. However, more tissue samples of varying pathology states and greater tissue area coverage (per sample) are needed to properly assess the ability of Raman spectroscopy and system transferability algorithms over multiple instrument sites.

We present a high-resolution, near-IR spectroscopic study of multiple outflows in the LkH$\alpha$ 234 star formation region using the Immersion GRating INfrared Spectrometer (IGRINS). Spectral mapping over the blueshifted emission of HH 167 allowed us to distinguish at least three separate, spatially overlapped, outflows in H${_2}$ and [Fe II] emission. We show that the H${_2}$ emission represents not a single jet, but complex multiple outflows driven by three known embedded sources: MM1, VLA 2, and VLA 3. There is a redshifted H${_2}$ outflow at a low velocity, $\VLSR$ $$ 100 {\kms} driven by VLA 3B is also detected in H${_2}$ emission, and confirm that this jet has a position angle about 240$\degree$. Spectra of the redshifted knots at 14$\arcsec$$-$65$\arcsec$ northeast of LkH$\alpha$ 234 are presented for the first time. These spectra also provide clues to the existence of multiple outflows. We detected high-velocity (50$-$120 {\kms}) H${_2}$ gas in the multiple outflows around LkH$\alpha$ 234. Since these gases move at speeds well over the dissociation velocity ($>$ 40 {\kms}), the emission must originate from the jet itself rather than H${_2}$ gas in the ambient medium. Also, position-velocity diagrams and excitation diagram indicate that emission from knot C in HH 167 come from two different phenomena, shocks and photodissociation., Comment: 32 pages, 12 figures, 2 tables, Accepted for publication in the Astrophysical Journal

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem, and we obtain new necessary and sufficient conditions to guarantee that the spectral mapping theorem holds for the B-Weyl spectrum and for polynomials in T. We also prove that the spectral mapping theorem holds for the B-Browder spectrum and for analytic functions on an open neighborhood of \sigma(T). As applications, we show that if T is algebraically M-hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f(T), where f \in H(T), the space of functions analytic on an open neighborhood of \sigma(T). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f(T), for each f \in H(\sigma(T)).

This work deals with spectral mapping theorems for neutron transport semigroups in unbounded geometries and L1 setting. The mathematical analysis relies on harmonic analysis of certain measure valued mappings related to Dyson-Phillips expansions and on some functional analytic results on the critical spectrum.

We give spectral mapping theorems for general neutron transport semigroups on unbounded convex domains. The mathematical analysis relies on the critical spectrum, some related stability results and Fourier analysis of transport equations.

We review the evidence that spectral curvature in the extended emission of radio galaxies is caused by synchrotron losses, and that the spatial variation can be interpreted to yield ages and expansion speeds. One of the biggest worries has been the true value of the magnetic field, but X-ray detections of inverse-Compton radiation are beginning to suggest that “minimum energy” estimates are remarkably accurate. A critical test is to compare model and observed spectra over a broad frequency range; to date this has has only been done for Cygnus A, and the results proved controversial. Here we discuss several more cases and begin to draw some general conclusions.Hotspots are usually well fitted by continuous injection models, as expected. In two cases the implied injection index is flatter than 0.5, too flat to be produced by standard Fermi acceleration in a non-relativistic shock. The bridge spectra are reasonably fitted by single-burst models, but in some objects the injection index is not constant across the lobes, showing instead a tendency to steepen in the inner bridge, where the break frequencies are lowest. The true spectral shape may be a more gradual curve than the standard models, possibly because of mixing of electron populations with different ages. Our results are limited by the inaccuracy of the absolute flux density scale, especially at low frequencies, and a fresh attack on the flux scale would be timely.

We present ~2'x2' spectral-maps of Orion BN/KL outflows taken with Herschel at ~12'' resolution. For the first time in the far-IR domain, we spatially resolve the emission associated with the bright H2 shocked regions "Peak 1" and "Peak 2" from that of the Hot Core and ambient cloud. We analyze the ~54-310um spectra taken with the PACS and SPIRE spectrometers. More than 100 lines are detected, most of them rotationally excited lines of 12CO (up to J=48-47), H2O, OH, 13CO, and HCN. Peaks 1/2 are characterized by a very high L(CO)/L(FIR)~5x10^{-3} ratio and a plethora of far-IR H2O emission lines. The high-J CO and OH lines are a factor ~2 brighter toward Peak 1 whereas several excited H2O lines are ~50% brighter toward Peak 2. A simplified non-LTE model allowed us to constrain the dominant gas temperature components. Most of the CO column density arises from Tk~200-500 K gas that we associate with low-velocity shocks that fail to sputter grain ice mantles and show a maximum gas-phase H2O/CO~10^{-2} abundance ratio. In addition, the very excited CO (J>35) and H2O lines reveal a hotter gas component (Tk~2500 K) from faster (v_S>25 km/s) shocks that are able to sputter the frozen-out H2O and lead to high H2O/CO>~1 abundance ratios. The H2O and OH luminosities cannot be reproduced by shock models that assume high (undepleted) abundances of atomic oxygen in the preshock gas and/or neglect the presence of UV radiation in the postshock gas. Although massive outflows are a common feature in other massive star-forming cores, Orion BN/KL seems more peculiar because of its higher molecular luminosities and strong outflows caused by a recent explosive event., 21 pages, 11 figures. Accepted for publication in ApJ (english not edited)

Using the new periodicity concept based on shifts, we construct a unified Floquet theory for homogeneous and nonhomogeneous hybrid periodic systems on domains having continuous, discrete or hybrid structure. New periodicity concept based on shifts enables the construction of Floquet theory on hybrid domains that are not necessarily additive periodic. This makes periodicity and stability analysis of hybrid periodic systems possible on non-additive domains. In particular, this new approach can be useful to know more about Floquet theory for linear $q$-difference systems defined on $\overline{q^{\mathbb{Z}}}:=\{q^{n}% :n\in\mathbb{Z}\} \cup \{0\}$ where $q>1$. By constructing the solution of matrix exponential equation we establish a canonical Floquet decomposition theorem. Determining the relation between Floquet multipliers and Floquet exponents, we give a spectral mapping theorem on closed subsets of reals that are periodic in shifts. Finally, we show how the constructed theory can be utilized for the stability analysis of dynamic systems on periodic time scales in shifts., 34 pages

We present mid-infrared continuum and emission line images of the Galactic oxygen-rich supernova remnant (SNR) G292.0+1.8, acquired using the MIPS and IRS instruments on the Spitzer Space Telescope. The MIPS 24 micron and 70 micron images of G292.0+1.8 are dominated by continuum emission from a network of filaments encircling the SNR. The morphology of the SNR, as seen in the mid-infrared, resembles that seen in X-rays with the Chandra X-ray Observatory. Most of the mid-infrared emission in the MIPS images is produced by circumstellar dust heated in the non-radiative shocks around G292.0+1.8, confirming the results of earlier mid-IR observations with AKARI. In addition to emission from hot dust, we have also mapped atomic line emission between 14 micron and 36 micron using IRS spectral maps. The line emission is primarily associated with the bright oxygen-rich optical knots, but is also detected from fast-moving knots of ejecta. We confirm our earlier detection of 15-25 micron emission characteristic of magnesium silicate dust in spectra of the radiatively shocked ejecta. We do not detect silicon line emission from any of the radiatively shocked ejecta in the southeast of the SNR, possibly because that the reverse shock has not yet penetrated most of the Si-rich ejecta in that region. This may indicate that G292.0+1.8 is less evolved in the southeast than the rest of the SNR, and may be further evidence in favor of an asymmetric SN explosion as proposed in recent X-ray studies of G292.0+1.8., 16 pages, 1 table, 7 figures, accepted for publication in ApJ

We give necessary and sufficient conditions for a Banach space operator with the single valued extension property (SVEP) to satisfy Weyl's theorem and $a$-Weyl's theorem. We show that if $T$ or $T^{\ast}$ has SVEP and $T$ is transaloid, then Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma (T))$. When $T^{\ast}$ has SVEP, $T$ is transaloid and $T$ is $a$-isoloid, then $a$-Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma (T))$. We also prove that if $T$ or $T^{\ast}$ has SVEP, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum., Comment: 23 pages

This paper is the first reported description of Raman Spectroscopy in the assessment of axillary lymph nodes in breast cancer patients. Raman Spectroscopy is an inelastic scattering spectroscopic technique appropriate for the assessment of unprocessed complex biological tissues. Spectra represent biochemical signatures of the tissue under scrutiny. The described method of Raman spectral mapping produces false-color spectral images of lymph node sections. These can be compared with standard histopathology slides and white light images of nodal tissue. This method has the potential to allow the detailed biochemical assessment of heterogeneous lymph node features, and to contribute towards the development of an optical diagnostic tool for use in a clinical setting.