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In this chapter we introduce another form of variational data assimilation theory known as the physical-space statistical analysis system , or PSAS . The physical space component of the name is in relation to other systems in the early-to-mid-1990s being formulated in spectral space, rather than in grid-point space. Another feature of PSAS-based data assimilation systems is that they are formulated in observation space. In this chapter we shall derive the 3D version of PSAS which we have already seen in Chapters 15 and 16 . Following the 3D version of PSAS, we present the 4D version which is based upon representer theory. Finally we shall summarize the proof that shows that the observation space version of VAR is the dual formulation, similar to the sense of duality in control theory, of the model space formulation.

In this article, we use quite elementary and simple ideas to rebuild and study the patch and flat topologies on the prime spectrum from a natural point of view (this new approach is based on the si...

The purpose of this paper is to study topological properties of both the set of all $k$-prime ideals and the set of all $k$-prime congruences for any commutative semiring with zero and identity. We first prove that the $k$-prime spectrum, i.e. the set of all $k$-prime ideals equipped with the Zariski topology is a spectral space, and then prove that the set of all $k$-prime congruences is homeomorphic to the $k$-prime spectrum with respect to their Zariski topologies.

The objective of this paper is to study topological properties of the prime spectrum of a unitary semimodule over a semiring with zero and nonzero identity. We define a top semimodule over a semiring, show that the prime spectrum is a T 0 -space, and prove that each irreducible closed subset of the prime spectrum has a generic point and that the prime spectrum is a compact space if the top semimodule is finitely generated. For a multiplication semimodule over a commutative semiring, we reveal the structure of the radical of a subsemimodule and prove that the multiplication semimodule is finitely generated iff the prime spectrum is a compact space, that in the prime spectrum, every basic open set is compact and the intersection of finitely many basic open sets is also compact, and that the prime spectrum is a spectral space if the multiplication semimodule is finitely generated.

In this paper, we introduce and study the notions of $ M $-strongly hollow and $ M $-PS-hollow ideals where $ M $ is a module over a commutative ring $ R $. These notions are generalizations of strongly hollow ideals. We investigate some properties and characterizations of $ M $-strongly hollow ($ M $-PS-hollow) ideals. Then we define and study a topology on the set of all $ M $-PS-hollow ideals of a commutative ring $ R $. We investigate when this topological space is irreducible, Noetherian, $ T_{0} $, $ T_{1} $ and spectral space.

The transfer set in calibration transfer can be further refined using a backward refinement (BR) method. Our BR includes many loops, and in each the sample can be removed which reduces the root mean square error of validation (RMSEV) most effectively. This procedure can be repeated many times so that many subsets of samples can be obtained. Then the sample subset with a small value of RMSEV can be chosen as the best sample subset. Two batches of data including corn and wheat datasets are used to test this method. The results show that compared with the ordinary Kennard–Stone method, BR can largely reduce errors when it is applied in calibration transfer methods, including: canonical correlation analysis combined with informative components extraction, direct standardization, piecewise direct standardization PDS and spectral space transformation.

This paper evaluates and modifies the so-called suboptimal control technique for turbulent skin friction reduction through a combination of low-order modelling and direct numerical simulation (DNS). In a previous study, Nakashima et al. (J. Fluid Mech., vol. 828, 2017, pp. 496–526) employed resolvent analysis to show that the efficacy of suboptimal control was mixed across spectral space when the streamwise wall shear stress (case ST) was used as a sensor signal, i.e. specific regions of spectral space showed drag increment. This observation suggests that drag reduction may be attained if control is applied selectively in spectral space. DNS results presented in the present study, however, do not show a significant effect on the flow with selective control. A posteriori analyses attribute this lack of efficacy to a much lower actuation amplitude in the simulations compared to model assumptions. Building on these observations, resolvent analysis is used to design and provide a preliminary assessment of modified control laws that also rely on sensing the streamwise wall shear stress. Control performance is then assessed by means of DNS. The proposed control laws generate as much as $10\,\%$ drag reduction, and these results are broadly consistent with resolvent-based predictions. The physical mechanisms leading to drag reduction are assessed via conditional sampling. It is shown that the new control laws effectively suppress the near-wall quasi-streamwise vortices. A physically intuitive explanation is proposed based on a separate evaluation of clockwise and anticlockwise vortices.

We study the set of localizations of an integral domain from a topological point of view, showing that it is always a spectral space and characterizing when it is a proconstructible subspace of the space of all overrings. We then study the same problems in the case of quotient rings, flat overrings and sublocalizations., Comment: to appera in the Journal of Pure and Applied Aplgebra

Modern monothetic hierarchical soil classification systems such as the Chinese Soil Taxonomy (CST) are semi‐quantitative and their future development is likely to trend towards a fully quantitative system using the concepts of numerical taxonomy. Previous researchers have calculated the taxonomic distances between individual soils based on soil physiochemical properties, not, however, based on spectra of full soil profiles with different horizons. We hypothesized that numerical taxonomy implemented by cluster analysis of the taxonomic distance matrix based on vis–NIR spectra would accord with some CST Orders assigned by pedologists, and not with others, depending on how closely spectral features represent the diagnostic features used in the classification. Taxonomic distances in spectral space were computed for all pairs of 191 profiles, resulting in a distance matrix on which hierarchical cluster analysis was performed. Different indices were calculated to determine the optimum number of clusters, resulting in four spectral soil classes. These were then compared with CST Orders assigned to the profiles by expert allocation. The numerical classes and CST Orders matched poorly because of the completely different classification philosophies behind numerical taxonomy and the CST, which is based largely on presumed genesis and uses sharp thresholds leading to very similar soils being allocated to different classes. Thus, we consider the numerical classification as information that is complementary to the monothetic hierarchical system. Numerical classification can reveal the taxonomic objective aspects of the relation between the defined classes and can suggest new groupings. HIGHLIGHTS: Taxonomic distances can serve as an objective measure of soil similarity. Soil spectra are a good candidate for numerical classification based on taxonomic distances. The conceptual basis of Chinese Soil Taxonomy (CST) does not match that of taxonomic distance. Numerical classification based on spectra can suggest revisions to the CST.

Natural convection (NC) in high permeable porous media is usually investigated using the Darcy–Lapwood–Brinkman model (DLB). The problem of the porous squared cavity is widely used as a common benchmark case for NC in porous media. The solutions to this problem with the DLB model are limited to steady-state conditions. In this paper, we developed a time-dependent high accurate solution based on the Fourier–Galerkin method (FG). The solution is derived considering two configurations dealing with unsteady and transient modes. The governing equations are reformulated using the stream function. The Temperature and the stream functions are expended as unknowns in space using Fourier series which are appropriately substituted in the equations. The equations are then projected to the spectral space using Fourier trigonometric trial functions. The obtained developed equations form a nonlinear differential algebraic system of equations. An appropriate technique is used to integrate the spectral system in time and to ensure high accuracy. The results of the FG method are compared to a finite element solution for different Rayleigh and Darcy numbers values. The transient and unsteady solutions are obtained with a feasible and low computational cost. The paper provides high accurate time-dependent solutions useful for benchmarking numerical models dealing with NC in porous media. The results of the developed solutions are efficient to gain physical insight into the time-dependent NC processes.

This paper proposed a weighted principal component analysis method based on cone response function to reserve more color information. To verify the advantages of the proposed method, Munsell spectra were used as training samples to determine the conversion model between high-dimensional spectral space and low-dimensional space, Munsell, ISO SOCS spectra and two multispectral images were used as test samples. Compared with the principal component analysis method, the proposed method can significantly improve the colorimetric accuracy at the expense of a small amount of spectral accuracy. In addition, compared with the other three weighted principal component analysis methods based on cone response function, this method has a lot of improvement in colorimetric accuracy.

The purpose of this paper is to study α -ideals in a more general context, in universal algebras having a constant 0 . Several characterizations are obtained for an ideal I of an algebra A to be an α -ideal. It is shown that the class of all α -ideals of an algebra A forms an algebraic lattice. Prime α -ideals and several related properties are investigated. Some properties of the spectral space of prime α -ideals equipped with the hull-kernel topology are derived.

v4 is the final version; International audience; In an earlier paper we established that every second countable, completely normal spectral space is homeomorphic to the ℓ-spectrum of some Abelian ℓ-group. We extend that result to ℓ-spectra of vector lattices over any countable totally ordered division ring k. Extending our original machinery, about finite lattices of polyhedra, from linear to affine and allowing relativizations to convex subsets, then invoking Baro's Normal Triangulation Theorem, we obtain the following result:Theorem. For every countable formally real field k, every second countable, completely normal spectral space is homeomorphic to the real spectrum of some commutative unital k-algebra.The countability assumption on k is necessary: there exists a second countable, completely normal spectral space that cannot be embedded, as a spectral subspace, into either the ℓ-spectrum of any right vector lattice over an uncountable directed partially ordered division ring, or the real spectrum of any commutative unital algebra over an uncountable field.

The category of distributive lattices is, in classical mathematics, antiequivalent to the category of spectral spaces. We give here some examples and a short dictionary for this antiequivalence. We propose a translation of several abstract theorems (in classical mathematics) into constructive ones, even in the case where points of a spectral space have no clear constructive content.

We correct three issues in the paper “The Zariski topology on sets of semistar operations without finite-type assumptions”.

A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections. Theorem. A topological space X is homeomorphic to the spectrum of some countable Abelian {\ell}-group with unit (resp., MV-algebra) iff X is spectral, has a countable basis of open sets, and for any points x and y in the closure of a singleton {z}, either x is in the closure of {y} or y is in the closure of {x}. We establish this result by proving that a countable distributive lattice D with zero is isomorphic to the lattice of all principal ideals of an Abelian {\ell}-group (we say that D is {\ell}-representable) iff for all a, b $\in$ D there are x, y $\in$ D such that a $\lor$ b = a $\lor$ y = b $\lor$ x and x $\land$ y = 0. On the other hand, we construct a non-{\ell}-representable bounded distributive lattice, of cardinality $\aleph$ 1 , with an {\ell}-representable countable L$\infty, \omega$-elementary sublattice. In particular, there is no characterization, of the class of all {\ell}-representable distributive lattices, in arbitrary cardinality, by any class of L$\infty, \omega$ sentences., Comment: Misprints v2: In Example 7.1, (a-mb)\wedge(b-mc) \leq 0 (i.e., \wedge instead of \vee).In Corollary 8.6, X, Y^-, and Y^+ are just elements of \Op(\mathcal{H}) (not necessarily basic open)

In order to solve the calibration transfer problem in near-infrared (NIR) spectroscopy, a fast and efficient method named principal components canonical correlation analysis (PC-CCA) has been proposed. PC-CCA can extract principal components of spectra from slave instrument with principal component analysis (PCA), and then transfer the slave principal components toward latent variables of partial least squares (PLS) model with canonical correlation analysis directly. Two NIR datasets of corn and tobacco samples measured with three and two spectrometers, respectively, were used to test the reliability of this method. The piecewise direct standardization (PDS), spectral space transformation (SST) and calibration transfer method based on canonical correlation analysis (CTCCA) methods were performed for comparative study of the proposed model transfer technique. The results of both datasets reveal that PC-CCA can drastically reduce time of transfer and lead to dozens-fold speedup. It can also reduce prediction errors and achieve the smallest root mean square errors of prediction (RMSEPs). For example, the spectra transfer from M5 to MP5, the comparative experiment results show that RMSEP with the proposed method PC-CCA is reduced from 2.5641 (without transfer) to 0.0934, and much smaller than that with PDS (0.1999), CTCCA (0.1377) and SST (0.1296) at N = 50. These advantages make PC-CCA a promising calibration transfer method in NIR application.

A classical soil line (SL) in the RED–NIR spectral space is specified by two coefficients “a” and “b.” In this form, it does not characterize soil types and subtypes. A multitemporal soil line (MSL) represents the major axis of the ellipse describing all possible pairs of RED–NIR values characterizing a bare soil surface for a given pixel of remote sensing images. The MSL in the RED–NIR spectral space is specified by several (N) coefficients. The resulting N-dimensional space of MSL coefficients makes it possible to give unique characteristics for each type and subtype of soils in the following zonal soil sequence: soddy-podzolic soils, light gray forest soils, gray forest soils, dark gray forest soils, podzolized chernozems, and leached chernozems. The analysis of variance allows us to state that the soils of this sequence significantly differ from one another in the characteristic sets of MSL coefficients. In other words, these coefficients characterize soil types and subtypes, and the MSL can be considered an empirical soil line (ESL) of the given type and subtype of soil. A classical SL is an integrity of ESLs of different soils within the given scene of remote sensing data.

Let Spec(T) be the spectrum of a tensor- triangulated category (T, circle times , 1). We show that there is a homeomorphism between the spectral space of radical thick tensor ideals in (T, circle times , 1) and the collection of open subsets of Spec(T) in inverse topology. In fact, we prove a more general result in terms of supports on (T, circle times , 1) and work by combining methods from commutative algebra, topology and tensor triangular geometry. (C) 2018 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

In this paper, we present a new mesh segmentation method that achieves visually meaningful segmentation by combining mesh saliency with spectral clustering. Our method solves the segmentation problem by embedding the original mesh model into spectral space. Firstly, the mesh concave regions are determined according to the minimum rule in visual theory, and then a Laplacian matrix is defined by considering the mesh saliency and curvature information. Next, we calculate the first k eigenvectors of the Laplacian matrix by eigen-decomposition process, and embed the original mesh into a k-dimensional spectral space. Finally, we can achieve the visually meaningful segmentation by utilizing the Gaussian Mixture method, and the initial cluster centers are decided by mesh saliency. The experimental results have demonstrated the effectiveness of the proposed segmentation method. Especially for the model with convex regions and branch components, our method can achieve better visual quality.

Calibration transfer is of great necessity for practical applications of near infrared (NIR) spectroscopy, since the original calibration model would become invalid when spectra are measured on different instruments or under different detection conditions. In the calibration transfer method of spectral space transformation (SST) that has shown excellent performance in practice, loadings of PCA have been proven to contain important information about differences between different instruments. Just like the loadings of PCA, the weight matrix of PLS contains much information, including the weights of absorbance at different wavelengths for each PLS factor. In this study, two new calibration transfer methods based on the weight matrix (CTWM) of partial least squares (PLS) were proposed. CTWM1 uses PLS to build a linear regression model between the secondary spectra of standardization samples and the low-dimensional information extracted from the primary spectra of standardization samples using the weight matrix of the original calibration model. CTWM2 retains the weight matrix but uses ordinary least squares (OLS) to update the y-weight vector using the secondary spectra and property values of standardization samples. CTWM1 can be applied to the occasion when both the primary and secondary spectra of standardization samples can be obtained, while CTWM2 can be applied to the occasion when the primary spectra of standardization samples cannot be obtained but the observed property values can be. The only parameter in CTWM1 and CTWM2 is the number of latent variables selected in the weight matrix and it can be set to a reasonable value according to the number of standardization samples and the number of latent variables selected in the original primary calibration model, which means that CTWM1 and CTWM2 have the advantage of easy implementation and are practical for calibration transfer. The performance of CTWM1 and CTWM2 has been tested on two NIR datasets. Compared with slope and bias correction (SBC), spectral space transformation (SST) and piecewise direct standardization (PDS), CTWM1 shows good performance and can give reliable and accurate results. Compared with recalibration using standardization samples, CTWM2 shows good performance when the difference in instrument responses is relatively small.

In this paper a pseudo-spectral incompressible smoothed particle hydrodynamics (FFT-ISPH) solver is presented. While the solution of the linear system arising from the pressure Poisson equation in physical space using iterative solvers in incompressible SPH is a viable solution, it is widely accepted that the computational cost of solving the pressure Poisson equation by iterative solvers is excessive and affects negatively on the efficiency of the solver. The proposed scheme is an intermediate between a fully spectral and a standard incompressible SPH solver. Herein, the solution of the pressure Poisson equation is performed in spectral space whereas the discretisation and time integration are performed in physical space. This results in a second and higher-order scheme with the classical first-order projection and a third-order Runge-Kutta time integration scheme, respectively. A detailed performance analysis shows gains in computational cost of two orders of magnitude. Further, it is demonstrated that the solution of the pressure Poisson equation in spectral space is independent of the number of neighbouring nodes, in contrast to the iterative solver whose cost increases by a factor of three with smoothing length to particle size ratio. Periodic, bounded, and mixed boundary conditions test cases have been used to demonstrate the applicability, accuracy and robustness of the scheme with second and fourth-order convergence rates. The scheme opens an avenue for simulations of high-order incompressible flows in SPH where filtering operations in the frequency domain can be performed straightforwardly.

The potential of the red-NIR spectral space has been frequently investigated for the sole estimation of the Leaf Area Index (LAI) or Soil Moisture (SM) values. However, the applicability of the space for simultaneous prediction of the LAI and SM has not yet been studied. In this study, a general model was first proposed for the contemporaneous estimation of the LAI and SM . Then, the correlations between the field LAI and SM data and ten parameters, extracted from the red-NIR space, were assessed to improve the accuracy of the general model. The results indicated that five of these ten parameters were more suitable for LAI estimation, while the other five provided more information for SM estimation. Using this information, a modified model was finally developed to estimate the LAI and SM contemporaneously, for which the results were more promising than those obtained from the general model. Using the modified model, the correlation coefficient (r) and the Root Mean Square Error (RMSE) for LAI esti...

We examine the effect of variable filter width in the partially integrated transport modeling (PITM) method in the framework of second moment closure (SMC) and the commutation errors arising from the non-commutativity of the filtering process with temporal or spatial derivatives. We propose a method to account for this effect in PITM numerical simulations. In particular, we model the commutation terms in a practical way to get tractable Navier–Stokes and turbulence equations. We apply a discrete approximation to the top hat filter to compute the commutation terms for the velocities and turbulent stresses by means of a superfilter width. A special attention is devoted to the treatment of the turbulent stresses. Moreover, a physical interpretation of the commutation terms is given in the spectral space involving the flux of energy transfer of the spectrum. For illustration purpose, we perform numerical simulations of the fully developed turbulent channel flow on several grids including a sudden grid step increase in the grid-size in the streamwise direction that is akin to grid discontinuities. The commutation errors are then distributed in a small delimited region of the field. As a result, we found that the impact of the commutation terms on the PITM solution remains moderate even if the grid-size variation is increasing up to 400% in the streamwise direction but the accounting for the commutation errors has however a slight beneficial effect on the solution by improving the flow prediction.

Closure models started from Chou’s work have been developed for more than 70 years, aiming at providing analytical tools to describe turbulent flows in the spectral space. In this study, a preliminary attempt is presented to introduce a closure model in the physical space, using the velocity structure functions as key parameters. The present closure model appears to qualitatively reproduce the asymptotic scaling behaviors at small and large scales, despite some inappropriate behaviors such as oscillations. Therefore, further improvements of the present model are expected to provide appropriate descriptions of turbulent flows in the physical space.

The second spectrum Specs(M) is the collection of all second elements of M. In this paper, we study the topology on Specs(M), which is a generalization of the Zariski topology on the prime spectrum of lattice modules. Besides some properties, Specs(M) is characterized and the interrelations between the topological properties of Specs(M) and the algebraic properties of M, are studied.

Within-class spectral variation, which is caused by varied imaging conditions, such as changes in illumination, environmental, atmospheric, and temporal conditions, significantly degrades the performance of hyperspectral image classification. Recent studies have shown that such spectral variation can be alleviated by exploring the low-rank property in the spectral domain, especially based on the low-rank subspace assumption. In this paper, the low-rank subspace assumption is approached by exploring the low-rank property in the local spectral domain. In addition, the low-rank property in the spatial domain is also explored to alleviate spectral variation. As a result, two novel spectral-spatial low-rank (SSLR) strategies are designed to alleviate spectral variation by exploring the low-rank property in both spectral and spatial domains. Experimental results on two benchmark hyperspectral datasets demonstrate that exploring the low-rank property in local spectral space can help to alleviate spectral variation and improve the performance of classification obviously for all tested data, while exploring the low-rank property in spatial space is more effective for images presenting large homogeneous areas.

Hyperspectral images (HSIs) usually contain hundreds of spectral bands. When they are used for classification tasks, HSIs may suffer from the curse of high dimensionality. To address this problem, the essential procedures of dimension reduction and feature extraction (FE) are employed. In this letter, we propose an FE method for HSIs using low-rank representation with neighborhood preserving regularization (LRR_NP). The proposed method can simultaneously employ locally spatial similarity and the spectral space structure, which comprises a union of multiple low-rank subspaces. The framework of LRR can structurally represent the union structure of a spectral space. Because spatial neighbor pixels always share high similarity in a feature space, an NP regularization item is introduced into the framework of LRR to consider the locally spatial correlation. Classification experiments are conducted on real HSI data sets; the results demonstrate that the features that are extracted by LRR_NP are more discriminative than the state-of-art methods, including both unsupervised methods and supervised methods.

Several papers deal with the problem of counting the number of idempotent endomorphisms of a structure S onto a substructure T. In this paper we consider the case when T is a projective lattice-ordered abelian group with a distinguished strong order unit, or equivalently, a projective MV-algebra. Suppose A is the image (=range) of an idempotent endomorphism of the free n-generator MV-algebra M ( [ 0 , 1 ] n ) of McNaughton functions on [ 0 , 1 ] n . We prove that the number r ( A ) of idempotent endomorphisms of M ( [ 0 , 1 ] n ) onto A is finite if, and only if, the maximal spectral space μ A is homeomorphic to a (Kuratowski) closed domain M of [ 0 , 1 ] n , in the sense that M = cl ( int ( M ) ) . Further, the closed domain condition is decidable and r ( A ) is computable, once an idempotent endomorphism of M ( [ 0 , 1 ] n ) onto A is explicitly given. Thus every finitely generated projective MV-algebra B comes equipped with a new invariant ι ( B ) = sup ⁡ { r ( A ) | A ≅ B , for A the image of an idempotent endomorphism of M ( [ 0 , 1 ] k ) } , and k the smallest number of generators of B. We compute ι ( B ) for many projective MV-algebras B existing in the literature. Various problems concerning idempotent endomorphisms of free MV-algebras are shown to be decidable. Via the Γ functor, our results and computations automatically transfer to finitely generated projective abelian l-groups with a distinguished strong unit.

A spectral space is localic if it corresponds to a frame under Stone Duality. This class of spaces was introduced by the author (under the name ’locales’) as the topological version of the classical frame theoretic notion of locales, see Johnstone and also Picado and Pultr). The appropriate class of subspaces of a localic space are the localic subspaces. These are, in particular, spectral subspaces. The following main questions are studied (and answered): Given a spectral subspace of a localic space, how can one recognize whether the subspace is even localic? How can one construct all localic subspaces from particularly simple ones? The set of localic subspaces and the set of spectral subspaces are both inverse frames. The set of localic subspaces is known to be the image of an inverse nucleus on the inverse frame of spectral subspaces. How can the inverse nucleus be described explicitly? Are there any special properties distinguishing this particular inverse nucleus from all others? Colimits of spectral spaces and localic spaces are needed as a tool for the comparison of spectral subspaces and localic subspaces.

We consider a closure operator $c$ of finite type on the space $SMod(\mathcal M)$ of thick $\mathcal K$-submodules of a triangulated category $\mathcal M$ that is a module over a tensor triangulated category $(\mathcal K,\otimes,1)$. Our purpose is to show that the space $SMod^c(\mathcal M)$ of fixed points of the operator $c$ is a spectral space that also carries the structure of a topological monoid., Comment: 10 pages

Let R be a commutative ring with identity and $$\hbox {Spec}^{s}(M)$$ denote the set all second submodules of an R-module M. In this paper, we investigate various properties of $$\hbox {Spec}^{s}(M)$$ with respect to different topologies. We investigate the dual Zariski topology from the point of view of separation axioms, spectral spaces and combinatorial dimension. We establish conditions for $$\hbox {Spec}^{s}(M)$$ to be a spectral space with respect to quasi-Zariski topology and second classical Zariski topology. We also present some conditions under which a module is cotop.

International audience; In-field hyperspectral imagery is a promising tool for crop phenotyping or monitoring. In association with partial least square regression (PLS-R), it allows building high spatial resolution maps of the chemical content of plant leaves. However, several optical phenomena must be taken into account, due to their influence on collected spectral data. The most challenging is multiple scattering, produced when a leaf is partly illuminated by light reflection or transmission from neighboring leaves. It can induce bias in prediction results. This paper presents a method for multi-scattering correction. Its development has been based on simulation tools: a 3D canopy model of winter wheat was combined with light propagation modeling, in order to simulate the apparent reflectance of every visible leaf in the canopy for a given actual reflectance. Leaf nitrogen content (LNC) prediction has been considered. A data set of reflectance spectra associated with LNC values has been issued from real leaf measurements. A theoretical disturbance subspace representing the spectrum dispersion in the spectral space due to multi-scattering has then been built by considering polynomial combinations of the initial spectra, and a projection along this subspace has been applied to every simulated spectra. Using this strategy, a PLS-R model built on initial spectra was still satisfactory when applied to simulated spectra with multiple scattering. The method has then been applied to real plants in greenhouse and field conditions, and its prediction results compared with those of a standard PLS-R, confirming its efficiency in the presence of various lighting environments.

This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian l-groups. As a first main result, we show that a topological space $X$ is the prime spectrum of an MV-algebra if and only if: (1) $X$ is spectral, and (2) the lattice of compact open subsets of $X$ is an epimorphic image of a lattice of "cylinder rational polyhedra" (a natural generalization of rational polyhedra) of some hypercube. As a second main result we extend our results to Abelian l-groups. That is, let $X$ be a spectral space and $K(X)$ the lattice of its compact open sets. The following are equivalent: (1) $X$ is the spectrum of some Abelian l-group; (2) $X$ is homeomorphic to $Spec(K(X))$ and $K(X)\cup\{\infty\}$ is isomorphic to the lattice of the compact open sets of a local MV-algebra, where $\infty>x$ for every $x\in K(X)$. Finally we axiomatize, in monadic second order logic, the lattices of cylinder rational polyhedra of dimension $1$ and $2$., The main result of the paper (Theorem 5.15) is flawed. We thank Fred Wehrung for pointing out the error

[EN] The objective of this work is the introduction and investigation of favourable time integration methods for the Gross-Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes the form of a nonlinear Schrödinger equation involving a space-time-dependent potential. A natural approach that combines commutator-free quasi-Magnus exponential integrators with operator splitting methods and Fourier spectral space discretisations is proposed. Furthermore, the special structure of the Hamilton operator permits the design of specifically tailored schemes. Numerical experiments confirm the good performance of the resulting exponential integrators., Part of this work was developed during a research stay at the Wolfgang Pauli Institute Vienna; the authors are grateful to the director Norbert Mauser and the staff members for their support and hospitality. Philipp Bader, Sergio Blanes, and Fernando Casas acknowledge funding by the Ministerio de Economía y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE).

The analysis of 34 cloudless fragments of Landsat 5, 7, and 8 images (1985–2014) on the territory of Plavsk, Arsen’evsk, and Chern districts of Tula oblast has been performed. It is shown that bare soil surface on the RED–NIR plots derived from the images cannot be described in the form of a sector of spectral plane as it can be done for the NDVI values. The notion of spectral neighborhood of soil line (SNSL) is suggested. It is defined as the sum of points of the RED–NIR spectral space, which are characterized by spectral characteristics of the bare soil applied for constructing soil lines. The way of the SNSL separation along the line of the lowest concentration density of points on the RED–NIR spectral space is suggested. This line separates bare soil surface from vegetating plants. The SNSL has been applied to construct soil line (SL) for each of the 34 images and to delineate bare soil surface on them. Distances from the points with averaged RED–NIR coordinates to the SL have been calculated using the method of moving window. These distances can be referred to as averaged spectral deviations (ASDs). The calculations have been performed strictly for the SNSL areas. As a result, 34 maps of ASDs have been created. These maps contain ASD values for 6036 points of a grid used in the study. Then, the integral map of normalized ASD values has been built with due account for the number of points participating in the calculation (i.e., lying in the SNSL) within the moving window. The integral map of ASD values has been compared with four traditional soil maps on the studied territory. It is shown that this integral map can be interpreted in terms of soil taxa: the areas of seven soil subtypes (soddy moderately podzolic, soddy slightly podzolic, light gray forest. gray forest, dark gray forest, podzolized chernozems, and leached chernozems) belonging to three soil types (soddy-podzolic, gray forest, and chernozemic soils) can be delineated on it.

In this study, a novel method for clustering hyperspectral images is proposed. The proposed method performs projected clustering in feature/spectral space and merges regions in image/spatial space. The novelty of the proposed method lies in the way in which spectral and spatial information is used, along with its inclusion in the projected clustering framework. The proposed method transfers clusters formed in feature space to image space by converting them into regions. Then in image space, regions are iteratively merged by making use of spatial adjacency and spectral similarity. To evaluate the effectiveness of the proposed method, experiments are conducted on three hyperspectral images. The proposed method is also compared with other partitional clustering methods. Results demonstrate that the proposed method has ability to achieve better performance in most cases.

Differential equations may possess coefficients that vary on a spectrum of scales. Because coefficients are typically multiplicative in real space, they turn into convolution operators in spectral space, mixing all wavenumbers. However, in many applications, only the largest scales of the solution are of interest and so the question turns to whether it is possible to build effective coarse-scale models of the coefficients in such a manner that the large scales of the solution are left intact. Here we apply the method of numerical homogenization to deterministic linear equations to generate sub-grid-scale models of coefficients at desired frequency cutoffs. We use the Fourier basis to project, filter and compute correctors for the coefficients. The method is tested in 1D and 2D scenarios and found to reproduce the coarse scales of the solution to varying degrees of accuracy depending on the cutoff. We relate this method to mode-elimination Renormalization Group (RG) and discuss the connection between accuracy and the cutoff wavenumber. The tradeoff is governed by a form of the uncertainty principle for convolutions, which states that as the convolution operator is squeezed in the spectral domain, it broadens in real space. As a consequence, basis sparsity is a high virtue and the choice of the basis can be critical., 24 pages, 6 figures, in press, Journal of Computational Physics

Near-infrared (NIR) has been successfully applied for rapid and nondestructive analyses in diverse fields, but the issue of calibration model transfer has not been clarified thus far. Typically, the calibration model developed for one instrument cannot be applied directly to other instruments in many cases. To achieve calibration model transfer between the master and slave instruments, an improved piecewise direct standardization designated Rank–Kennard–Stone-PDS (Rank-KS-PDS) is proposed in this paper. An optimal sample selection method (Rank-KS) is proposed to select the transfer samples. The process of sample selection is based simultaneously on the distribution of spectral space and of property space. Thus, the samples selected by Rank-KS have greater representativeness and wider coverage. The Rank-KS-PDS method has been used to predict the alkaloid and glycoside content in tobacco. Comparative studies of calibration model transfer with the proposed method, random selection (RS), and KS have also been performed. For the prediction model of alkaloids, the comparative experiment results have verified the reduction in the number of transfer samples required from 12 to 8 and in the root-mean-square error of prediction (RMSEP) from 0.1440 to 0.1250 with the proposed method. For the prediction model of glycosides, the accuracy of model transfer was found to improve significantly, accompanied by a reduction in RMSEP from 1.6945 to 1.5850 without any increase in the number of transfer samples. With the proposed Rank-KS-PDS method, the selected transfer samples based on one property can be directly applied to the calibration model transfer of other properties with satisfactory results.

Spectral unmixing estimates the abundance of each endmember at every pixel of a hyperspectral image. Each material in traditional unmixing algorithms is represented through a constant spectral signature. However, endmember variability always exists due to environmental, atmospheric, and temporal conditions, which leads to poor accuracy of the estimated abundances. This paper proposes a new unmixing algorithm based on a new linear transformation called endmember orthonormal mapping (EOM) to overcome the aforementioned problem. The EOM transformation maps original spectral space to a new EOM space to reduce endmember variability. In the original spectral space, each material is represented by a set of spectra (endmember set) which is extracted using the automated endmember bundles (AEB) method. The EOM transforms each endmember set to a vector in the EOM space so that these vectors are orthonormal. On account of orthonormalized endmembers, the condition number of the mixing matrix in the EOM space reduces. Furthermore, we consider the noise term as an additional virtual endmember set mapped to a vector that is orthogonal to other endmembers. As a result, a promising unmixing accuracy is obtained through applying the least squares abundance estimation in the subspace orthogonal to noise. Experimental results of both synthetic and real hyperspectral images demonstrate that the proposed algorithms provide much enhanced performance compared with the state-of-the-art algorithms.

Hyperspectral Imaging is one of the most popular technologies in plant phenotyping. Various kinds of hyperspectral imaging systems have been developed in the past years. However, there are always significant differences in sensors and imaging environmental conditions between different facilities, which makes it difficult to share image processing algorithms and plant features prediction models. Calibration transfer between the imaging systems is critically important. The white referencing calibration has been proved effective in removing the major lighting intensity differences, but significant spectral differences between systems still exist. In this study, we considered the calibration transfer as a spectral adjustment process and compared four different methods (DS, PDS, DPDS and SST) for adjusting the spectra. To perform the calibration transfer, maize plants were imaged using push-broom style, top-view VNIR hyperspectral camera in two greenhouse phenotyping facilities (inter-facility) and within the same facility (intra-facility) under different imaging conditions. The suggested spectral adjustment methods were tested using master partial least squares regression (PLSR) based calibration models developed for predicting the relative water content (RWC) and nitrogen content (N) of maize plants. Results showed that spectral space transformation (SST) decreased the RMSEv from 9.450% and 10.636% to 3.280% and 2.424% for the predicted RWC of intra and inter-facility transfer, respectively, while the corresponding measures achieved by direct standardization (DS) were 3.412% and 3.105%, respectively. In case of N predictions, similar results were observed for DS and SST. The other two methods (PDS and DPDS) reduced the prediction error for RWC and N, but their performance was not on par with DS and SST. The results indicated that the proposed methods, especially DS and SST were able to alleviate the perturbations inherited in the spectra and thus can help to avoid the need for time-consuming, labor-intensive and costly full recalibration process that arose due to the intra or inter-facility variations.

Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper, we introduce and study a new topology on Cl.Specg(M), the collection of all graded classical prime submodules of M, called the Zariski-like topology. Then we investigate the relationship between algebraic properties of M and topological properties of Cl.Specg(M). Moreover, we study Cl.Specg(M) from point of view of spectral space.

Given an arbitrary spectral space $X$, we consider the set ${\boldsymbol{\mathcal{X}}}(X)$ of all nonempty subsets of $X$ that are closed with respect to the inverse topology. We introduce a Zariski-like topology on ${\boldsymbol{\mathcal{X}}}(X)$ and, after observing that it coincides the upper Vietoris topology, we prove that ${\boldsymbol{\mathcal{X}}}(X)$ is itself a spectral space, that this construction is functorial, and that ${\boldsymbol{\mathcal{X}}}(X)$ provides an extension of $X$ in a more `complete' spectral space. Among the applications, we show that, starting from an integral domain $D$, ${\boldsymbol{\mathcal{X}}}(\mathrm{Spec}(D))$ is homeomorphic to the (spectral) space of all the stable semistar operations of finite type on $D$., to apper in the Rocky Mountain J. Math

In the current work, we attempt to leverage the fewer wavelengths and samples to develop a classification model for classifying hard and soft blueberries using near infrared (NIR) data. To do this, random frog selection and active learning approaches are used in the spectral space and the sample queue, respectively. To reduce the spectral number, a random frog spectral selection approach was applied to collect wavelengths informative with hardness. Prediction model based on 22 selected spectra gave slightly better results than that based on the full spectra. In terms of the selection operation in the sample space, the query by committee was validated to be suitable for blueberry hardness classification with the accuracy, precision and recall of 78%, 74% and 98% when taking only 25 sample queries. Its standard deviation curves of performance metrics are also located in regions of low values (around 0.05) and fluctuated steadily in shape, winning over those of the other 4 active learning strategies and random method. In summary, the respective uses of random frog and query by committee in the NIR spectral vector and the sample queue showed the considerable potential for establishing a simple but robust classifier for hard and soft blueberries with very low labeling cost.