Whole Set Of Solutions Research Articles

Retrieval and analysis of the composition of an aerosol mixture through Mie–Raman–fluorescence lidar observations

Abstract. In the atmosphere, aerosols can originate from numerous sources, leading to the mixing of different particle types. This paper introduces an approach to the partitioning of aerosol mixtures in terms of backscattering coefficients. The method utilizes data collected from the Mie–Raman–fluorescence lidar, with the primary input information being the aerosol backscattering coefficient (β), particle depolarization ratio (δ), and fluorescence capacity (GF). The fluorescence capacity is defined as the ratio of the fluorescence backscattering coefficient to the particle backscattering coefficient at the laser wavelength. By solving a system of equations that model these three properties (β, δ and GF), it is possible to characterize a three-component aerosol mixture. Specifically, the paper assesses the contributions of smoke, urban, and dust aerosols to the overall backscattering coefficient at 532 nm. It is important to note that aerosol properties (δ and GF) may exhibit variations even within a specified aerosol type. To estimate the associated uncertainty, we employ the Monte Carlo technique, which assumes that GF and δ are random values uniformly distributed within predefined intervals. In each Monte Carlo run, a solution is obtained. Rather than relying on a singular solution, an average is computed across the whole set of solutions, and their dispersion serves as a metric for method uncertainty. This methodology was tested using observations conducted at the ATOLL (ATmospheric Observation at liLLe) observatory, Laboratoire d'Optique Atmosphérique, University of Lille, France.

Inertia matrix identification of combined spacecraft using a deep neural network with optimized network structure

After the space robot captures the out-of-control spacecraft, they form a combined spacecraft. For the parameter identification problem of combined spacecraft with complex angular rate measurement noise, deep learning could provide a new solution, due to its ability to extract robust features from noisy data. However, the deep neural network (DNN) has substantial computing resource requirements, which restricts its application in spacecraft systems. To solve the above problems, an inertia matrix identification approach combining a structure optimization strategy of DNN is proposed to identify the inertia matrix in the presence of complex measurement noise using the DNN with optimized structure. The attitude dynamics of combined spacecraft are firstly derived, and the identification problem considering complex measurement noise is described. Based on the dynamics model, the construction process of DNN for accurately identifying the inertia matrix is proposed. A structure optimization strategy of DNN is then designed to optimize the structure of DNN for increasing computing efficiency. This strategy provides a whole set of solutions including the structure selection of the initial DNN, the redundant neurons pruning process, and the end-of-optimization condition. By combining the proposed construction process of DNN, the structure of the DNN is finally optimized by iterating the structured pruning and retraining step. The performance of the proposed DNN with optimized network structure is verified on a testing set with complex measurement noise. The results show that the proposed approach can accurately identify the inertia matrix of combined spacecraft, and meanwhile greatly improve the computing efficiency of DNN.

Graphical and Computational Tools to Guide Parameter Choice for the Cluster Weighted Robust Model

The Cluster Weighted Robust Model (CWRM) is a recently introduced methodology to robustly estimate mixtures of regressions with random covariates. The CWRM allows users to flexibly perform regression clustering, safeguarding it against data contamination and spurious solutions. Nonetheless, the resulting solution depends on the chosen number of components in the mixture, the percentage of impartial trimming, the degree of heteroscedasticity of the errors around the regression lines and of the clusters in the explanatory variables. Therefore, an appropriate model selection is crucially required. Such a complex modeling task may generate several “legitimate” solutions: each one derived from a distinct hyperparameters specification. The present article introduces a two step-monitoring procedure to help users effectively explore such a vast model space. The first phase uncovers the most appropriate percentages of trimming, whilst the second phase explores the whole set of solutions, conditioning on the outcome derived from the previous step. The final output singles out a set of “top” solutions, whose optimality, stability and validity is assessed. Novel graphical and computational tools—specifically tailored for the CWRM framework—will help the user make an educated choice among the optimal solutions. Three examples on real datasets showcase our proposal in action. Supplementary files for this article are available online.

Extended lattice Gelfand–Dickey hierarchy

The lattice Gelfand–Dickey (GD) hierarchy is a lattice version of the GD hierarchy. A special case is the lattice KdV hierarchy. Inspired by recent work of Buryak and Rossi, we propose an extension of the lattice GD hierarchy. The extended system has an infinite number of logarithmic flows alongside the usual flows. We present the Lax, Sato and Hirota equations and a factorization problem of difference operators that captures the whole set of solutions. The construction of this system resembles the extended 1D and bigraded Toda hierarchy, but exhibits several novel features as well.

Course Recommendation based on Sequences: An Evolutionary Search of Emerging Sequential Patterns

To provide a good study plan is key to avoid students’ failure. Academic advising based on student’s preferences, complexity of the semester, or even background knowledge is usually considered to reduce the dropout rate. This article aims to provide a good course index to recommend courses to students based on the sequence of courses already taken by each student. Hence, unlike existing long-term course planning methods, it is based on graduate students to model the course and not on external factors that might introduce some bias in the process. The proposal includes a novel sequential pattern mining algorithm, called (ES)^2P (Evolutionary Search of Emerging Sequential Patterns), that properly identifies paths followed by good students and not followed by not so good students, as a long-term course planning approach. A major feature of the proposed (ES)^2P algorithm is its ability to extract the best k solutions, that is, those with a best recommendation index score instead of returning the whole set of solutions above a predefined threshold. A real study case is performed including more than 13,000 students belonging to 13 faculties to demonstrate the usefulness of the proposal not only to recommend study plans but also to give advices at different stages of the students’ learning process.

Increasing the technological process’ energy efficiency of the of applying multifunctional wear-and-heat-resistant protective coatings by electro-acoustic spraying

The article is devoted to measures improving the energy efficiency of the technological process of applying wear-resistant multifunctional composite protective coatings by the method of electroacoustic spraying. The proposed technique for increasing energy efficiency is based on the use of methods for optimal planning of an experiment using non-orthogonal matrices for planning an experiment and the obtained interpolation-regression equations for the output of the main parameters of the quality of processed products, micro-hardness and roughness, which were tested using the adequacy criterion. The optimization technique based on the criterion of minimizing the reduced energy costs, the technological process of applying wear-resistant protective coatings is based on the determination of the main modes of electro-acoustic spraying according to the specified quality parameters of the required surface layer of the processed mechanical engineering products, micro-hardness and roughness. As a result, from the whole set of solutions of the corresponding equations, we get the opportunity to select those operating modes of the electro-acoustic spraying station that correspond to the minimum values of the amplitude of ultrasonic vibrations and the voltage supplied to the inter-electrode space.

Solving Generalized Equations with Bounded Variables and Multiple Residuated Operators

This paper studies the resolution of sup-inequalities and sup-equations with bounded variables such that the sup-composition is defined by using different residuated operators of a given distributive biresiduated multi-adjoint lattice. Specifically, this study has analytically determined the whole set of solutions of such sup-inequalities and sup-equations. Since the solvability of these equations depends on the character of the independent term, the resolution problem has been split into three parts distinguishing among the bottom element, join-irreducible elements and join-decomposable elements.

Families of solutions of algebraic Riccati equations

We consider Homogeneous Algebraic Riccati Equations in the general situation when the matrix of the dynamics can be mixed. We show that in this case the equation may have infinitely many families of solutions. An analysis of these families is carried over and an explicit formula is derived. When the matrix of the dynamics is unmixed, this formula provides an explicit parametrization of the whole set of the solutions. We finally show that there are situations in which the equation has spurious solutions that do not belong to any of the families and derive sufficient conditions under which, on the contrary, the union of the families covers the whole set of solutions.

A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

Abstract We consider settings for which one needs to perform multiple flow simulations based on the Navier–Stokes equations, each having different initial condition data, boundary condition data, forcing functions, and/or coefficients such as the viscosity. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple right-hand-side vectors. Rigorous analyses are given proving the conditional stability and establishing error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.

Different equivalent approaches to the geodetic reference system

A new approach to the determination of least-squares solutions for the rank deficient model is presented, which, in place of the classical minimal constraints, utilizes the expression of the excess parameters as a function of a subset of unknown parameters describing the model without ambiguities. The role of the lack of reference system definition in the resulting rank deficiency is explained, first for the original non-linear model case, and is specialized next to the linearized model. The whole set of solutions is parameterized in terms of two matrices defining the linearized relation from the model describing parameters to the remaining ones. Particular values are obtained for the case of solution with minimal trace of its covariance matrix, as well for the solution for minimum norm. Finally, the connection with the existing classical approach is established, while the approach is further elaborated in terms of the full-rank decompositions of the model design matrix.

Countable families of solutions of a limit stationary semilinear fourth-order Cahn-Hilliard-type equation I. Mountain pass and Lusternik-Schnirel’man patterns in R N $\mathbb{R}^{N}$

Solutions of the stationary semilinear Cahn-Hilliard-type equation $$-\Delta^{2} u - u -\Delta\bigl(|u|^{p-1}u\bigr)=0 \quad \mbox{in }\mathbb{R}^{N}, \mbox{with } p>1, $$ which are exponentially decaying at infinity, are studied. Using the mounting pass lemma allows us to determinate the existence of a radially symmetric solution. On the other hand, the application of Lusternik-Schnirel’man (L-S) category theory shows the existence of, at least, a countable family of solutions. However, through numerical methods it is shown that the whole set of solutions, even in 1D, is much wider. This suggests that, actually, there exists, at least, a countable set of countable families of solutions, in which only the first one can be obtained by the L-S min-max approach.

A tale of two oscillators

It is not always easy to find problems in classical mechanics that can be solved with the use of knowledge in quantum mechanics. In this short note we present such an example. The problem is that of a bead which is free to move on an infinite rigid rod, where the rod is constrained to rotate with an increasing angular speed. The bead is also connected to the center of the rod by a spring. The question is to examine the whole set of solutions as time goes to infinity; particularly focusing on the possibility of having vanishing radial position. We humans are all equipped with instinctive intuitive knowledge about mechanical systems because we live in a macroscopic world that their physics dominates. Yet in the example we shall study, we will see that our generic intuition might fail to predict a set of zero measure in the solution set. One might say that our intuition is of course only based on generic situations.

Stabilization by deflation for sparse dynamical systems without loss of sparsity

Multiple-input, multiple-output models for coupled systems in structural dynamics including unbounded domains, like soil or fluid, are characterized by sparse system-matrices and unstable parts in the whole set of solutions due to spurious modes. Spectral shifting with deflation can stabilize these unstable parts; however the originally sparse system-matrices become fully populated when this procedure is applied. This paper presents a special consecutive treatment of the deflated system without losing the numerical advantages from sparsity. The procedure starts with an LU-decomposition of the sparse undeflated system and continues with restricting the solution space with respect to deflation using the same LU-decomposition. An example from soil–structure interaction shows the benefits of this consecutive treatment.

Non-linear dynamics in pulse combustor: A review

The state of the art of non-linear dynamics applied to pulse combustor theoretically and experimentally is reviewed. Pulse combustors are a class of air-breathing engines in which pulsations in combustion are utilized to improve the performance. As no analytical solution can be obtained for most of the nonlinear systems, the whole set of solutions can be investigated with the help of dynamical system theory. Many studies have been carried out on pulse combustors whose dynamics include limit cycle behaviour, Hopf bifurcation and period-doubling bifurcation. The dynamic signature has also been used for early prediction of extinction.

Establishing Nash equilibria of strategic games: a multistart Fuzzy Adaptive Simulated Annealing approach

This paper's proposal is to show some significant results obtained by the application of the optimization algorithm known as Fuzzy Adaptive Simulated Annealing (Fuzzy ASA) to the task of finding all Nash equilibria of normal form games. To that end, a special version of Fuzzy ASA, that utilizes space-filling curves to find good seeds, is applied to several well-known strategic games, showing its effectiveness in obtaining all Nash equilibria in all cases. The results are compared to previous work that also used computational intelligence techniques in order to solve the same problem but could not find all equilibria in all tests. Game theory is a very important subject, modeling interactions between generic agents, and Nash equilibrium represents a powerful concept portraying situations in which joint strategies are optimal in the sense that no player can benefit from changing her/his strategy while the other players do not change their strategies as well. So, new techniques are always welcome, mainly those that can find the whole set of solutions for a given strategic game.

Using concept lattice theory to obtain the set of solutions of multi-adjoint relation equations

An important goal in the fuzzy relation equations framework is to obtain the whole set of solutions. This paper introduces algebraic properties in the multi-adjoint concept lattice setting. Moreover, using the relationship between these concept lattices and multi-adjoint relation equations, the complete set of solutions of these general equations is characterized and computed, avoiding the necessity of considering minimal solutions.

Bifurcation curves of a diffusive logistic equation with harvesting orthogonal to the first eigenfunction

We study the global bifurcation curves of a diffusive logistic equation, when harvesting is orthogonal to the first eigenfunction of the Laplacian, for values of the linear growth up to λ2+δ, examining in detail their behavior as the linear growth rate crosses the first two eigenvalues. We observe some new behavior with regard to earlier works concerning this equation. Namely, the bifurcation curves suffer a transformation at λ1, they are compact above λ1, there are precisely two families of degenerate solutions with Morse index equal to zero, and the whole set of solutions below λ2 is not a two dimensional manifold.

Solution of Dirac equation in external Yang–Mills gauge field

The exact solution of the Dirac equation in the external non-Abelian SU(N) gauge field, which is governed by the Yang–Mills equations and in the form of a plane wave on the light cone, is obtained. The whole set of solutions for both particles and anti-particle is constructed. On a basis of the derived solution the fermion current in such field is calculated.

Organic Intellectuals in the Dark Ages?

I am going to begin with intellectuals: not a word we are all that comfortable with, or even familiar with, in Britain. Raymond Williams noted that that plural noun from the early nineteenth century when it entered the English language tended to be pejorative, and that 'unfavourable uses were domi nant until the mid-twentieth century... and still [1976] persist'.1 And in the early twenty-first century? I'm afraid this is one of those terms and concepts to which there still clings inescapably a whiff of un-Britishness. Intellectuals, like intelligentsia (a rare borrowing from Russian, in the nineteenth cen tury), may have positive connotations in other languages, but not in British English. Stefan Collini has recently re-examined British intellectuals in modern times. He's sharply criticized the notion that academic professionalization and specialization since the mid-nineteenth century broke the links between scholars and the general reader. Anxiety that the learned have lost contact with any but the learned is, as Collini says, probably as old as learning itself. But the general reader may be a mythical beast. In any case, the eighteenth century's cosy contacts between learned men and other elite men have yielded to a very much more diverse culture that is more democratic, and the gap between the specialist and a wider public, rather than widening to impassability, has been dramatically narrowed by university expansion. Collini argues that 'the sheer size and diversity of the academic world, broadly defined' (that is, to include all institutions of tertiary education) has meant that 'the university is coming to be seen as the arena within which reflection on the cultural and intellectual life of society... is largely expected to take place'.2 This may still be a lot truer in America than in England but the trend is there. As more and more aspects of culture are taught and studied in universities, so more and more cultural criticism goes on there as well. What we now have in our universities, then, is an 'academic public sphere'. So what some traditionalists deplore as dumbing down is in fact an opening out, a bridging of that old anxiety-provoking gap not a problem but a whole set of solutions. In all this, there's a lot of good fresh thinking, and a lot of humanity. But solutions bring new problems. Here are three in Collini's brief analysis. First it's almost wholly Anglo-American hence Anglophone in its frame of reference. And when I say Anglo, I mean it Scotland (which to its honour has refused to charge fees for university education) doesn't get a mention. But how well does this analysis translate to other parts of the world with

Nonuniqueness of inverted core‐mantle boundary flows and deviations from tangential geostrophy

The task of finding a flow at the top of the core that fits geomagnetic observations at the Earth's surface is an underdetermined problem. Weighted regularized least squares together with the tangential geostrophy hypothesis strongly constrain the flow, inhibiting both medium‐ to small‐scale flow features from emerging and low‐latitude currents from crossing the geographic equator. In this study, we solve the inverse problem in the frozen flux approximation, using a weak regularization and alleviating the tangential geostrophy constraint. Two classes of solutions are found (I and II), both fitting the data within the errors but nonetheless showing very distinct features. Using, as a further criterion, the ability of the flow solutions to reproduce the decade variations of the length of day does not allow us, in general, to give a clear preference to one class over another. For typical flows from the whole set of solutions described above, we identify the regions on the core‐mantle boundary (CMB) where tangential geostrophy is violated, and we quantify those deviations. Assuming that they are due to Lorentz forces acting on the fluid, we compute preliminary charts of electrical density currents (J) on the CMB, consistent with the frozen flux and insulating mantle hypotheses. The most pronounced features in these charts are concentrated over the hemisphere roughly centered beneath the Atlantic Ocean. In light of thermodynamical arguments, class II flows seem closer to real surface core flows. We compute an upper limit JRMS ≈ 0.4 A/m2 for geomagnetically “visible” density currents related to these flows.