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In this paper, a new general approach is developed to construct and study Lebesgue‐type decompositions of linear operators or relations T$T$ in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue‐type decompositions than what has been studied in the literature so far. The key point is that it allows a nontrivial interaction between the closable and the singular components of T$T$. The motivation to study such decompositions comes from the fact that they naturally occur in the corresponding Lebesgue‐type decomposition for pairs of quadratic forms. The approach built in this paper uses so‐called complementation in Hilbert spaces, a notion going back to de Branges and Rovnyak. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, {(G(t),F(t)),t≥0}$\lbrace (G(t), F(t)),t\ge 0\rbrace$, that is, called the resolvent family for the linear evolution process with memory, the F(t)$F(t)$ is called the memory effect family. In this paper, we prove that the families G(t)$G(t)$ and F(t)$F(t)$ are exponentially bounded, and the family (G(t),F(t))$(G(t),F(t))$ associate with an operator pair (A,L)$(A, L)$ that is called generator of the resolvent family. Using (A,L)$(A,L)$, we derive associated differential equation with memory and representation of F(t)$F(t)$ via L. These results give necessary conditions of the well‐posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on (A,L)$(A,L)$. The obtained results can be directly applied to linear delay differential equation, integro‐differential equation and functional differential equations. [ABSTRACT FROM AUTHOR]

Different numerical algorithms for the solution of a class of unsteady convection‐diffusion‐reaction (CDR) equations are presented and compared in this paper. The fully implicit in time discretizations are usually preferred because they are unconditionally stable for linear problems. However, when implicit discretization is used for nonlinear problems, iterations over the nonlinearity have to be performed. Picard (simple linearization) or Newton's methods can be used for this purpose. An alternative to the fully implicit discretization is fractional time‐step methods, e.g. splitting with respect to physicochemical processes. The study of the latter class of discretization is especially interesting in the case when only the reactive term contains nonlinearity, while the convection and diffusion operators are linear. The CDR models used to describe processes in catalytic filters belong to this class. Numerical experiments for CDR equation with controllable stiffness of reaction term for different transport regimes, which are described by Peclet and Damkohler numbers are demonstrated and analyzed. [ABSTRACT FROM AUTHOR]

This paper is concerned with a mixed type differential operator Lu=kyuxx−uyy+byux+qyu,$$ Lu=k(y){u}_{xx}-{u}_{yy}+b(y){u}_x+q(y)u, $$which is initially defined with C0,π∞Ω‾$$ {C}_{0,\pi}^{\infty}\left(\overline{\Omega}\right) $$, where Ω‾={x,y:−π≤x≤π,−∞

Semi‐implicit (SI) time‐stepping schemes for atmosphere and ocean models require elliptic solvers that work efficiently on modern supercomputers. This paper reports our study of the potential computational savings when using mixed precision arithmetic in the elliptic solvers. Precision levels as low as half (16 bits) are used and a detailed evaluation of the impact of reduced precision on the solver convergence and the solution quality is performed. This study is conducted in the context of a novel SI shallow‐water model on the sphere, purposely designed to mimic numerical intricacies of modern all‐scale weather and climate (W&C) models. The governing algorithm of the shallow‐water model is based on the non‐oscillatory MPDATA methods for geophysical flows, whereas the resulting elliptic problem employs a strongly preconditioned non‐symmetric Krylov‐subspace Generalized Conjugated‐Residual (GCR) solver, proven in advanced atmospheric applications. The classical longitude/latitude grid is deliberately chosen to retain the stiffness of global W&C models. The analysis of the precision reduction is done on a software level, using an emulator, whereas the performance is measured on actual reduced precision hardware. The reduced‐precision experiments are conducted for established dynamical‐core test‐cases, like the Rossby‐Haurwitz wavenumber 4 and a zonal orographic flow. The study shows that selected key components of the elliptic solver, most prominently the preconditioning and the application of the linear operator, can be performed at the level of half precision. For these components, the use of half precision is found to yield a speed‐up of a factor 4 compared to double precision for a wide range of problem sizes. Plain Language Summary: Numerical models of the Earth system are very important to predict weather and climate (W&C). The models are computationally expensive and run on supercomputers. If the models are enabled to run faster, model simulations can be enhanced—for example, via a higher resolution—and the accuracy of predictions can be improved. One way to enable the models to run faster is via the use of very low numerical precision. For example, via the use of so‐called half precision which is representing all variables of the simulation with only 16 bits, instead of the default value of 64 bits, reducing computational time by up to a factor of four. In this paper, we investigate whether numerical precision can be reduced down to half precision for parts of the linear solvers which are used in some W&C models. Linear solvers help to progress the model state in time and are responsible for a large fraction of the overall computational cost of model simulations. Key Points: A detailed study of mixed‐precision for linear solvers for weather and climate models is performed, including half precision with 16 bitsPrecision can be reduced in important parts of the solver but a naive approach to reduce precision everywhere does not workCompute time can be reduced by a factor four compared to double precision for parts that work in half precision [ABSTRACT FROM AUTHOR]

This paper deals with the approximations of the functions by generalized Durrmeyer operators of Szász–Mirakjan, which are linear positive operators. Several approximation results are presented well, and we estimate the approximation properties along with the order of approximation and the convergence theorem of the proposed operators. For an explicit explanation of the operators, we determine the properties using the weight function. A quantitative approach is discussed for the operators; quantitative Voronovskaya type and Grüss type theorems are established, showing the operators' more efficient work. We investigate the A$$ A $$‐statistical convergence properties for the said operators, including the rate of approximation in a statistical sense. An important property for the rate of convergence of the operators is obtained in terms of the function with a derivative of the bounded variation. At last, the graphical representations and numerical analysis are discussed and shown to support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we study a porous thermoelastic system with microtemperature and strong time delay acting on the volume fraction equation. The thermal effect of microtemperature is based on the Lord–Shulman theory (J Mech Phys Solids. 15(5) (1967), 299–309.), while the strong delay is motivated by Makheloufi's et al. recent work (Math Meth Appl Sci. 44 (2021), 6301–6317.). To prove the well‐posedness of the system, lack of exponential stability and the polynomial decay with optimal rate, we use the semigroup theory of linear operators. [ABSTRACT FROM AUTHOR]

For field development and drilling decisions, production assets and reservoir engineers require dynamic reservoir properties, such as saturation and pressure changes of a reservoir from the pre‐production virgin state. To date, geophysicists have produced time‐lapse (4D) seismic attributes (mostly on stacked seismic data) rather than dynamic parameters directly. In this paper, we present a new method to estimate saturation and pressure properties from time‐lapse seismic data to provide to reservoir engineers. This new three‐step method is demonstrated over the Edvard Grieg field in the North Sea. We can realize this method thanks to advanced seismic multi‐component acquisition via PP and PS seismic data and processing that allows accurate estimation of amplitude‐variation‐with‐offset parameters P‐ and S‐impedances. With time‐lapse P‐ and S‐impedances optimally resolved, we estimate a stable set of axes identifying water saturation increase (water replacing oil), gas saturation increase (gas injection or gas out of solution), pressure increase (at injectors) and pressure decrease (at producers). Once these axes are obtained, we convert every 4D P‐ and S‐impedance data points into 4D pseudo‐saturation and pseudo‐pressure using a transformation of coordinates. We next use the rock physics relationships of this field to show that a linear relationship can be used to map any 4D change in the field from impedances to saturations and pressures. Key locations in the field with largest saturation and pressure changes are used to find calibration values at these extreme points. Next, from the pseudo‐seismic 4D data, a linear mapping is used to calculate actual reservoir property changes (fractions for saturation and bars for pressure). This allows us to obtain fieldwide dynamic values for water and gas saturations and injection and production related pressure changes. The results are shown, and dynamic changes are interpreted on the Edvard Grieg field data. [ABSTRACT FROM AUTHOR]

Let c=(Cn)n≥0$$ c={\left({C}_n\right)}_{n\ge 0} $$ be the Catalan sequence and T$$ T $$ a linear and bounded operator on a Banach space X$$ X $$ such 4T$$ 4T $$ is a power‐bounded operator. The Catalan generating function is defined by the following Taylor series: C(T):=∑n=0∞CnTn.$$ C(T):= \sum \limits_{n=0}^{\infty }{C}_n{T}^n. $$Note that the operator C(T)$$ C(T) $$ is a solution of the quadratic equation TY2−Y+I=0$$ T{Y}^2-Y+I=0 $$. In this paper, we define powers of the Catalan generating function C(T)$$ C(T) $$ in terms of the Catalan triangle numbers. We obtain new formulae that involve Catalan triangle numbers: the spectrum of c∗j$$ {c}^{\ast j} $$ and the expression of c−∗j$$ {c}^{-\ast j} $$ for j≥1$$ j\ge 1 $$ in terms of Catalan polynomials (∗$$ \ast $$ is the usual convolution product in sequences). In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future. [ABSTRACT FROM AUTHOR]

In this paper we show that for a vector space Vd$V_d$ of dimension d$d$ there exists a linear map detS2:Vd⊗d(2d−1)→k$det^{S^2}:V_d^{\otimes d(2d-1)}\rightarrow k$ with the property that detS2(⊗1⩽i

In this paper, the operator spectrum theory is applied to study the general stabilisation issues for mean‐field stochastic Markov jump systems (MF‐SMJSs), where multiple delays, multiplicative noises and homogeneous Markov chain exist simultaneously. The innovative contributions are described as follows. On the one hand, a feasible model augmented strategy is adopted to transform the dynamics into an auxiliary delay‐free form. By introducing a delay‐dependent linear Lyapunov operator (DDLLO), the Lyapunov/spectrum stabilising conditions are constructed, which are both necessary and sufficient. On the other hand, in terms of spectral analysis technique, the notions of interval stabilisation and essential destabilisation are generalised to MF‐SMJSs for the first time. The necessary and sufficient stabilisation conditions are derived, respectively, which can be verified availably by LMI feasibility tests. To confirm the effectiveness of the theoretic results, two illustrative examples are included. [ABSTRACT FROM AUTHOR]

Hyperspectral image classification has gained great progress in recent years based on deep learning model and massive training data. However, it is expensive and unpractical to label hyperspectral image data and implement model in constrained environment. To address this problem, this paper proposes an effective ghost module based spectral network for hyperspectral image classification. First, Ghost3D module is adopted to reduce the size of model parameter dramatically by redundant feature maps generation with linear transformation. Then Ghost2D module with channel‐wise attention is used to explore informative spectral feature representation. For large field covering, the non‐local operation is utilized to promote self‐attention. Compared with the state‐of‐the‐art hyperspectral image classification methods, the proposed approach achieves superior performance on three hyperspectral image data sets with fewer sample labelling and less resource consumption. [ABSTRACT FROM AUTHOR]

In this paper, the linear differential expression of order n≥2$$ n\ge 2 $$ with distribution coefficients of various singularity orders is considered. We obtain the associated matrix for the regularization of this expression. Furthermore, we present new statements of inverse spectral problems that consist in the recovery of differential operators from the Weyl matrix on the half‐line and on a finite interval. The uniqueness theorems for these two inverse problems are proved by developing the method of spectral mappings. To the best of the author's knowledge, inverse problems for higher order differential operators with distribution coefficients on the half‐line have not been studied before. For the finite interval case, we for the first time consider the issue of recovering coefficients of the boundary conditions. [ABSTRACT FROM AUTHOR]

The paper is largely concerned with the possibility of obtaining a series representation for a compact linear map T acting between Banach spaces. It is known that, using the notions of j−$j-$eigenfunctions and j−$j-$ eigenvalues, such a representation is possible under certain conditions on T. Particular cases discussed include those in which T can be factorized through a Hilbert space, or has certain s‐numbers that are fast‐decaying. The notion of p‐compactness proves to be useful in this context; we give examples of maps that possess this property. [ABSTRACT FROM AUTHOR]

Given a bounded Lipschitz domain Ω⊂Rn$\Omega \subset \mathbb {R}^n$, Rychkov showed that there is a linear extension operator E$\mathcal {E}$ for Ω$\Omega$, which is bounded in Besov and Triebel‐Lizorkin spaces. In this paper, we introduce some new estimates for the extension operator E$\mathcal {E}$ and give some applications. We prove the equivalent norms ∥f∥Apqs(Ω)≈∑|α|≤m∥∂αf∥Apqs−m(Ω)$\Vert f\Vert _{\mathcal A_{pq}^s(\Omega)}\approx \sum _{|\alpha |\le m}\Vert \partial ^\alpha f\Vert _{\mathcal A_{pq}^{s-m}(\Omega)}$ for general Besov and Triebel‐Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives on Ω¯c$\overline{\Omega }^c$ up to the boundary. [ABSTRACT FROM AUTHOR]

This paper considers the Cauchy problem for the non‐linear dynamic string equation of Kirchhoff‐type with time‐varying coefficients. The objective of this work is to develop a time‐domain discretization algorithm capable of approximating a solution to this initial‐boundary value problem. To this end, a symmetric three‐layer semi‐discrete scheme is employed with respect to the temporal variable, wherein the value of a non‐linear term is evaluated at the middle node point. This approach enables the numerical solutions per temporal step to be obtained by inverting the linear operators, yielding a system of second‐order linear ordinary differential equations. Local convergence of the proposed scheme is established, and it achieves quadratic convergence regarding the step size of the discretization of time on the local temporal interval. We have conducted several numerical experiments using the proposed algorithm for various test problems to validate its performance. It can be said that the obtained numerical results are in accordance with the theoretical findings. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to approximate the fractional calculus (FC) operator with general analytic kernel by using auxiliary newly defined linear positive operators. For this purpose, we introduce the Stancu variant of modified Bernstein–Kantorovich operators and investigate their simultaneous approximation properties. Then we construct new operators by means of these auxiliary operators, and based on the obtained results, we prove the main theorems on the approximation of the general FC operators. We also obtain some quantitative estimates for this approximation in terms of modulus of continuity and Lipschitz class functions. Additionally, we exhibit our approximation results for the well‐known FC operators such as Riemann–Liouville integral, Caputo derivative, Prabhakar integral, and Caputo–Prabhakar derivative. [ABSTRACT FROM AUTHOR]

Linear operators with off‐diagonal decay appear in many areas of mathematics including harmonic and numerical analysis, and their stability is one of the basic assumptions. In this paper, we consider a family of localized integral operators in the Beurling algebra with kernels having mild singularity near the diagonal and certain Hölder continuity property, and prove that their weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other on a space of homogeneous type with Ahlfors regular measure. [ABSTRACT FROM AUTHOR]

Machine learning algorithms are widely applied to create powerful prediction models. With increasingly complex models, humans' ability to understand the decision function (that maps from a high‐dimensional input space) is quickly exceeded. To explain a model's decisions, black‐box methods have been proposed that provide either non‐linear maps of the global topology of the decision boundary, or samples that allow approximating it locally. The former loses information about distances in input space, while the latter only provides statements about given samples, but lacks a focus on the underlying model for precise 'What‐If'‐reasoning. In this paper, we integrate both approaches and propose an interactive exploration method using local linear maps of the decision space. We create the maps on high‐dimensional hyperplanes—2D‐slices of the high‐dimensional parameter space—based on statistical and personal feature mutability and guided by feature importance. We complement the proposed workflow with established model inspection techniques to provide orientation and guidance. We demonstrate our approach on real‐world datasets and illustrate that it allows identification of instance‐based decision boundary structures and can answer multi‐dimensional 'What‐If'‐questions, thereby identifying counterfactual scenarios visually. [ABSTRACT FROM AUTHOR]

In this paper, we study an age‐structured reaction‐diffusion‐advection population model. First, we use a non‐densely defined operator to the linear age‐structured reaction‐diffusion‐advection population model in a patchy environment. By spectral analysis, we obtain the asynchronous exponential growth of the population model. Then we consider nonlinear death rate and birth rate, which all depend on the function related to the generalized total population, and we prove the existence of a steady state of the system. Finally, we study the age‐structured reaction‐diffusion‐advection population model in non‐autonomous situations. We give the comparison principle and prove the eventual compactness of semiflow by using integrated semigroup. We also prove the existence of compact attractors under the periodic situation. [ABSTRACT FROM AUTHOR]

This paper proposes a data‐driven sensor fault detection and isolation approach for the general class of nonlinear systems. The proposed method uses deep neural network architecture to obtain an invariant set of basis functions for the Koopman operator to form a linear predictor for a nonlinear system. Then, the obtained Koopman predictor has been used in a geometric framework for sensor fault detection and isolation purposes without relying on a priori knowledge about the underlying dynamics as well as requiring faulty data, leading to a data‐driven sensor fault detection and isolation framework for nonlinear systems. Finally, the approach's efficacy is demonstrated using simulation case study on a two‐degree of freedom robot arm. [ABSTRACT FROM AUTHOR]

Summary: With the popularity of cloud computing, more and more users store sensitive information in cloud servers. In order to protect the data over the cloud server, symmetric encryption with keyword search has been developed and the phrase search has been proposed subsequently to overcome the inefficiency produced by single/multi‐keyword search. However, most existing phrase search schemes fails to support fault‐tolerant search which is essential to users. Therefore, this paper proposes a fault‐tolerant dynamic phrase search scheme with forward privacy and backward privacy (FTPS). Piecewise Linear Chaotic Map and minhash function are used to blur information, and Bloom filter based index is constructed to realize efficient search and dynamic update simultaneously. Security analysis proves that FTPS can properly preserve the privacy of search user, and experimental results show that FTPS is practical. [ABSTRACT FROM AUTHOR]

This paper studies the stabilization of the discrete‐time bilinear system z(k+1)=Az(k)+ukBz(k), where A and B are bounded linear operators on a pre‐Hilbert space H, by means of a new nonlinear state feedback control which is uniformly bounded. First, sufficient conditions for uniform exponential stabilization are given, and the decay rate of the state is estimated. Then global weak stabilization is investigated under less stringent assumptions. In the case of the finite dimension, easily verified conditions for uniform stabilization are given. Moreover, a decomposition method allows additional stabilization results. Finally, illustrative examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we approach the two weighted boundedness of commutators via matrix weights. This approach provides both a sufficient and a necessary condition for the two weighted boundedness of commutators with an arbitrary linear operator in terms of one matrix weighted norm inequalities for this operator. Furthermore, using this approach, we surprisingly provide conditions that almost characterize the two matrix weighted boundedness of commutators with CZOs and completely arbitrary matrix weights, which is even new in the fully scalar one weighted setting. Finally, our method allows us to extend the two weighted Holmes/Lacey/Wick results to the fully matrix setting (two matrix weights and a matrix symbol), completing a line of research initiated by the first two authors. [ABSTRACT FROM AUTHOR]

This paper discusses an application of partial tensor Golub–Kahan bidiagonalization to the solution of large‐scale linear discrete ill‐posed problems based on the t‐product formalism for third‐order tensors proposed by Kilmer and Martin (M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641‐658). The solution methods presented first reduce a given (large‐scale) problem to a problem of small size by application of a few steps of tensor Golub–Kahan bidiagonalization and then regularize the reduced problem by Tikhonov's method. The regularization operator is a third‐order tensor, and the data may be represented by a matrix, that is, a tensor slice, or by a general third‐order tensor. A regularization parameter is determined by the discrepancy principle. This results in fully automatic solution methods that neither require a user to choose the number of bidiagonalization steps nor the regularization parameter. The methods presented extend available methods for the solution for linear discrete ill‐posed problems defined by a matrix operator to linear discrete ill‐posed problems defined by a third‐order tensor operator. An interlacing property of singular tubes for third‐order tensors is shown and applied. Several algorithms are presented. Computed examples illustrate the advantage of the tensor t‐product approach, in comparison with solution methods that are based on matricization of the tensor equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a τ$$ \tau $$‐discrete semigroup {Tτn}n∈ℕ0$$ {\left\{{T}_{\tau}^n\right\}}_{n\in {\mathbb{N}}_0} $$ generated by a closed linear operator A$$ A $$ in a Banach space X$$ X $$. We show that {Tτn}n∈ℕ0$$ {\left\{{T}_{\tau}^n\right\}}_{n\in {\mathbb{N}}_0} $$ allows us to write the solution to an abstract discrete difference equation of first order as a discrete variation of parameters formula. Moreover, we study the main properties of {Tτn}n∈ℕ0$$ {\left\{{T}_{\tau}^n\right\}}_{n\in {\mathbb{N}}_0} $$ and its relation with the well‐known notion of discrete semigroups. Finally, we characterize uniform exponential stability of a C0$$ {C}_0 $$‐semigroup {T(t)}t≥0$$ {\left\{T(t)\right\}}_{t\ge 0} $$ in terms of the τ$$ \tau $$‐discrete semigroup {Tτn}n∈ℕ0$$ {\left\{{T}_{\tau}^n\right\}}_{n\in {\mathbb{N}}_0} $$. [ABSTRACT FROM AUTHOR]

Let Hc$$ {H}_c $$ be a (2n)×(2n)$$ (2n)\times (2n) $$ symmetric tridiagonal matrix with diagonal elements c∈ℝ$$ c\in \mathbb{R} $$ and off‐diagonal elements one, and S$$ S $$ be a (2n)×(2n)$$ (2n)\times (2n) $$ diagonal matrix with the first n$$ n $$ diagonal elements being plus ones and the last n$$ n $$ being minus ones. Davies and Levitin studied the eigenvalues of a linear pencil Ac=Hc−λS$$ {\mathcal{A}}_c={H}_c-\lambda S $$ as 2n$$ 2n $$ approaches to infinity. It was conjectured by DL that for any n∈ℕ$$ n\in \mathbb{N} $$ the non‐real eigenvalues λ$$ \lambda $$ of Ac$$ {\mathcal{A}}_c $$ satisfy both |λ+c|<2$$ \mid \lambda +c\mid <2 $$ and |λ−c|<2$$ \mid \lambda -c\mid <2 $$. The conjecture has been verified numerically for a wide range of n$$ n $$ and c$$ c $$, but so far the full proof is missing. The purpose of the paper is to support this conjecture with a partial proof and several numerical experiments which allow to get some insight in the behaviour of the non‐real eigenvalues of Ac$$ {\mathcal{A}}_c $$. We provide a proof of the conjecture for n≤3$$ n\le 3 $$, and also in the case where |λ+c|=|λ−c|$$ \mid \lambda +c\mid =\mid \lambda -c\mid $$. In addition, numerics indicate that some phenomena may occur for more general linear pencils. [ABSTRACT FROM AUTHOR]

The design of minimum phase finite impulse response (FIR) filters is considered. The study demonstrates that the residual errors achieved by current state‐of‐the‐art design methods are nowhere near the smallest error possible on a finite resolution digital computer. This is shown to be due to conceptual errors in the literature pertaining to what constitutes a factorable linear phase filter. This study shows that factorisation is possible with a zero residual error (in the absence of machine finite resolution error) if the linear operator or matrix representing the linear phase filter is positive definite. Methodology is proposed able to design a minimum phase filter that is optimal—in the sense that the residual error is limited only by the finite precision of the digital computer, with no systematic error. The study presents practical application of the proposed methodology by designing two minimum phase Chebyshev FIR filters. Results are compared to state‐of‐the‐art methods from the literature, and it is shown that the proposed methodology is able to reduce currently achievable residual errors by several orders of magnitude. [ABSTRACT FROM AUTHOR]

Lateral interaction in the biological brain is a key mechanism that underlies higher cognitive functions. Linear self‐organising map (SOM) introduces lateral interaction in a general form in which signals of any modality can be used. Some approaches directly incorporate SOM learning rules into neural networks, but incur complex operations and poor extendibility. The efficient way to implement lateral interaction in deep neural networks is not well established. The use of Laplacian Matrix‐based Smoothing (LS) regularisation is proposed for implementing lateral interaction in a concise form. The authors' derivation and experiments show that lateral interaction implemented by SOM model is a special case of LS‐regulated k‐means, and they both show the topology‐preserving capability. The authors also verify that LS‐regularisation can be used in conjunction with the end‐to‐end training paradigm in deep auto‐encoders. Additionally, the benefits of LS‐regularisation in relaxing the requirement of parameter initialisation in various models and improving the classification performance of prototype classifiers are evaluated. Furthermore, the topologically ordered structure introduced by LS‐regularisation in feature extractor can improve the generalisation performance on classification tasks. Overall, LS‐regularisation is an effective and efficient way to implement lateral interaction and can be easily extended to different models. [ABSTRACT FROM AUTHOR]

Let S$S$ be a compact orientable surface of genus g$g$ with marked points in the interior. Franks–Handel (Proc. Amer. Math. Soc. 141 (2013) 2951–2962) proved that if n<2g$n<2g$ then the image of a homomorphism from the mapping class group Mod(S)${\rm Mod}(S)$ of S$S$ to GL(n,C)${\rm GL}(n,{\mathbb {C}})$ is trivial if g⩾3$g\geqslant 3$ and is finite cyclic if g=2$g=2$. The first result is our own proof of this fact. Our second main result shows that for g⩾3$g\geqslant 3$ up to conjugation there are only two homomorphisms from Mod(S)${\rm Mod}(S)$ to GL(2g,C)${\rm GL}(2g,{\mathbb {C}})$: the trivial homomorphism and the standard symplectic representation. Our last main result shows that the mapping class group has no faithful linear representation in dimensions less than or equal to 3g−3$3g-3$. We provide many applications of our results, including the finiteness of homomorphisms from mapping class groups of nonorientable surfaces to GL(n,C)${\rm GL}(n,{\mathbb {C}})$, the triviality of homomorphisms from the mapping class groups to Aut(Fn)${\rm Aut}(F_n)$ or to Out(Fn)${\rm Out}(F_n)$, and homomorphisms between mapping class groups. We also show that if the surface S$S$ has r$r$ marked point but no boundary components, then Mod(S)${\rm Mod}(S)$ is generated by involutions if and only if g⩾3$g\geqslant 3$ and r⩽2g−2$r\leqslant 2g-2$. [ABSTRACT FROM AUTHOR]

We describe a proof‐of‐concept development and application of a phase‐averaging technique to the nonlinear rotating shallow‐water equations on the sphere, discretised using compatible finite‐element methods. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables), so that larger timesteps may be taken. We overcome the two key technical challenges that stand in the way of studying the phase averaging and advancing its implementation: (1) we have developed a stable matrix exponential specific to finite elements and (2) we have developed a parallel finite averaging procedure. Following recent studies, we consider finite‐width phase‐averaging windows, since the equations have a finite timescale separation. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method, which we use here to compute our solutions. Here, we focus on the stability and accuracy of the numerical solution. We confirm that there is an optimal averaging window, in agreement with theory. Critically, we observe that the combined time discretisation and averaging error is much smaller than the time discretisation error in a semi‐implicit method applied to the same spatial discretisation. An evaluation of the parallel aspects will follow in later work. [ABSTRACT FROM AUTHOR]

We consider the well‐posedness of the second‐order degenerate differential equations (Mu′)′(t)+Λu′(t)+∫−∞ta(t−s)u′(s)ds=Au(t)+∫−∞tb(t−s)Bu(s)ds+f(t)(P)$$\begin{eqnarray*} &&\hspace*{18pc} (Mu^{\prime })^{\prime }(t) +\Lambda u^{\prime }(t)+\int _{-\infty }^t a(t-s)u^{\prime }(s)ds\\ &&\hspace*{18pc}\quad = Au(t)+ \int _{-\infty }^t b(t-s) Bu(s) ds + f(t)\qquad\qquad\qquad \mathrm{{(P)}} \end{eqnarray*}$$with infinite delay on [0, 2π] in Lebesgue–Bochner spaces Lp(T;X)$L^p(\mathbb {T}; X)$ and periodic Besov spaces Bp,qs(T;X)$B_{p,q}^s (\mathbb {T}; X)$, where A,B,Λ$A,\ B,\ \Lambda$, and M are closed linear operators in a Banach space X satisfying D(A)∩D(B)⊂D(M)∩D(Λ)$D(A)\cap D(B)\subset D(M) \cap D(\Lambda)$ and the kernels a,b∈L1(R+)$ a, b\in L^1(\mathbb {R}_+)$. Using known operator‐valued Fourier multiplier theorems, we are able to give necessary and sufficient conditions for the well‐posedness of (P) in Lp(T;X)$L^p(\mathbb {T}; X)$ and Bp,qs$B_{p,q}^s$(T;X)$(\mathbb {T}; X)$. These results are applied to examine some concrete examples. [ABSTRACT FROM AUTHOR]

In this work, we introduce a strongly continuous one‐parameter family of bounded linear operators that completely describes the well‐posedness of a second order abstract differential delay equation in the initial history space Lp([−r,0];X)$L^p([-r,0];X)$, r>0$r>0$. This family, which satisfies a specific functional equation is applied to characterize the mild solution of the considered second order delay equation. [ABSTRACT FROM AUTHOR]

Non‐linearity is an inherent feature of practical systems. Although there have been significant advances in the control of nonlinear systems, the proposed methods often require considerable computational resources or rely on local linearization around equilibrium points. The Koopman operator is an infinite‐dimensional linear operator that fully captures a system's non‐linear dynamics. However, one of the major problems is identifying a Koopman finite dimensional linear model for a nonlinear system. Initiated by the Willems' fundamental Lemma, a class of data‐driven control methods has been developed for linear systems without the need to identify the system's matrices. Motivated by these two ideas, a data‐driven Koopman‐based predictive control scheme for non‐linear systems is proposed for unknown disturbed non‐linear systems utilising a finite‐length dataset. Then, considering the uncertainty in the Koopman state variables, a robust data‐driven Koopman predictive control structure is presented. Also, the results led to the design of a data‐driven Koopman predictive control strategy with terminal components to ensure the closed‐loop stability of nonlinear systems. The proposed scheme is tested on the distributed‐parameter model of the ACUREX solar collector field (located at Almería, Spain) to regulate the field outlet temperature around a desired value. Finally, simulation results show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Summary: This article introduces a robust and affordable method to compute nullspace and generalized inverse of finite element operators involved in dual domain decomposition methods. The methodology relies on the operator partial factorization and on the analysis of a well chosen Schur complement. The sparse linear operator is interpreted as a network and graph centrality measures are used to select the condensation variables. Eigenvector, Katz and Page Rank centralities are evaluated. An extension to deal with symmetric indefinite systems arising from mixed finite elements is also presented. The approach is assessed on highly heterogeneous problems and one industrial application is presented: the numerical homogenization of solid propellant. [ABSTRACT FROM AUTHOR]

Using operator‐valued Lp$$ {L}^p $$‐Fourier multiplier theorem, weighted Sobolev spaces, and the Carleman transform, we characterize the well‐posedness of second‐order degenerate differential equations with finite delay (Mu)′′(t)=Au(t)+Fut+Gut′+f(t)$$ {(Mu)}^{\prime \prime }(t)= Au(t)+F{u}_t+G{u}_t^{\prime }+f(t) $$ on Lp(ℝ;X)$$ {L}^p\left(\mathbb{R};X\right) $$, where A:D(A)⊆X→X$$ A:D(A)\subseteq X\to X $$ and M:D(M)⊆X→X$$ M:D(M)\subseteq X\to X $$ are closed linear operators defined on a Banach space X$$ X $$, the operators F$$ F $$ and G$$ G $$ are in B(Lp([−τ,0];X);X)$$ B\left({L}^p\left(\left[-\tau, 0\right];X\right);X\right) $$ for some fixed τ>0$$ \tau >0 $$, and ut(s)=u(t+s),ut′(s)=u′(t+s)$$ {u}_t(s)=u\left(t+s\right),{u}_t^{\prime }(s)={u}^{\prime}\left(t+s\right) $$ when t∈ℝ$$ t\in \mathbb{R} $$ and s∈[−τ,0]$$ s\in \left[-\tau, 0\right] $$. These results are used to study the well‐posedness of the associated second‐order neutral degenerate differential equations. [ABSTRACT FROM AUTHOR]

The present work considers two important convergence techniques, namely, deferred type statistical convergence and P$$ P $$‐summability method in respect of linear positive operators. With regard to these techniques, following closely the ideas developed in the articles (Appl. Math. Lett. 18, 2005, 1339‐1344, and Sau Paulo J. Math. Sci. 13, 2019, 696‐707), we state and prove two general non‐trivial Korovkin‐type approximation results for positive linear operators. Further, we define an operator based on multivariate q$$ q $$‐Lagrange–Hermite polynomials and exhibit the applicability of the above theorems to these operators. [ABSTRACT FROM AUTHOR]

Assessing the conditions for persistence of spatially structured populations, especially those that are exploited by humans or threatened by global change, is of critical importance to inform management and conservation efforts. Observations for entire metapopulations are usually incomplete and rarely, if ever, sufficiently long to deduce population persistence from spatial patterns of abundance. Instead, insights based on metapopulation theory are often used for interpreting the demographic trajectories of real populations and for informing management decisions. The classical theoretical tool used to assess conditions for metapopulation persistence is the "invasibility criterion," which characterizes the asymptotic, or long‐term, stability of a small colonizing population. Essentially, when the linear operator governing the metapopulation dynamics of an invasion event has a positive eigenvalue, recovery and resistance to extinction (resilience) are implied. The converse, however, is not necessarily the case—an invasion may grow over multiple generations, even when the eigenvalues indicate that extinction will eventually occur, a situation referred to here as "reactive persistence." For the management, restoration, and conservation of real metapopulations subject to continual disturbance, this transient behavior is often more relevant than the asymptotic behavior over long time scales. We develop the theoretical tools for assessing reactive persistence, demonstrating how the conditions for asymptotic and reactive persistence differ in both the patch‐occupancy models suited to many terrestrial populations and those where local patch extinctions can be disregarded in the dynamics, often suited to marine species. After presenting the mathematical basis for generalizing the invasibility criterion to include reactive persistence, we illustrate how these concepts and tools can be applied in practice, using as a case study the population ecology and restoration of the seagrass Zostera muelleri (Irmisch ex Ascherson, 1867) in the Port of Gladstone in the Great Barrier Reef World Heritage Area Australia. It is shown how the analysis of the transient dynamics of the Z. muelleri metapopulation can be used to guide restoration efforts. Moreover, it is demonstrated that these reactive persistence concepts provide a more appropriate basis for site prioritization for restoration interventions than traditional stability analysis. [ABSTRACT FROM AUTHOR]

Linear ill‐posed problems often are analyzed in function spaces using tools from functional analysis, while their numerical solution typically is computed by first discretizing the problem and then applying tools from finite‐dimensional linear algebra. The Chebfun package makes it possible to solve linear ill‐posed problems without explicit discretization. This work discusses the use of Tikhonov regularization with a fairly general linear regularization operator within the Chebfun framework. [ABSTRACT FROM AUTHOR]

Random linear mappings are widely used in modern signal processing, compressed sensing, and machine learning. These mappings may be used to embed the data into a significantly lower dimension while at the same time preserving useful information. This is done by approximately preserving the distances between data points, which are assumed to belong to ℝn. Thus, the performance of these mappings is usually captured by how close they are to an isometry on the data. Gaussian linear mappings have been the object of much study, while the sub‐Gaussian settings is not yet fully understood. In the latter case, the performance depends on the sub‐Gaussian norm of the rows. In many applications, e.g., compressed sensing, this norm may be large, or even growing with dimension, and thus it is important to characterize this dependence. We study when a sub‐Gaussian matrix can become a near isometry on a set, show that previous best‐known dependence on the sub‐Gaussian norm was suboptimal, and present the optimal dependence. Our result not only answers a remaining question posed by Liaw, Mehrabian, Plan, and Vershynin in 2017, but also generalizes their work. We also develop a new Bernstein‐type inequality for subexponential random variables, and a new Hanson‐Wright inequality for quadratic forms of sub‐Gaussian random variables, in both cases improving the bounds in the sub‐Gaussian regime under moment constraints. Finally, we illustrate popular applications such as Johnson‐Lindenstrauss embeddings, null space property for 0‐1 matrices, randomized sketches, and blind demodulation, whose theoretical guarantees can be improved by our results (in the sub‐Gaussian case). © 2021 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We study the semiclassical limit of the sine‐Gordon (sG) equation with below threshold pure impulse initial data of Klaus‐Shaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of Dubrovin, Grava, and Klein, we found that in a Oϵ4/5 neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are universally described by the Painlevé I tritronquée solution. A linear map can be explicitly made from the tritronquée solution to this neighborhood. Under this map: away from the tritronquée poles, the first correction of sG is universally given by the real part of the Hamiltonian of the tritronquée solution; localized defects appear at locations mapped from the poles of the tritronquée solution; the defects are proved universally to be a two‐parameter family of special localized solutions on a periodic background for the sG equation. We are able to characterize the solution in detail. Our approach is the rigorous steepest descent method for matrix Riemann‐Hilbert problems, substantially generalizing [5] to establish universality beyond the context of solutions of a single equation. © 2021 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

The nonlinear mechanical behavior of intact rock and rock mass is characterized by irreversible deformation, associated with strain hardening, strain softening and degradation of stiffness. In the case of stratified rock types like shales or phyllites, the mechanical behavior additionally depends on the loading direction with respect to the orientation of the principal material directions, resulting in inherent anisotropic material behavior, or in many cases more specifically in transversely isotropic material behavior. On the basis of a critical review of several approaches for modeling nonlinear anisotropic material behavior, proposed in the literature, the linear mapping of the Cauchy stress tensor into a fictitious isotropic configuration, originally proposed by Boehler [1], was identified as the best choice for the application in 3D finite element simulations. Within this framework, an isotropic damage plasticity model for rock, proposed by Unteregger et al. [2], extended to inherent transversely isotropic behavior is presented. It is implemented into a FE‐program by means of the return mapping algorithm and it is regularized by an over‐nonlocal implicit gradient enhancement for ensuring mesh‐insensitive results. For validation, numerical simulations of triaxial compression tests on Tournemire shale, documented in Niandou et al. [3], are performed at both the integration point level and the structural level. The performance of the constitutive model is assessed by its ability to predict the complex direction‐dependent mechanical behavior of transversely isotropic rock and the failure modes observed in triaxial compression tests with different confining pressures and different inclination angles of the stratification planes with respect to the direction of axial loading. [ABSTRACT FROM AUTHOR]

Let I1=[a0,a1),...,Ik=[ak−1,ak)$I_1=[a_0,a_1),\ldots ,I_{k}= [a_{k-1},a_k)$ be a partition of the interval I=[0,1)$I=[0,1)$ into k$k$ subintervals. Let f:I→I$f:I\rightarrow I$ be a map such that each restriction f|Ii$f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any f$f$ admits at most k$k$ periodic orbits, where the upper bound is sharp. We are also interested in the dynamics of piecewise linear λ$\lambda$‐affine maps, where 0<λ<1$0<\lambda <1$. Let b1,...,bk$b_1,\ldots ,b_k$ be real numbers and let Fλ:I→R$F_\lambda : I\rightarrow \mathbb {R}$ be a function such that each restriction Fλ|Ii(x)=λx+bi$F_\lambda |_{I_i}(x)=\lambda x +b_i$. Under a generic assumption on the parameters a1,...,ak−1,b1,...,bk$a_1,\ldots ,a_{k-1},b_1,\ldots ,b_k$, we prove that, up to a zero Hausdorff dimension set of slopes λ$\lambda$, the ω$\omega$‐limit set of the piecewise λ$\lambda$‐affine maps fλ:x∈I→Fλ(x)(mod1)$f_\lambda :x\in I \rightarrow F_\lambda (x)\pmod {1}$ at every point equals a periodic orbit and there exist at most k$k$ periodic orbits. Moreover, let E(k)$\mathfrak {E}^{(k)}$ be the exceptional set of parameters λ,a1,...,ak−1,b1,...,bk$\lambda ,a_1,\ldots ,a_{k-1},b_1,\ldots ,b_k$ which define non‐asymptotically periodic f$f$, we prove that E(k)$\mathfrak {E}^{(k)}$ is a Lebesgue null measure set whose Hausdorff dimension is large or equal to k$k$. [ABSTRACT FROM AUTHOR]

Combining historical control data with current control data may reduce the necessary study size of a clinical trial. However, this only applies when the historical control data are similar enough to the current control data. Several Bayesian approaches for incorporating historical data in a dynamic way have been proposed, such as the meta‐analytic‐predictive (MAP) prior and the modified power prior (MPP). Here we discuss the generalization of the MPP approach for multiple historical control groups for the linear regression model. This approach is useful when the controls differ more than in a random way, but become again (approximately) exchangeable conditional on covariates. The proposed approach builds on the approach previously developed for binary outcomes by some of the current authors. Two MPP approaches have been developed with multiple controls. The first approach assumes independent powers, while in the second approach the powers have a hierarchical structure. We conducted several simulation studies to investigate the frequentist characteristics of borrowing methods and analyze a real‐life data set. When there is between‐study variation in the slopes of the model or in the covariate distributions, the MPP approach achieves approximately nominal type I error rates and greater power than the MAP prior, provided that the covariates are included in the model. When the intercepts vary, the MPP yields a slightly inflated type I error rate, whereas the MAP does not. We conclude that our approach is a worthy competitor to the MAP approach for the linear regression case. [ABSTRACT FROM AUTHOR]

Purpose: Image‐guided radiotherapy (IGRT) research sometimes involves simulated changes to patient positioning using retrospectively collected clinical data. For example, researchers may simulate patient misalignments to develop error detection algorithms or positioning optimization algorithms. The Brainlab ExacTrac system can be used to retrospectively "replay" simulated alignment scenarios but does not allow export of digitally reconstructed radiographs (DRRs) with simulated positioning variations for further analysis. Here we describe methods to overcome this limitation and replicate ExacTrac system DRRs by using projective geometry parameters contained in the ExacTrac configuration files saved for every imaged subject. Methods: Two ExacTrac DRR generators were implemented, one with custom MATLAB software based on first principles, and the other using libraries from the Insight Segmentation and Registration Toolkit (ITK). A description of perspective projections for DRR rendering applications is included, with emphasis on linear operators in real projective space P3${\mathbb{P}^3}$. We provide a general methodology for the extraction of relevant geometric values needed to replicate ExacTrac DRRs. Our generators were tested on phantom and patient images, both acquired in a known treatment position. We demonstrate the validity of our methods by comparing our generated DRRs to reference DRRs produced by the ExacTrac system during a treatment workflow using a manual landmark analysis as well as rigid registration with the elastix software package. Results: Manual landmarks selected between the corresponding DRR generators across patient and phantom images have an average displacement of 1.15 mm. For elastix image registrations, we found that absolute value vertical and horizontal translations were 0.18 and 0.35 mm on average, respectively. Rigid rotations were within 0.002 degrees. Conclusion: Custom and ITK‐based algorithms successfully reproduce ExacTrac DRRs and have the distinctive advantage of incorporating any desired 6D couch position. An open‐source repository is provided separately for users to implement in IGRT patient positioning research. [ABSTRACT FROM AUTHOR]

For nonlinear dispersive systems, the nonlinear Schrödinger (NLS) equation can usually be derived as a formal approximation equation describing slow spatial and temporal modulations of the envelope of a spatially and temporally oscillating underlying carrier wave. Here, we justify the NLS approximation for a whole class of quasilinear dispersive systems, which also includes toy models for the waterwave problem. This is the first time that this is done for systems, where a quasilinear quadratic term is allowed to effectively lose more than one derivative. With effective loss, we here mean the loss still present after making a diagonalization of the linear part of the system such that all linear operators in this diagonalization have the same regularity properties. [ABSTRACT FROM AUTHOR]

Using a nonlocal version of the center‐manifold theorem and a normal form reduction, we prove the existence of small‐amplitude generalized solitary‐wave solutions and modulated solitary‐wave solutions to the steady gravity‐capillary Whitham equation with weak surface tension. Through the application of the center‐manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four‐dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity‐capillary water wave problem, the associated linear operator is seen to undergo either a reversible 02+(ik0)$0^{2+}(\text{i}k_0)$ bifurcation or a reversible (is)2$(\text{i}s)^2$ bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second‐order or third‐order terms. These truncated systems relate directly to systems obtained in the study of the full gravity‐capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity‐capillary Whitham equation due to reversibility. Consequently, this work illuminates further connections between the gravity‐capillary Whitham equation and the full two‐dimensional gravity‐capillary water wave problem. [ABSTRACT FROM AUTHOR]