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We show abstract versions for Banach couples of several limiting compact interpolation theorems established by Edmunds and Opic for couples of [ABSTRACT FROM AUTHOR]

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,−∞

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]

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]

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]

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]

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]

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]

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]

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]

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]

Summary: This paper presents algorithm improvements that reduce the numerical noise and increase the numerical stability of the material point method formulation. Because of the linear mapping required in each time step of the material point method algorithm, a possible mismatch between the number of material points and grid nodes leads to a loss of information. These null‐space related errors may accumulate and affect the numerical solution. To remove the null‐space errors, the presented algorithm utilizes a null‐space filter. The null‐space filter shown removes the null‐space errors, resulting from the rank deficient mapping and the difference between the number of material points and the number of nodes. The presented algorithm enhancements also include the use of the explicit generalized‐α integration method, which helps optimizing the numerical algorithmic dissipation. This paper demonstrates the performance of the improved algorithms in several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, output regulation for linear delta operator systems subject to actuator saturation is investigated by state feedback. The relation between the regulatable regions and the null controllable region is described in this paper. A set of all initial conditions of a plant and a exosystem is called asymptotically regulatable region for which output regulation is possible. An asymptotically regulatable region is characterized according to a null controllable region of an anti-stable subsystem of the plant. Feedback laws are constructed to solve the problems on output regulation. A numerical example is given to illustrate the effectiveness and potential for the developed techniques. Copyright © 2016 John Wiley & Sons, Ltd. [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]

This paper addresses the stabilization problem of a microscale beam system subject to a delay. Several situations are considered depending whether the delay occurs as a boundary or interior/distributed term. In both cases, the microbeam system is shown to be well posed in the sense of semigroups theory of linear operators. More importantly, using the energy method, the exponential stability is established as long as the parameter of the delay term is small. [ABSTRACT FROM AUTHOR]

This paper investigates the qualitative properties near a degenerate fixed point of a discrete predator–prey model with cooperative hunting derived from the Lotka–Volterra model where the eigenvalues of the corresponding linear operator are ±1. Applying the theory of the normal form and the Takens's theorem, we change the problem of this discrete model into the one of an ordinary differential system. With the technique of desingularization to blow up the degenerate equilibrium of the ordinary differential system, we obtain its qualitative properties. Utilizing the conjugacy between the discrete model and the time‐one mapping of the vector field, we obtain the qualitative structures of this discrete model. [ABSTRACT FROM AUTHOR]

Parallel‐in‐time methods, such as multigrid reduction‐in‐time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time‐dependent partial differential equations (PDEs) in modern high‐performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection‐dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant‐coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically nonscalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse‐grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. One of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse‐grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge–Kutta methods combined with upwind‐finite‐difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed‐ups over sequential time‐stepping. [ABSTRACT FROM AUTHOR]

In this paper, we consider a control system represented by a second‐order evolution impulsive problems with delay and deviated arguments in a Banach space X. We used the strongly continuous cosine family of linear operators and fixed‐point method to study the exact controllability. Also, we study the trajectory controllability of the considered control problem. Finally, an example is provided to illustrate the application of the obtained abstract results. [ABSTRACT FROM AUTHOR]

We discuss at first in this paper the Gauge equivalence among several u‐linear Hamiltonian operators and present explicitly the associated Gauge transformation of Bäcklund type among them. We then establish the sufficient and necessary conditions for the linear superposition of the discussed u‐linear operators and matrix differential operators with constant coefficients of arbitrary order to be Hamiltonian, which interestingly shows that the resulting Hamiltonian operators survive only up to the third differential order. Finally, we explore a few illustrative examples of integrable hierarchies from Hamiltonian pairs embedded in the resulting Hamiltonian operators. [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]

This paper provides various 'contractivity' results for linear operators of the form [ABSTRACT FROM AUTHOR]

Piecewise linear one-dimensional maps have been proposed as the basis for low-power analog and mixed-signal true random number generators (TRNGs). Recent research has moved towards conceiving maps that operate robustly under the consideration of parameter variations. In this paper, we introduce an oscillator circuit mapping a low-complexity map known as the skew-tent. This oscillator is employed as the basis for a TRNG scheme. Simulation results in TSMC 0.18 μ m validate the chaotic oscillator and the randomness of the TRNG scheme is verified with the NIST test suite 800-22. Copyright © 2016 John Wiley & Sons, Ltd. [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]

Instrumental variables are widely used in applied econometrics to achieve identification and carry out estimation and inference in models that contain endogenous explanatory variables. In most applications, the function of interest (e.g., an Engel curve or demand function) is assumed to be known up to finitely many parameters (e.g., a linear model), and instrumental variables are used to identify and estimate these parameters. However, linear and other finite-dimensional parametric models make strong assumptions about the population being modeled that are rarely if ever justified by economic theory or other a priori reasoning and can lead to seriously erroneous conclusions if they are incorrect. This paper explores what can be learned when the function of interest is identified through an instrumental variable but is not assumed to be known up to finitely many parameters. The paper explains the differences between parametric and nonparametric estimators that are important for applied research, describes an easily implemented nonparametric instrumental variables estimator, and presents empirical examples in which nonparametric methods lead to substantive conclusions that are quite different from those obtained using standard, parametric estimators. [ABSTRACT FROM AUTHOR]

We prove a Desch–Schappacher type perturbation theorem for one‐parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi‐continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the Banach algebra L(E) of bounded, linear operators on a Banach space E, and semigroups on the space of bounded and continuous functions over a Polish space induced by jointly continuous semiflows. For both of these classes we present an application of our abstract perturbation theorem. [ABSTRACT FROM AUTHOR]

In this paper, information about the instability of equilibrium solutions of a nonlinear family of localized reaction‐diffusion equations in dimension one is provided. More precisely, explicit formulas to the equilibrium solutions are computed and, via analytic perturbation theory, the exact number of positive eigenvalues of the linear operator associated to the stability problem is analyzed. In addition, sufficient conditions for blow up of the solutions of the equation are also discussed. [ABSTRACT FROM AUTHOR]

Summary: This paper addresses a modified singularity removal technique for the eigenvalue or optimal mode problems in pipe flow using a pseudospectral method. The current approach results in the linear stability operator to be devoid of any unstable physically spurious modes, and thus, it provides higher numerical stability during time‐based integration. The correctness of the numerical operator is established by calculating the known eigenvalues of pipe Poiseuille flow. Subsequently, the optimal modes are determined with Farrell's approach and compared with the existing literature. The usefulness of this approach is further demonstrated in the time‐based numerical integration of the linearized Navier‐Stokes operator for the adjoint method–based optimal mode determination. The numerical scheme is implemented with the radial velocity‐radial vorticity formulation. Even number of Chebyshev‐Lobatto grid points are distributed over the domain r∈[−1,1] omitting the centerline, which also efficiently provides higher resolution near the wall boundary. The boundary conditions are imposed with homogeneous wall boundary conditions, whereas the analytic nature of a proper set of base functions enforces correct centerline conditions. The resulting redundancy introduced in the process is eliminated with the proper usage of parity. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a general family of two‐dimensional wave equations, which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated the existence problem of point transformations that lead mappings between linear and nonlinear members of particular families and determined the structure of the nonlinear terms of linearizable equations. We have also given examples about some equivalence transformations between linear and nonlinear equations and obtained exact solutions of nonlinear equations via the linear ones. [ABSTRACT FROM AUTHOR]

The present paper deals with a new positive linear operator which gives a connection between the Bernstein operators and their genuine Bernstein‐Durrmeyer variants. These new operators depend on a certain function τ defined on [0,1] and improve the classical results in some particular cases. Some approximation properties of the new operators in terms of first and second modulus of continuity and the Ditzian‐Totik modulus of smoothness are studied. Quantitative Voronovskaja–type theorems and Grüss‐Voronovskaja–type theorems constitute a great deal of interest of the present work. Some numerical results that compare the rate of convergence of these operators with the classical ones and illustrate the relevance of the theoretical results are given. [ABSTRACT FROM AUTHOR]

In this paper, we present a practically robust method for computing foldover‐free volumetric mappings with hard linear constraints. Central to this approach is a projection algorithm that monotonically and efficiently decreases the distance from the mapping to the bounded conformal distortion mapping space. After projection, the conformal distortion of the updated mapping tends to be below the given bound, thereby significantly reducing foldovers. Since it is non‐trivial to define an optimal bound, we introduce a practical conformal distortion bound generation scheme to facilitate subsequent projections. By iteratively generating conformal distortion bounds and trying to project mappings into bounded conformal distortion spaces monotonically, our algorithm achieves high‐quality foldover‐free volumetric mappings with strong practical robustness and high efficiency. Compared with existing methods, our method computes mesh‐based and meshless volumetric mappings with no prescribed conformal distortion bounds. We demonstrate the efficacy and efficiency of our method through a variety of geometric processing tasks. [ABSTRACT FROM AUTHOR]

Let T be the Volterra operator on L2[0, 1]Tf(x)=∫0xf(t)dt, where f ∈ L2[0, 1], 0 ≤ x ≤ 1. It is well known that ‖n!Tn‖ = O(1/n!). In a recent paper [1], D. Kershaw has proved that limn→∞‖n!Tn‖=1/2, a result which was first conjectured by Lao and Whitley in [2]. It is easy to prove that limn→∞ sup‖n!Tn‖≤1/2, For completeness, we give the proof using the familiar Schmidt norm estimate for the norm of an integral operator (see Section 2 below). Kershaw proves that lim infn→∞‖n!Tn‖≥1/2 by analysing the special positivity preserving properties of T*T. He uses one of the many abstract theorems on eigenvalues and eigenfunctions of compact operators which preserve a cone. In this paper we shall reprove (1), giving a short and direct proof of (2). 1991 Mathematics Subject Classification 47G10, 45-04. [ABSTRACT FROM AUTHOR]

Summary: This paper studies the problem of recursive state estimation of stochastic linear systems with nonlinear measurements. The main idea is to rewrite the measurement map in a linear form by considering, as system output, a vector of "virtual" measurements. The result is a linear system with a non‐Gaussian and nonstationary output noise. State estimation is therefore obtained using a Kalman filter or, alternatively, a quadratic filter, suitably designed for non‐Gaussian systems. This work provides two sufficient conditions for the application of the virtual measurement approach and shows its effectiveness in the case of the maneuvering target tracking problem. [ABSTRACT FROM AUTHOR]

Summary: Today, effective implementations of digital image correlation (DIC) are based on iterative algorithms with constant linear operators. A relevant idea of the classic finite element (or, more generally, global) DIC solver consists in replacing the gradient of the deformed state image with that of the reference image, so as to obtain a constant operator. Different arguments (small strains, small deformations, equality of the two gradients close to the solution, etc) have been given in the literature to justify this approximation, but none of them are fully accurate. Indeed, the convergence of the optimization algorithm has to be investigated from its ability to produce descent directions. Through such a study, this paper attempts to explain why this approximation works and what is its domain of validity. Then, an inverse compositional Gauss‐Newton implementation of finite element DIC is proposed as a cost‐effective and mathematically sound alternative to this approximation. [ABSTRACT FROM AUTHOR]

In this paper, we study well‐posedness and asymptotic stability of a wave equation with a general boundary control condition of diffusive type. We prove that the system lacks exponential stability. Furthermore, we show an explicit and general decay rate result, using the semigroup theory of linear operators and an estimate on the resolvent of the generator associated with the semigroup. [ABSTRACT FROM AUTHOR]

In this paper, we will modify the Milnor–Thurston map, which maps a one‐dimensional mapping to a piece‐wise linear of the same entropy, and study its properties. This will allow us to give a simple proof of monotonicity of topological entropy for real polynomials and better understand when a one‐dimensional map can and cannot be approximated by hyperbolic maps of the same entropy. In particular, we will find maps of particular combinatorics which cannot be approximated by hyperbolic maps of the same entropy. [ABSTRACT FROM AUTHOR]