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Smart voice systems, such as voice assistants and smart speakers, are integral to domains such as smart homes, customer service, healthcare, and smart learning. The effectiveness of these systems relies on user comprehension performance, which is crucial for enhancing user experience. In this study, the primary factors influencing comprehension performance in multilanguage smart voice systems are examined, and the efficacy of various analytical methods, including LASSO regression, SEM, PLS-SEM, CNN, and BiLSTM, are assessed by identifying and improving these factors. Using a diverse dataset from human–computer interaction experiments made publicly available on GitHub, these five methods are applied to discern the impact of environmental and user-specific factors on comprehension. The key findings indicate the following: 1) Noise types and noise sound levels markedly affect comprehension. Noise sound level exhibited an inverted U-shape curve (parameter: 0.088) due to the low and high levels of noise. Certain rhythmic noises, such as those from clocks (parameter: 0.033), enhance comprehension by fostering a conducive auditory environment. 2) Analytical method comparisons reveal that while LASSO regression (MSE = 0.026), SEM, and PLS-SEM effectively map the linear relationships and pathways affecting comprehension, deep learning approaches such as CNN and BiLSTM (MSE = 0.019) excel at handling complex, multidimensional data, offering superior predictive performance.3) In a non-native language environment, the evaluation of user comprehension models is notably different from that in native language settings (native R2: 0.545; non-native R2: 0.347). Specifically, in non-native language environments, the variables and mechanisms influencing user comprehension models are clearer, more controllable, and more susceptible to proficiency levels (parameter: 0.164). This comprehensive study presents a novel comparison of traditional statistical and machine learning methods in analyzing smart voice system interaction across languages. These findings emphasize the significance of tailoring smart voice systems to user diversity in language proficiency, age, and educational background and suggest optimizing these systems under varied environmental conditions to improve comprehension and overall effectiveness. The insights from this study are critical for policymakers and designers aiming to refine the adaptability and user-centric nature of smart voice systems. [ABSTRACT FROM AUTHOR]

In this paper we propose some Harris-like criteria in order to study the long time behavior of general positive and periodic semiflows. These criteria allow us to obtain new existence results of principal eigenelements, and their exponential attractiveness. We present applications to two biological models in a space-time varying environment: a non local selection-mutation equation and a growth-fragmentation equation. The particularity of this article is to study some inhomogeneous problems that are periodic in time, as it appears for instance when the environment changes, due for instance to the seasonal cycle or circadian rhythms. [ABSTRACT FROM AUTHOR]

The subjective number-space mapping and, especially, its evolution in young children has been the subject of intense controversy among different competing models. Many studies point out that: (i) young children's innate estimates follow a logarithmic mapping (Weber-Fechner law) and (ii) driven by education, children evolve into a linear mapping. In this paper we show, in consonance with other investigations, that innate numerical intuitions of young children are in fact a linear mapping, and not a logarithmic one. We found that young children extrapolate linearly from a reference point. The apparent logarithmic mapping is an artifact produced by placing a huge range of numbers within a small number line. We show how the appearance of the mapping can be modulated by changing the size and contour conditions of the number lines used in experiments. [ABSTRACT FROM AUTHOR]

On Hom-Lie algebras and superalgebras, we introduce the notions of biderivations and linear commuting maps, and compute them for some typical Hom-Lie algebras and superalgebras, including the q-deformed W(2,2) algebra, the q-deformed Witt algebra and superalgebra. [ABSTRACT FROM AUTHOR]

We study the approximate controllability of the discrete fractional systems of order 1 < α < 2 (∗) C ∇ α u n = A u n + B v n + f (n , u n) , n ≥ 2 , subject to the initial states u 0 = x 0 , u 1 = x 1 , where A is a closed linear operator defined in a Hilbert space X, B is a bounded linear operator from a Hilbert space U into X , f : N 0 × X → X is a given sequence and C ∇ α u n is an approximation of the Caputo fractional derivative ∂ t α of u at t n : = τ n , where τ > 0 is a given step size. To do this, we first study resolvent sequences { S α , β n } n ∈ N 0 generated by closed linear operators to obtain some subordination results. In addition, we discuss the existence of solutions to (∗) and next, we study the existence of optimal controls to obtain the approximate controllability of the discrete fractional system (∗) in terms of the resolvent sequence { S α , β n } n ∈ N 0 for some α , β > 0. Finally, we provide an example to illustrate our results. [ABSTRACT FROM AUTHOR]

While the asymptotic Borel mapping, sending a function into its series of asymptotic expansion in a sector, is known to be surjective for arbitrary openings in the framework of ultraholomorphic classes associated with sequences of rapid growth, there is no general procedure to construct extension operators in this case. We do provide such operators in complex sectors for some particular classes considered by S. Pilipović, N. Teofanov and F. Tomić in the ultradifferentiable setting. Although these classes are, in their words, "beyond Gevrey regularity", in some cases they keep the property of stability under differentiation, which is crucial for our technique, based on formal Borel- and truncated Laplace-like transforms with suitable kernels. [ABSTRACT FROM AUTHOR]

Resolvent analysis is a powerful tool that can reveal the linear amplification mechanisms between the forcing inputs and the response outputs about a base flow. These mechanisms can be revealed in terms of a pair of forcing and response modes and the associated energy gains (amplification magnitude) at a given frequency. The linear relationship that ties the forcing and the response is represented through the resolvent operator (transfer function), which is constructed through spatially discretizing the linearized Navier–Stokes operator. One of the unique strengths of resolvent analysis is its ability to analyze statistically stationary turbulent flows. In light of the increasing interest in using resolvent analysis to study a variety of flows, we offer this guide in hopes of removing the hurdle for students and researchers to initiate the development of a resolvent analysis code and its applications to their problems of interest. To achieve this goal, we discuss various aspects of resolvent analysis and its role in identifying dominant flow structures about the base flow. The discussion in this paper revolves around the compressible Navier–Stokes equations in the most general manner. We cover essential considerations ranging from selecting the base flow and appropriate energy norms to the intricacies of constructing the linear operator and performing eigenvalue and singular value decompositions. Throughout the paper, we offer details and know-how that may not be available to readers in a collective manner elsewhere. Towards the end of this paper, examples are offered to demonstrate the practical applicability of resolvent analysis, aiming to guide readers through its implementation and inspire further extensions. We invite readers to consider resolvent analysis as a companion for their research endeavors. [ABSTRACT FROM AUTHOR]

We study the nonlinear dynamics of perturbed, spectrally stable T-periodic stationary solutions of the Lugiato–Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. It is known that for each N ∈ N , such a T-periodic wave train is asymptotically stable against NT-periodic, i.e. subharmonic, perturbations, in the sense that initially nearby data will converge at an exponential rate to a (small) spatial translation of the underlying wave. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential rates of decay depend on N and, in fact, tend to zero as N → ∞ , leading to a lack of uniformity in the period of the perturbation. In recent work, the authors performed a delicate decomposition of the associated linearized solution operator and obtained linear estimates which are uniform in N. The dynamical description suggested by this uniform linear theory indicates that the corresponding nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation of the underlying wave. However, such a modulated perturbation is readily seen to satisfy a quasilinear equation, yielding an inherent loss of regularity. We regain regularity by transferring a nonlinear damping estimate, which has recently been obtained for the LLE in the case of localized perturbations to the case of subharmonic perturbations. Thus, we obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in N. This in turn yields an improved nonuniform subharmonic stability result providing an N-independent ball of initial perturbations which eventually exhibit exponential decay at an N-dependent rate. Finally, we argue that our results connect in the limit N → ∞ to previously established stability results against localized perturbations, thereby unifying existing theories. [ABSTRACT FROM AUTHOR]

We introduce a new embedding technique based on a nested barycentric coordinate system. We show that our embedding can be used to transform the problems of polyhedron approximation, piecewise linear classification and convex regression into one of finding a linear classifier or regressor in a higher dimensional (but nevertheless quite sparse) representation. Our embedding maps a piecewise linear function into an everywhere-linear function, and allows us to invoke well-known algorithms for the latter problem to solve the former. We explain the applications of our embedding to the problems of approximating separating polyhedra—in fact, it can approximate any convex body and unions of convex bodies—as well as to classification by separating polyhedra, and to piecewise linear regression. [ABSTRACT FROM AUTHOR]

We introduce an enhanced inertial proximal minimization algorithm tailored for a category of structured nonconvex and nonsmooth optimization problems. The objective function in question is an aggregation of a smooth function with an associated linear operator, a nonsmooth function dependent on an independent variable, and a mixed function involving two variables. Throughout the iterative procedure, parameters are selected employing a straightforward approach, and weak inertial terms are incorporated into two subproblems within the update sequence. Under a set of lenient conditions, we demonstrate that the sequence engendered by our algorithm is bounded. Furthermore, we establish the global and strong convergence of the algorithmic sequence, contingent upon the assumption that the principal function adheres to the Kurdyka–Łojasiewicz (KL) property. Ultimately, the numerical outcomes corroborate the algorithm's feasibility and efficacy. [ABSTRACT FROM AUTHOR]

The optical microscope algorithm (OMA) is a metaheuristic algorithm that draws inspiration from the magnifying functionality of optical microscopes. This study introduces an enhanced variant of OMA, termed PMSOMA, designed to mitigate the original version's limitations, notably its slow convergence rates and vulnerability to local optima. PMSOMA integrates a piecewise linear chaotic map to refine population initialization and augment diversity, alongside a sparse adaptive exploration mechanism to bolster search efficacy. The performance of PMSOMA was rigorously tested using a suite of 50 benchmark functions, the CEC2017 test suite, feature selection datasets, and three classical engineering challenges. The empirical findings confirm that PMSOMA surpasses both the original OMA and competing algorithms by delivering superior solutions, accelerating convergence, and demonstrating enhanced robustness in convergence. [ABSTRACT FROM AUTHOR]

In this research, we introduce a novel optimization algorithm termed the dual-relaxed inertial alternating direction method of multipliers (DR-IADM), tailored for handling nonconvex and nonsmooth problems. These problems are characterized by an objective function that is a composite of three elements: a smooth composite function combined with a linear operator, a nonsmooth function, and a mixed function of two variables. To facilitate the iterative process, we adopt a straightforward parameter selection approach, integrate inertial components within each subproblem, and introduce two relaxed terms to refine the dual variable update step. Within a set of reasonable assumptions, we establish the boundedness of the sequence generated by our DR-IADM algorithm. Furthermore, leveraging the Kurdyka–Łojasiewicz (KŁ) property, we demonstrate the global convergence of the proposed method. To validate the practicality and efficacy of our algorithm, we present numerical experiments that corroborate its performance. In summary, our contribution lies in proposing DR-IADM for a specific class of optimization problems, proving its convergence properties, and supporting the theoretical claims with numerical evidence. [ABSTRACT FROM AUTHOR]

The roots of the indicial polynomial constructed for a given linear ordinary differential operator provide information on the singularities of the solutions of the corresponding homogeneous differential equation. Operators and equations whose coefficients are formal Laurent series are discussed. Solutions of the same type are also considered. Under these assumptions, the structure of the indicial polynomial of the product of differential operators is described. This structural (multiplicative) property is preserved in the case of convergent series. [ABSTRACT FROM AUTHOR]

Quasiderivations of the universal enveloping algebra were first introduced by D. Gurevich, P. Pyatov, and P. Saponov in their study of reflection equation algebras; they are linear operators on that satisfy certain algebraic relations, which generalise the usual Leibniz rule. In this note, we show that the iterated action of the operator equal to a linear combination of the quasiderivations on a certain set of generators of the center of (namely on the symmetrised coefficients of the characteristic polynomial) produces commuting elements. The resulting algebra coincides with the quantum Mischenko–Fomenko algebra in , introduced earlier by Tarasov, Rybnikov, Molev, and others. [ABSTRACT FROM AUTHOR]

The authors consider variational inequalities in the Hilbert space and two algorithms for their approximate solution: extrapolation from the past and operator extrapolation. Both algorithms have less computationally expensive iterations than the the classical extragradient algorithm: only one operator computation rather than two is needed. Non-asymptotic linear convergence rate estimates are proved for the variational inequalities with the Lipschitz continuous operators satisfying the generalized strict monotonicity condition. The results are new and improve the available estimates. [ABSTRACT FROM AUTHOR]

Introduction: The EQ-5D-Y-3L is a generic measure of health-related quality of life in children and adolescents. Although the Brazilian-Portuguese EQ-5D-Y-3L version is available, there is no value set for it, hampering its use in economic evaluations. This study aimed to elicit a Brazilian EQ-5D-Y-3L value set based on preferences of the general adult population. Methods: Two independent samples of adults participated in an online discrete choice experiment (DCE) survey and a composite time trade-off (cTTO) face-to-face interview. The framing was "considering your views for a 10-year-old child". DCE data were analyzed using a mixed-logit model. The 243 DCE predicted values were mapped into the observed 28 cTTO values using linear and non-linear mapping approaches with and without intercept. Mapping approaches' performance was assessed to estimate the most valid method to rescale DCE predicted values using the model fit (R2), Akaike Information Criteria (AIC), root mean squared error (RMSE), and mean absolute error (MAE). Results: A representative sample of 1376 Brazilian adults participated (DCE, 1152; cTTO, 211). The linear mapping without intercept (R2 = 96%; AIC, − 44; RMSE, 0.0803; MAE, − 0.0479) outperformed the non-linear without intercept (R2 = 98%; AIC, − 63; RMSE, 0.1385; MAE, − 0.1320). Utilities ranged from 1 (full health) to − 0.0059 (the worst health state). Highest weights were assigned to having pain or discomfort (pain/discomfort), followed by walking about (mobility), looking after myself (self-care), doing usual activities (usual activities), and feeling worried, sad, or unhappy (anxiety/depression). Conclusion: This study elicited the Brazilian EQ-5D-Y-3L value set using a mixed-logit DCE model with a power parameter based on a linear mapping without intercept, which can be used to estimate the quality-adjusted life-years for economic evaluations of health technologies targeting the Brazilian youth population. [ABSTRACT FROM AUTHOR]

In the paper we give new asymptotic evaluations for sequences of linear positive operators V n : C 2 π R → C 2 π R . Our method of the proof is entirely different than those known in this area. As applications we complete and extend known asymptotic evaluations in this topic. [ABSTRACT FROM AUTHOR]

We introduce a new optimization approach to solving systems of split equality problems in real Hilbert spaces. We use the inertial method in order to improve the convergence rate of the proposed algorithms. Our algorithms do not depend on the norms of the bounded linear operators which appear in each split equality problem of the system under consideration. This is also a strong point of our algorithms because it is known that it is difficult to compute or estimate the norm of a linear operator in the general case. [ABSTRACT FROM AUTHOR]

We introduce the dual-Hilbert space and study the basic properties of a dual operator and its generalized inverse on this space. We provide upper bounds on the perturbation of the dual Moore–Penrose inverse of the dual operator if the dual operator is injective or surjective. If the null space or range space of the perturbed dual operator is invariant, stable perturbations are used to give the perturbation bounds for the dual Moore–Penrose inverse. Additionally, given the aforementioned conditions, perturbation bounds for the least squares solution are provided. The upper bounds on the distance between the solution of a perturbed least squares problem and the set of all of its unperturbed solutions under the dual operator norm are also presented. [ABSTRACT FROM AUTHOR]

In this article we focus mainly on the class of almost periodic functions in view of the Lebesgue measure (briefly: μ -a.p. functions) and on some if its subclasses. We are going to deal with autonomous superposition operators acting in the space of μ -a.p. functions. We will indicate necessary and sufficient conditions under which the autonomous superposition operator maps the space under consideration into itself as well as conditions under which it is continuous. As a corollary from these results we indicate when the autonomous superposition operator defined on that space is a bijection. Next, we will analyse in detail the situation when the composition of μ -a.p. function with a continuous function or with a homeomorphism gives a Stepanov almost periodic function. As an application of our results we indicate a subclass of μ -a.p. functions for which linear differential equations with a non-homogeneous term belonging to this subclass may not have μ -a.p. solutions. [ABSTRACT FROM AUTHOR]

Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the more general problem of computing the image under the homology functor of a continuous semi-algebraic map f : X → Y between closed and bounded semi-algebraic sets. For every fixed ℓ ≥ 0 we give an algorithm with singly exponential complexity that computes bases of the homology groups H i (X) , H i (Y) (with rational coefficients) and a matrix with respect to these bases of the induced linear maps H i (f) : H i (X) → H i (Y) , 0 ≤ i ≤ ℓ . We generalize this algorithm to more general (zigzag) diagrams of continuous semi-algebraic maps between closed and bounded semi-algebraic sets and give a singly exponential algorithm for computing the homology functors on such diagrams. This allows us to give an algorithm with singly exponential complexity for computing barcodes of semi-algebraic zigzag persistent homology in small dimensions. [ABSTRACT FROM AUTHOR]

We prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from H α (R , U) to H β (R , U) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail. [ABSTRACT FROM AUTHOR]

In recent years, the study of bounded linear operators in several variables has received great interest from many authors, including the second author's previous contributions. In our present work, we define a new class of multivariable operator theory, which we have called M-hyponormal tuple. We present some algebraic and spectral properties associated with them. [ABSTRACT FROM AUTHOR]

A novel method for separating the matrix spectrum by a straight line based on a fractional linear transformation is proposed. This method has a number of advantages over the approaches based on an exponential transformation; more precisely, the range of its application is wider and the number of iterations needed for its convergence is much lower. The proposed method is used to study flutter problems for an infinite strip under various edge fastening conditions, which, after suitable discretization of differential operators, are reduced to spectral problems for linear operators. The study of stability regions by the method of spectrum dichotomy by the imaginary axis makes it possible to construct neutral curves in the plane of parameters of the flutter problem. [ABSTRACT FROM AUTHOR]

The main aim of the paper is to determine some basic properties of the essential pseudospectrum of a bounded linear operator A defined in a Banach space X. We also prove two different versions of the essential pseudospectral mapping theorem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic properties of a particular class of state-dependent sweeping processes. While extensive research has been conducted on the existence and uniqueness of solutions for sweeping processes, there is a scarcity of studies addressing their behavior in the limit of large time. Additionally, we introduce novel algorithms designed for the resolution of quasi-variational inequalities. As a result, we introduce a new derivative-free algorithm to find zeros of nonsmooth Lipschitz continuous mappings with a linear convergence rate. This algorithm can be effectively used in nonsmooth and nonconvex optimization problems that do not possess necessarily second-order differentiability conditions of the data. [ABSTRACT FROM AUTHOR]

We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior a hom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨ · ⟩ and the corresponding linear elliptic operator - ∇ · a ∇ . In line with the theory of homogenization, the method proceeds by computing d = 3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨ · ⟩ . We make this point by investigating the bias (or systematic error), i.e., the difference between a hom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O (L - 1) . In case of a suitable periodization of ⟨ · ⟩ , we rigorously show that it is O (L - d) . In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨ · ⟩ of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function. [ABSTRACT FROM AUTHOR]

We propose a hybrid inertial self-adaptive algorithm for solving the split feasibility problem and fixed point problem in the class of demicontractive mappings. Our results are very general and extend several related results existing in the literature from the class of nonexpansive or quasi-nonexpansive mappings to the larger class of demicontractive mappings. Examples to illustrate numerically the effectiveness of the new analytical results are presented. [ABSTRACT FROM AUTHOR]

This paper studies the basic topological properties of weighted composition operators on the weighted sequence spaces , with , given by a weight sequence of positive reals such as boundedness, compactness, compactness of differences of two operators, formulas for their essential norms, and description of closed range operators. Previously these properties were studied by Luan and Khoi in the case of Hilbert space . Their methods can be also applied with some minor modifications to the case of Banach spaces with . They based essentially on using the dual spaces of continuous linear functionals and, consequently, cannot be applied to the quasi-Banach case . Moreover, some of them do not work even in . Motivated by these reasons, we develop a more universal approach that allows studying the whole scale of spaces . To this end, we establish the necessary and sufficient conditions for a linear operator to be compact on an abstract quasi-Banach sequence space. These conditions are new even in the case of Banach spaces. Moreover, we introduce the new characteristic, the -essential norm of a continuous linear operator on a quasi-Banach space . This characteristic measures the distance in the operator metric, between and the set of all -compact operators on . Here an operator is -compact on if is compact and coordinatewise continuous on . We show that for with the essential and -essential norms of a weighted composition operator coincide, whereas for we do not know whether the same is true or not. Our main results for weighted composition operators in are as follows: We provide criteria for an operator to be bounded, compact, or closed range, and completely describe the pairs of operators with compact difference; as well as some exact formula for the -essential norm. Some key aspects of our approach can be used for other operators and scales of spaces. [ABSTRACT FROM AUTHOR]

The article develops the theory of stability of linear operator schemes for operator inequalities and nonlinear nonstationary initial–boundary value problems of mathematical physics with nonlinearities of unbounded growth. Based on sufficient conditions for the stability of A.A. Samarskii's two- and three-level difference schemes, the corresponding a priori estimates for operator inequalities are obtained under the condition of the criticality of the difference schemes under consideration, i.e., when the difference solution and its first time derivative are nonnegative at all nodes of the grid domain. The results obtained are applied to the analysis of the stability of difference schemes that approximate the Fisher and Klein–Gordon equations with nonlinear right-hand sides. [ABSTRACT FROM AUTHOR]

Collage Theorem provides a bound for the distance between an element of a given space and a fixed point of a self-map on that space, in terms of the distance between the point and its image. We give in this paper some results of Collage type for Reich mutual contractions in b-metric and strong b-metric spaces. We give upper and lower bounds for this distance, in terms of the constants of the inequality involved in the definition of the contractivity. Reich maps contain the classical Banach contractions as particular cases, as well as the maps of Kannan type, and the results obtained are very general. The middle part of the article is devoted to the invertibility of linear operators. In particular we provide criteria for invertibility of operators acting on quasi-normed spaces. Our aim is the extension of the Casazza-Christensen type conditions for the existence of inverse of a linear map defined on a quasi-Banach space, using different procedures. The results involve either a single map or two operators. The latter case provides a link between the properties of both mappings. The last part of the article is devoted to study the construction of fractal curves in Bochner spaces, initiated by the first author in a previous paper. The objective is the definition of fractal curves valued on Banach spaces and Banach algebras. We provide further results on the fractal convolution of operators, defined in the same reference, considering in this case the nonlinear operators. We prove that some properties of the initial maps are inherited by their convolutions, if some conditions on the elements of the associated iterated function system are satisfied. In the last section of the paper we use the invertibility criteria given before in order to obtain perturbed fractal spanning systems for quasi-normed Bochner spaces composed of Banach-valued integrable maps. These results can be applied to Lebesgue spaces of real valued functions as a particular case. [ABSTRACT FROM AUTHOR]

Stechkin's problem of the best approximation of differentiation operators by bounded linear operators on the half-line in the uniform norm is studied. The structure of the best approximation operator is investigated, and its relationship to the spline dual (in the sense of N. P. Kuptsov) to the extremal spline in the Landau–Kolmogorov inequality on the half-line is examined. [ABSTRACT FROM AUTHOR]

Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical ranges, or to detect gaps in essential spectra and the corresponding failure of the finite section method. Despite new results on computing spectra and the substantial interest in these geometric problems, there remain no general methods able to compute such geometric features of spectra of infinite-dimensional operators. We provide the first algorithms for the computation of many of these long-standing problems (including the above). As demonstrated with computational examples, the new algorithms yield a library of new methods. Recent progress in computational spectral problems in infinite dimensions has led to the solvability complexity index (SCI) hierarchy, which classifies the difficulty of computational problems. These results reveal that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale's comprehensive program on the foundations of computational mathematics initiated in the 1980s. We classify the computation of geometric features of spectra in the SCI hierarchy, allowing us to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that our algorithms are optimal. We also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications. [ABSTRACT FROM AUTHOR]

Images are a crucial component in contemporary data transmission. Numerous images are transmitted daily through the open-source network. This paper presents a multi-image encryption scheme that utilises flip-shift-rotate synchronous-permutation-diffusion (FSR-SPD) processes to ensure the security of multiple images in a single encryption operation. The proposed encryption technique distinguishes itself from current multi-image encryption methods by utilising SPD operation and rapid FSR-based pixel-shuffling and diffusion operation. The SPD is a cryptographic technique that involves the simultaneous application of permutation and diffusion methods. The FSR-based process involves the manipulation of pixels through three different operations, namely flipping, shifting, and rotating. In the process of encryption, the image components of red, green, and blue colours are merged into a single composite image. The large image is partitioned into non-overlapping blocks of uniform size. The SPD technique is employed to tackle each specific block. The encryption method is efficient and expeditious as it exhibits high performance with both FSR and SPD procedures. The method employs a single, fixed-type, one-dimensional, piecewise linear chaotic map (PWLCM) for both the permutation and diffusion phases, resulting in high efficiency in both software and hardware. The proposed method is assessed using key space, histogram variance, neighbouring pixel correlation, information entropy, and computational complexity. The proposed method has a much bigger key space than the comparative method. Compared to comparison approaches, the suggested solution reduces encrypted picture histogram variance by 6.22% and neighbouring pixel correlations by 77.78%. Compared to the comparison technique, the proposed scheme has a slightly higher information entropy of 0.0025%. Other multiple-color image encryption methods are more computationally intensive than the suggested method. Computer simulations, security analysis, and comparison analysis evaluated the proposed methodology. The results show it outperforms multiple images encrypting methods. [ABSTRACT FROM AUTHOR]

The Steklov eigenvalues of the Lamé operator arise from the analysis of surface waves and vibration modes in a linearly elastic structure. In this paper, based on the Lagrange conforming finite element method, we propose and analyze an a posteriori error indicator of the residual type for the Steklov eigenvalue problem of the Lamé operator in linear elasticity. The a posteriori error indicator is computed locally from the approximate eigenpair. We prove the reliability and the efficiency of the error indicator for eigenfunctions as well as the reliability of the error indicator for eigenvalues. Finally, we present numerical experiments to validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a modification of general linear positive operators introduced by Ibragimov and Gadjiev in 1970 is constructed. It is shown that this modification preserves exponential mappings and also contains modified Bernstein-, Szász- and Baskakov-type operators as special cases. The convergence properties of corresponding operators on [ 0 , ∞) and in exponentially weighted spaces are investigated. Finally, the quantitative Voronovskaja theorem in terms of modulus of continuity for functions having exponential growth is examined. [ABSTRACT FROM AUTHOR]

The paper provides a characterization of linear operators preserving column majorization of the (0, 1)-vectors. In addition, such operators are characterized explicitly in the case where they are given by special matrices, namely, by (±1)-matrices of order not exceeding 10 or (0,±1)-matrices of order not exceeding 5. Several related combinatorial matrix theory results also are proved. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of three-composite nonconvex nonsmooth optimization problems, where one of nonsmooth functions is further composed with linear operator. Based on the variable smoothing method, as well as first-order methods with suitable majorization techniques, we propose a proximal variable smoothing gradient (ProxVSG) method for solving this kind of problem. The ProxVSG can be implemented efficiently, thanks to the fact that at each iteration, one just separately computes the proximal mapping of each nonsmooth function, rather than that of the sum of these nonsmooth functions. Furthermore, within our broad and flexible analysis framework, we propose a new proximal variable smoothing incremental aggregated gradient (ProxVSIAG) generalizations of the ProxVSG. In ProxVSIAG, an incremental aggregated estimate of the gradient is used, instead of the full gradient. Under suitable assumptions, we prove a complexity of O (ϵ - 3) to achieve an ϵ -approximate solution. Preliminary numerical experiments show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Let A be a unital Banach algebra with unit e, M be a Banach A-bimodule, and w ∈ A . In this paper, we characterize those continuous linear maps δ : A → M that satisfy one of the following conditions: δ (a b) = δ (a) b + a δ (b) , 2 δ (w) = δ (a) b + a δ (b) , δ (a b) = δ (a) b + a δ (b) - a δ (e) b , for any a , b ∈ A with a b = b a = w , where w is either a separating point with w ∈ Z (A) or an idempotent. [ABSTRACT FROM AUTHOR]

Our main purpose in this paper is to construct a predual of Morrey–Lorentz space via the block spaces, defined in [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1999, 28(1): 31–40]. As a direct application of duality, we obtain the Morrey–Lorentz boundedness of linear Calderón–Zygmund operators. Moreover, we study a weak Hardy factorization in terms of linear Calderón–Zygmund operators in Morrey–Lorentz spaces. As a consequence of this result, we obtain a characterization of functions in BMO(ℝn) through the boundedness of commutator [b, T], where T is a homogeneous Calderón–Zygmund operator. Finally, we prove a Morrey–Lorentz compactness characterization of [b, T]. Precisely, [b, T] is a compact operator on Morrey–Lorentz spaces if and only if b ∈ CMO(ℝn). [ABSTRACT FROM AUTHOR]

Assume that X is a real or complex Hilbert space, T a linear relation in X and B a bounded linear operator in X , whose adjoints are denoted by T ∗ and B ∗ , respectively. It is shown in this note that if the following four linear relations T B B ∗ T ∗ , B ∗ T ∗ T B , B T T ∗ B ∗ and T ∗ B ∗ B T are selfadjoint in X then T must be a closed linear relation. [ABSTRACT FROM AUTHOR]

We study linear maps between preduals of JBW-algebras that preserve orthogonal decompositions of functionals. We generalize and strengthen results of Bunce and Wright (Pac J Math 158(2):265–272, 1993) obtained for von Neumann algebras by characterizing continuous orthogonally decomposable linear maps between preduals of JBW-algebras. In particular, we show that Banach space adjoint of such morphism Φ is typically of the form Φ ∗ (y) = c π (y) , where c is a central positive element and π is a Jordan homomorphism. Further we show that the order topology on preduals of JBW-algebras equals to the norm topology. This allows to relax continuity in characterization of orthogonally decomposable isomorphisms and show that order and orthogonality relation in preduals are complete invariants of JBW-algebras. [ABSTRACT FROM AUTHOR]

Globally liver diseases are the most life-threatening diseases, and according to global cancer statistics, liver cancer is the most common. Early detection of liver cancer can prevent millions of patients' mortality every year. Automatic liver cancer detection will help radiologists to determine the tumour identification and its severity, and it is also helpful to reduce the occurrence of errors which results in a reduction in the number of deaths from liver cancer. It gives more accurate results in less time, saving the radiologist's effort and time. The proposed model focused on improving the segmenting of liver images and then classifying the liver tumours from the CT images. The present study suggests the hyper tangent Fuzzy C-Means (HTFCM) to segment the liver images. It used Hyper tangent distance to calculate the data point distance from the cluster centres and obtained segmentation results almost closer to the ground truth liver images. Due to the fuzziness in the liver images, all state-of-the-art models except the proposed model cannot precisely locate the tumours. This study solved the issue of linear mapping using fuzzy logic, improved the classification accuracy, and reduced the processing time of early diagnosis of liver diseases. The proposed model improves the classification accuracy to 99.58% and reduces the processing time by 2-25 s to classify the liver tumours. [ABSTRACT FROM AUTHOR]

Images are considered crucial for conveying information due to their visualisation properties and capacity to hold a large amount of data. To safeguard image data from potential risks of information leakage, numerous image encryption algorithms have been developed. These algorithms often make use of chaotic maps, known for their high unpredictability, ergodicity, and sensitivity to parameters and initial values. This paper introduces a new chaos-based digital image encryption algorithm to protect digital images from unauthorised access or attacks. The algorithm heavily relies on permutation and diffusion processes. Specifically, it employs two rounds of pixel permutation using a Tinkerbell chaotic map sequence on the source image and two rounds of pixel diffusion using the Linear Feedback Shift Register and Logistic Map sequences on the permuted image. The encryption algorithm's effectiveness is thoroughly examined through various cryptanalysis techniques, including key space analysis, information entropy, correlation coefficient, differential attack, key sensitivity, histogram analysis, occlusion attack, noise attack, and encryption execution time. The experimental results are then compared with the most recent literature to demonstrate the algorithm's reliability and its ability to withstand diverse attacks. Notably, the proposed algorithm stands out as a secure and viable encryption solution, requiring less computational time than previous studies, thus making it a more practical option for real-world applications. [ABSTRACT FROM AUTHOR]

A monomial in finite collection of homogeneous polynomials is a point-wise product of these polynomials each raised to some power. It was established in [14] and [15] that, given a finite set of positive linear operators acting from an Archimedean vector lattice into an f-algebra with multiplicative unit, the corresponding monomial is orthogonally additive polynomial if and only if the collection of these operators is a disjointness preserving set. It is shown in this paper that a similar result is also true in the case when the target space is a uniformly complete vector lattice. Moreover, the result remains valid if we replace the linear operators with homogeneous orthogonally additive polynomials. The correct definition of monomials in homogeneous polynomials is guaranteed by the homogeneous functional calculus and the concavification procedure. [ABSTRACT FROM AUTHOR]

This paper deals with the equation - Δ u + μ u = f on high-dimensional spaces R m , where the right-hand side f (x) = F (T x) is composed of a separable function F with an integrable Fourier transform on a space of a dimension n > m and a linear mapping given by a matrix T of full rank and μ ≥ 0 is a constant. For example, the right-hand side can explicitly depend on differences x i - x j of components of x. Following our publication (Yserentant in Numer Math 146:219–238, 2020), we show that the solution of this equation can be expanded into sums of functions of the same structure and develop in this framework an equally simple and fast iterative method for its computation. The method is based on the observation that in almost all cases and for large problem classes the expression ‖ T t y ‖ 2 deviates on the unit sphere ‖ y ‖ = 1 the less from its mean value the higher the dimension m is, a concentration of measure effect. The higher the dimension m, the faster the iteration converges. [ABSTRACT FROM AUTHOR]

This study aimed to investigate the degree to which an L-fuzzy subset of a vector space could be considered an L-fuzzy vector subspace. Characterizations of this relationship were provided, and a novel approach to the fuzzification of a vector space was introduced. In the case where L was a complete distributive lattice, an L-fuzzy convex structure was obtained using L-fuzzy vector subspace degrees, thereby forming an L-fuzzy convex space. Additionally, it was proved that a linear mapping between vector spaces precisely corresponded to an L-fuzzy convexity-preserving mappings and L-fuzzy convex-to-convex mappings. [ABSTRACT FROM AUTHOR]

The extremal structure of zero-neighbourhoods of a topological module is analyzed reaching unexpected conclusions when the module topology is not Hausdorff. These results motivate us to introduce the notion of metric modules, which are modules endowed with a translation-invariant metric, turning them into an (additive) topological group. We study the central and diametral points of additively symmetric subsets and find examples of convex sets which are not symmetric translates (translates of addivitely symmetric subsets). As a consequence of all of these, it seems natural to transport the well-known Bishop-Phelps property from the category of real topological vector spaces to general topological modules over topological rings. Then we stick to particular topological rings, the unital C ∗ -algebras, showing that the subset of positive elements lying below the unity is an effect algebra. We also prove that every continuous linear operator on a Hausdorff locally convex topological vector space that commutes with all continuous linear projections of one-dimensional range is a multiple of the identity. Finally, we discuss how to transport the previous result to C ∗ -algebras. [ABSTRACT FROM AUTHOR]

We consider the nonlinear equation B(λ)x = R(x, λ) + b(λ), where R(0, 0) = 0, b(0) = 0, the linear operator B(λ) has a bounded inverse operator for S ∋ λ → 0, and S is an open set, 0 ∈ ∂S. We examine the existence of a small continuous solution of the maximal order of smallness x(λ) → 0 as S ∋ λ → 0. A constructive method of constructing this solution is presented. [ABSTRACT FROM AUTHOR]

In this paper, we examine differential equations with nonclassical initial conditions and noninvertible operators in their principal parts. We find necessary and sufficient conditions for the existence of unbounded solutions with a pole of order p at points where the operator in the principal part of the differential equation is noninvertible. Based on the alternative Lyapunov–Schmidt method and Laurent expansions, we propose a two-stage method for constructing expansion coefficients of the solution in a neighborhood of a pole. We develop the techniques of skeleton chains of linear operators in Banach spaces and discuss its applications to the statement of initial conditions for differential equations. The results obtained develop the theory of degenerate differential equations. Illustrative examples are given. [ABSTRACT FROM AUTHOR]