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The calculation of molecular excited states is critically important to decipher a plethora of molecular properties. In this paper, we develop an equation of motion formalism on top of a bi-exponentially parameterized ground state wavefunction toward the determination of excited states. While the ground state bi-exponential parameterization ensures an accurate description of the wavefunction through the inclusion of high-rank correlation effects, the excited state is parameterized by a novel linear response operator with an effective excitation rank beyond two. To treat the ground and excited states in the same footings, in addition to the conventional one- and two-body response operators, we introduced certain two-body "generalized" response operators with an effective excitation rank of one. We introduce a projective formulation for determining the perturbed amplitudes for the set of "generalized" operators. Our formulation entails a significantly small number of unknown parameters and is shown to be highly accurate compared to allied methods for several difficult chemical systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of gaps in the essential spectra of self-adjoint relations under non-negative relatively compact perturbation in Hilbert spaces. A stability result about semi-boundedness of self-adjoint relations under relatively bounded perturbation is given. It is shown that the gaps in the essential spectra of self-adjoint relations are invariant under non-negative relatively compact perturbation. The results obtained in the present paper generalize the corresponding results for linear operators to linear relations. [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]

The aim of this paper is to study the asymptotic convergence of the sequences. Concretely, we will give the asymptotically statistical convergence of sequences. The asymptotically statistical convergence is studied in many papers including papers [3], [5], [23], [26], [29], [39], [41]. In this paper, we introduce and investigate the concepts of asymptotically deferred statistical equivalent functions of order $ \alpha $ and strong asymptotically deferred statistical equivalent functions of order $ \alpha $ defined on discrete countable amenable semigroups, which generalizes some of the results known in the literature. This is achieved by introducing the Folner sequences and sequences of non-negative integers. Based on this concept we apply it to the approximation theory, and we have proved the Korovkin type theorem and study the rate of convergence for positive linear operators which are deferred statistically convergent of order $ \alpha $. [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]

The present paper deals with the approximation properties of the linear positive operators, including Δh -Gould-Hopper Appell polynomials. We investigate some theorems for convergence of the operators and their approximation degrees with the help of the classical approach, Peetre's K-functional, Lipschitz class and Voronovskajatype theorem. In the last section of the paper, we introduce the Kantorovich form of the operators and examine the approximation degree. [ABSTRACT FROM AUTHOR]

In this paper, we investigate various stabilisation problems of a hybrid system of elasticity, consisting of a thin tape, which moves axially between two sets of rollers. We present a control solution, which works simultaneously, for three variants of the tape model. In particular, we design boundary feedback control laws via Lyapunov's second method. The resulting closed-loop system for the general case is then analysed, where we prove the existence and uniqueness of solution using the semigroup theory of linear operators; furthermore, we prove the exponential decay of the solution via a frequency domain theorem. The stabilisation of the two other special cases is derived implicitly from the general case proofs presented in the paper. Finally, the paper is concluded by presenting computer simulations to support the theoretical results. [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 explore the spectral properties of unbounded generalized Fredholm operators acting on a non-reflexive Banach space X. The results are formulated in terms of some topological conditions made on X or on its dual X * . In addition, we introduce the concept of the so-called g-g-Riesz linear operators as an extension of Riesz operators. The obtained results are used to discuss the incidence of the behavior of generalized essential spectra. Furthermore, a relation between the generalized essential spectrum and the left (resp. the right) essential spectrum by means of g-Riesz perturbation is provided. [ABSTRACT FROM AUTHOR]

Let H be a real or complex Hilbert space and $ {\mathcal B}(H) $ B (H) the algebra of all bounded linear operators on H. Recall that a map $ \delta :{\mathcal B}(H)\to {\mathcal B}(H) $ δ : B (H) → B (H) is called an inner Jordan $ * $ ∗ -derivation if there exists some $ T\in \mathcal B(H) $ T ∈ B (H) such that $ \delta (A)=AT-TA^* $ δ (A) = AT − T A ∗ for all $ A\in {\mathcal B}(H) $ A ∈ B (H). In this paper, it is proved that inner Jordan $ * $ ∗ -derivations are the only additive maps δ of $ {\mathcal B}(H) $ B (H) with the property that $ \delta (P)=\delta (P)P^*+P\delta (P) $ δ (P) = δ (P) P ∗ + Pδ (P) for all idempotent operators $ P\in {\mathcal B}(H) $ P ∈ B (H) if $ \dim H=\infty $ dim ⁡ H = ∞ , which is satisfied by additive local Jordan $ * $ ∗ -derivations. For the finite dimensional case, additional conditions are required for δ to be an inner Jordan $ * $ ∗ -derivation. As applications, it is shown that, for any given $ C,D\in {\mathcal B}(H) $ C , D ∈ B (H) , δ satisfies $ \delta (A)B^*+B\delta (A)+\delta (B)A^*+A\delta (B)=D $ δ (A) B ∗ + Bδ (A) + δ (B) A ∗ + Aδ (B) = D for all $ A,B\in {\mathcal B}(H) $ A , B ∈ B (H) with AB + BA = C if and only if δ is an inner Jordan $ * $ ∗ -derivation and $ D=\delta (C) $ D = δ (C). Also, several known results are generalized. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we will consider parallel summable operators on Hilbert space. Operators A and B are said to be parallel summable if and . We will prove that the parallel sum can be represented as A: B = A(A + B)† B without additional assumption of the closedness of the range of the operator A + B. Furthermore, we will derive the equalities C (A : B) = C A : C B and (A : B) : C = A : (B : C ) under weaker conditions than the ones represented in [X. Tian, S. Wang, C. Deng, On parallel sum of operators, Linear Algebra Appl. 603 (2020) 57–83] and [W. Luo, C. Song, Q. Xu, The parallel sum for adjointable operators on Hilbert C* -modules, Acta Math. Sin. 62 (2019) 541–552]. Finally, we will extend some recent result for Hermitian positive semi-definite matrices to bounded linear operators. [ABSTRACT FROM AUTHOR]

Let (X , ≤) be a locally finite pre-ordered set and R be a commutative ring with unity. In this paper we apply the theory of zero product determined algebras to show that each linear map on the incidence algebra I (X , R) which is derivable at zero is a generalized derivation and every local derivation on I (X , R) is a derivation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish several versions of Paley–Wiener-type theorems for the canonical Fourier–Bessel transform, which is a Bessel type of linear canonical transform indexed by a matrix parameter m ∈ SL(2, ℝ). [ABSTRACT FROM AUTHOR]

In this paper, we present a new encryption and decryption algorithm that includes Fibonacci numbers, linear mappings, and change of basis. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, all biderivations of solvable Lie algebras of maximal rank g=Q⊕N are characterized. Namely, we consider a solvable Lie algebra of the form g=Q⊕N , where Q is the maximal torus subalgebra of g, N is the nilradical of g and dimQ=dimN/N2 .In the case dimQ=N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed-point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, with special cases being able to be found in many important engineering practical applications, such as image recognizing, signal processing, and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results. [ABSTRACT FROM AUTHOR]

A map Φ between matrices is said to be zero product preserving if $$\begin{align*} \Phi(A)\Phi(B) = 0 \quad {\rm whenever}\ AB = 0. \end{align*}$$ Φ (A) Φ (B) = 0 whenever AB = 0. In this paper, we give concrete descriptions of an additive/linear zero product preserver $ \Phi : \mathbf{M}_n(\mathbb {F}) \rightarrow \mathbf{M}_r(\mathbb {F}) $ Φ : M n (F) → M r (F) between matrix algebras of different dimensions over an arbitrary field $ \mathbb {F} $ F , and $ n\geq 2 $ n ≥ 2. In particular, we show that if Φ is linear and preserves zero products then $$\begin{align*} \Phi(A)= S\begin{pmatrix} R_1 \otimes A & 0 \\ 0 & \Phi_0(A)\end{pmatrix} S^{-1}, \end{align*}$$ Φ (A) = S ( R 1 ⊗ A 0 0 Φ 0 (A) ) S − 1 , for some invertible matrices $ R_1 $ R 1 in $ \mathbf{M}_k(\mathbb {F}) $ M k (F) , S in $ \mathbf{M}_r(\mathbb {F}) $ M r (F) and a zero product preserving linear map $ \Phi _0: \mathbf{M}_n(\mathbb {F}) \rightarrow \mathbf{M}_{r-nk}(\mathbb {F}) $ Φ 0 : M n (F) → M r − nk (F) into nilpotent matrices. If $ \Phi (I_n) $ Φ (I n) is invertible, then $ \Phi _0 $ Φ 0 is vacuous. In general, the structure of $ \Phi _0 $ Φ 0 could be quite arbitrary, especially when $ \Phi _0(\mathbf{M}_n({\mathbb F})) $ Φ 0 (M n (F)) has trivial multiplication, i.e. $ \Phi _0(X)\Phi _0(Y) = 0 $ Φ 0 (X) Φ 0 (Y) = 0 for all X, Y in $ \mathbf{M}_n(\mathbb {F}) $ M n (F). We show that if $ \Phi _0(I_n) = 0 $ Φ 0 (I n) = 0 or $ r-nk \le n+1 $ r − nk ≤ n + 1 , then $ \Phi _0(\mathbf{M}_n({\mathbb F})) $ Φ 0 (M n (F)) indeed has trivial multiplication. More generally, we characterize subspaces $ \mathbf{V} $ V of square matrices satisfying XY = 0 for any $ X, Y \in \mathbf{V} $ X , Y ∈ V . Similar results for double zero product preserving maps are obtained. [ABSTRACT FROM AUTHOR]

The Koopman operator framework holds promise for spectral analysis of nonlinear dynamical systems based on linear operators. Eigenvalues and eigenfunctions of the Koopman operator, the so-called Koopman eigenvalues and Koopman eigenfunctions, respectively, mirror global properties of the system's flow. In this paper, we perform the Koopman analysis of the singularly perturbed van der Pol system. First, we show the spectral signature depending on singular perturbation: how two Koopman principal eigenvalues are ordered and what distinct shapes emerge in their associated Koopman eigenfunctions. Second, we discuss the singular limit of the Koopman operator, which is derived through the concatenation of Koopman operators for the fast and slow subsystems. From the spectral properties of the Koopman operator for the singularly perturbed system and the singular limit, we suggest that the Koopman eigenfunctions inherit geometric properties of the singularly perturbed system. These results are applicable to general planar singularly perturbed systems with stable limit cycles. [ABSTRACT FROM AUTHOR]

This research explores the interplay between violator spaces and greedoids—two distinct theoretical frameworks developed independently. Violator spaces were introduced as a generalization of linear programming, while greedoids were designed to characterize combinatorial structures where greedy algorithms yield optimal solutions. These frameworks have, until now, existed in isolation. This paper bridges the gap by showing that greedoids can be defined using a modified violator operator. The established connections not only deepen our understanding of these theories but also provide a new characterization of antimatroids. [ABSTRACT FROM AUTHOR]

In this paper, we introduce several new types of partial fractional derivatives in the continuous setting and the discrete setting. We analyze some classes of the abstract fractional differential equations and the abstract fractional difference equations depending on several variables, providing a great number of structural results, useful remarks and illustrative examples. Concerning some specific applications, we would like to mention here our investigation of the fractional partial differential inclusions with Riemann–Liouville and Caputo derivatives. We also establish the complex characterization theorem for the multidimensional vector-valued Laplace transform and provide certain applications. [ABSTRACT FROM AUTHOR]

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its domain, null-space, and projector operator; establish the interrelations between its different realizations; and present a generalized fractional Taylor formula involving the generic fractional derivative. Then, we consider the fractional relaxation equation containing the generic fractional derivative, derive a closed-form formula for its unique solution, and study its complete monotonicity. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every bounded linear operator on a separable Hilbert space has a non-trivial invariant subspace. This answers the well-known invariant subspace problem. [ABSTRACT FROM AUTHOR]

In this paper, we employ a generalization of the Boas–Bellman inequality for inner products, as developed by Mitrinović–Pečarić–Fink, to derive several upper bounds for the 2 p -th power with p ≥ 1 of the numerical radius of the off-diagonal operator matrix 0 A B * 0 for any bounded linear operators A and B on a complex Hilbert space H. While the general matrix is not symmetric, a special case arises when B = A * , where the matrix becomes symmetric. This symmetry plays a crucial role in the derivation of our bounds, illustrating the importance of symmetric structures in operator theory. [ABSTRACT FROM AUTHOR]

This paper presents the Mann-type inertial accelerated subgradient extragradient algorithm with non-monotonic step sizes for solving the split equilibrium and fixed point problems relating to pseudomonotone and Lipschitz-type continuous bifunctions and nonexpansive mappings in the framework of real Hilbert spaces. By sufficient conditions on the control sequences of the parameters of concern, the strong convergence theorem to support the proposed algorithm, which involves neither prior knowledge of the Lipschitz constants of bifunctions nor the operator norm of the bounded linear operator, is demonstrated. Some numerical experiments are performed to show the efficacy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Piezoelectric ceramic, an indispensable micro-nano actuator, has advantages of small size, high sensitivity and strong controllability, which is widely applied in various fields like micro-nano processing, medicine, chemistry and materials. However, its nonlinear and asymmetric hysteresis features directly affect its positioning accuracy. The classical and generalized Prandtl-Ishlinskii models with linear symmetry operators are commonly used to deal with these issues, but they still face difficulties in asymmetric systems. Therefore, a piecewise modified Prandtl-Ishlinskii (PMPI) model is proposed to compensate nonlinear and asymmetric hysteresis in this paper. The new nonlinear operator and piecewise nonlinear envelope function are introduced into the PMPI model. Independent right and left parts are adopted to describe hysteresis increasing and decreasing stages, while the threshold and weight vectors correspond to them. Weight vectors are extended and a new diagonal matrix is constructed for avoiding excessive parameters and computation. Simulations and experiments are shown that the PMPI model has good compensation performance and effectiveness, which has better fitting, higher accuracy, fewer parameters and computation, and can be well applied to the precise nano-positioning systems with less nonlinear and asymmetric errors. [ABSTRACT FROM AUTHOR]

This paper focuses on filiform Lie algebras Q2m in matrix version. We give the matrix forms of automorphisms (derivations) of Q2m. We prove that linear 2-local automorphisms (derivations) are automorphisms (derivations). Using the results of derivations of Q2m, we describe biderivations on Q2m. As application, we characterize linear commuting maps and the commutative Post-Lie algebra structures on Q2m. [ABSTRACT FROM AUTHOR]

AbstractLet m, n be integers with m>n . Denote Mm,n by the set of all m×n matrices over the field F of characteristic zero. Let I be an n×m matrix with (i,i) -position 1 for any 1≤i≤n , and 0 in other position. Define a bracket [A,B]=AIB−BIA , where A,B∈Mm,n . Then Mm,n with this bracket is a Lie algebra, called non-square linear Lie algebra, denoted by gl(m×n,F) . In this paper, all derivations and biderivations of gl(m×n,F) are determined. As applications of biderivations, the linear commuting maps and commutative post-Lie algebra structures on gl(m×n,F) are given. [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]

In this paper, we study the notions of Browder operator and Browder S-spectrum of bounded right linear operator defined over the right quaternionic Hilbert space. Some properties of Browder operator and stability of the ascent, descent and Browder S-spectrum have been investigated in the right quaternionic setting. We also characterize the property of invariant Browder operators and study the spectral mapping theorem of Browder S-spectrum for self-adjoint operators in quaternionic setting. This investigation concludes by exploring the Browder S-spectrum of the sum of two bounded linear operators. [ABSTRACT FROM AUTHOR]

Two rigorous mathematical foundations for relativistic quantum field theory are local nets of von Neumann algebras and Wightman field theory. We recall that local nets of von Neumann algebras are local nets of bounded operator algebras, whereas Wightman field theory makes use of unbounded linear operators. In this paper, we introduce local nets of unbounded operator algebras and show how these can be connected with Wightman field theory. This is motivated by the fact that observables in quantum mechanics are unbounded operators in general. [ABSTRACT FROM AUTHOR]

This paper investigates the Li–Yorke chaos in linear systems with weak topology on Hilbert spaces. A weak topology induced by bounded linear functionals is first constructed. Under this weak topology, it is shown that the weak Li–Yorke chaos can be equivalently measured by an irregular or a semi-irregular vector, which are utilized to establish criteria for the weak Li–Yorke chaos of diagonalizable operators, Jordan blocks, and upper triangular operators. In particular, for a linear operator that can be decomposed into a direct sum of finite-dimensional Jordan blocks, it is Li–Yorke chaotic in weak topology if its point spectrum contains a pair of real opposite eigenvalues with absolute values not less than 1, or a pair of complex conjugate eigenvalues with moduli not less than 1. Interestingly, as a specific example of upper triangular operator, the existence of Li–Yorke chaos in weak topology can be derived for a class of linear operators expressed as the direct sum of finite-dimensional Jordan blocks and a strongly irreducible operator. [ABSTRACT FROM AUTHOR]

In this paper, we aimed to identify the dynamics of a crude distillation unit (CDU) using closed-loop data with NARX−NN and the Koopman operator in both linear (KL) and bilinear (KB) forms. A comparative analysis was conducted to assess the performance of each method under different experimental conditions, such as the gain, a delay and time constant mismatch, tight constraints, nonlinearities, and poor tuning. Although NARX−NN showed good training performance with the lowest Mean Squared Error (MSE), the KB demonstrated better generalization and robustness, outperforming the other methods. The KL observed a significant decline in performance in the presence of nonlinearities in inputs, yet it remained competitive with the KB under other circumstances. The use of the bilinear form proved to be crucial, as it offered a more accurate representation of CDU dynamics, resulting in enhanced performance. [ABSTRACT FROM AUTHOR]

This paper introduces a novel data-driven approximation method for the Koopman operator, called the RC-HAVOK algorithm. The RC-HAVOK algorithm combines Reservoir Computing (RC) and the Hankel Alternative View of Koopman (HAVOK) to reduce the size of the linear Koopman operator with a lower error rate. The accuracy and feasibility of the RC-HAVOK algorithm are assessed on Lorenz-like systems and dynamical systems with various nonlinearities, including the quadratic and cubic nonlinearities, hyperbolic tangent function, and piece-wise linear function. Implementation results reveal that the proposed model outperforms a range of other data-driven model identification algorithms, particularly when applied to commonly used Lorenz time series data. [ABSTRACT FROM AUTHOR]

Let n ∈ N 0 , ψ be an analytic self-map on D and u be an analytic function on D. The single operator D u , ψ n acting on various spaces of analytic functions has been a subject of investigation for many years. It is defined as (D u , ψ n f) (z) = u (z) f (n) (ψ (z)) , f ∈ H (D) . However, the study of the operator P u → , ψ k , which represents a finite sum of these operators with varying orders, remains incomplete. The boundedness, compactness and essential norm of the operator P u → , ψ k on the Bloch space are investigated in this paper, and several characterizations for these properties are provided. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra B n of order 2 n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ (P) of complement-closed and convex subposets P of B n. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V (P) on P. We then write down the adjacency operator A of the graph Γ (P) as the composition of two linear operators on V (P) , namely, the operator that multiplies elements of V (P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P , then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ (P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property. [ABSTRACT FROM AUTHOR]

The research of biomedical data is crucial for disease diagnosis, health management, and medicine development. However, biomedical data are usually characterized by high dimensionality and class imbalance, which increase computational cost and affect the classification performance of minority class, making accurate classification difficult. In this paper, we propose a biomedical data classification method based on feature selection and data resampling. First, use the minimal-redundancy maximal-relevance (mRMR) method to select biomedical data features, reduce the feature dimension, reduce the computational cost, and improve the generalization ability; then, a new SMOTE oversampling method (Spectral-SMOTE) is proposed, which solves the noise sensitivity problem of SMOTE by an improved spectral clustering method; finally, the marine predators algorithm is improved using piecewise linear chaotic maps and random opposition-based learning strategy to improve the algorithm's optimization seeking ability and convergence speed, and the key parameters of the spectral-SMOTE are optimized using the improved marine predators algorithm, which effectively improves the performance of the over-sampling approach. In this paper, five real biomedical datasets are selected to test and evaluate the proposed method using four classifiers, and three evaluation metrics are used to compare with seven data resampling methods. The experimental results show that the method effectively improves the classification performance of biomedical data. Statistical test results also show that the proposed PRMPA-Spectral-SMOTE method outperforms other data resampling methods. [ABSTRACT FROM AUTHOR]

The present paper continues investigation of narrow operators on complex vector lattices started in a recent paper by Dzhusoeva, Huang, Pliev and Sukochev. We answer their questions and strengthen one of their main results by proving that the extension T_{\Bbb {C}}\colon E_{\Bbb {C}}\to F_{\Bbb {C}} of a linear operator T \colon E \to F acting from a Dedekind complete real Riesz space E to a real Banach lattice F to their complexifications is narrow if and only if T is. [ABSTRACT FROM AUTHOR]

The construction of the fourth-order 2-point Fractional Block Backward Differentiation Formula (2FBBDF(4)) to solve the fractional order differential equations (FDEs) is presented in this paper. The method is developed using the fractional linear multistep method (FLMM) linked with the linear difference operator. This paper aims to approximate the fractional order problems via 2FBBDF(4), which is normally used to solve ordinary differential equations. The criteria for the stability of the method are analyzed in order to solve FDE problems. Consequently, the method is determined to be A-stable for different values of α within the interval (0, 1). Then, Newton's iteration is implemented in this method to solve the problems. Multiple numerical examples of linear, nonlinear, and system FDEs are provided for the scenario where the order α lies within the range of 0 and 1. Ultimately, the numerical results confirm that the proposed method performs at a similar level to the existing methods. [ABSTRACT FROM AUTHOR]

In this article, the symmetric operator L corresponding to the boundary value problem is represented as the difference of two commuting operators A and  B.  The uniqueness of the solution is guaranteed if the spectra of the operators A and B do not intersect and the domain of the operator B is given by non-degenerate boundary conditions. In contrast to the existing papers, the criterion for the uniqueness of the boundary value problem formulated in this paper is satisfied even when the system of root functions of the operator B does not form a basis in the corresponding space. At the same time, only the closedness of the linear operator A is required. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a general maximum principle for discrete fractional stochastic difference system of mean‐field type. The admissible control domain is nonconvex. We give Malliavin calculus for discrete‐time case to deal with the fractional terms. The maximum principle of general type is derived by classical variation and linear operator methods. In addition, a linear‐quadratic problem is solved to illustrate the main result and we also figure out a numerical result in this case. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic p ≥ 0. Assume F , F (1) , F (2) , and (F (1)) p are the free group on n > 1 generators, the first and second derived subgroups of F , and the subgroup of F (1) generated by all p -powers, respectively. In this paper, among other results, we show that for any word w ∈ F (1) ∖ F (2) (F (1)) p , the corresponding word map w : ∏ n PSL 2 (K) → PSL 2 (K) is surjective. We show that the class of such words w contains, for instance, all (weakly) Engel words. [ABSTRACT FROM AUTHOR]

This paper introduces a new idea to obtain a better order of approximation for the Baskakov operator. We conclude two new operators from orders one and two of the Baskakov type. Also, we prove some directed results concerning the rate of convergence of these operators. We apply the Maple software to demonstrate how the operators converge to a specific function. [ABSTRACT FROM AUTHOR]

This paper presents the nonlinear maximum product type of univariate and bivariate Bernstein--Stancu operators and uses new definitions to investigate the approximation properties. The order of approximation obtained with the nonlinear maximum product type of operator sequences would be better than the degree of approximation of the known linear operator sequences. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space and let T represent a bounded linear operator on H. In this paper we introduce, a new class of non normal operators, the (M, k)-quasi paranormal operator. An operator T is said to be a (M,k)-quasi paranormal operator, for a non negative integer k and a real positive number M if it satisfies ||Tk+1x||2 < M||Tk+2x|| ||Tkx||, for all x ∈ H. This new class of operators is generalization of some of the non normal operators, such as, the k-quasi paranormal and M-paranormal operators. We prove the basic properties, the structural and spectral properties and also the matrix representation of this new class of operators. [ABSTRACT FROM AUTHOR]

This paper considers a system with one robot and n safety units (one of which works while the others remain on standby), which is described by an integro-deferential equation. The system can fail in the following three ways: fails with an incident, fails safely and fails due to the malfunction of the robot. Using the C 0 – semigroups theory of linear operators, we first show that the system has a unique non-negative, time-dependent solution. Then, we obtain the exponential convergence of the time-dependent solution to its steady-state solution. In addition, we study the asymptotic behavior of some time-dependent reliability indices and present a numerical example demonstrating the effects of different parameters on the system. [ABSTRACT FROM AUTHOR]

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

In this paper, we consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with a linear mapping. Besides, we are interested in its application to the graph Fourier basis problem. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many favorable properties. Thanks to this and the scheme for updating the smoothing parameter, we show that any accumulation point of the solution sequence generated by the proposed algorithms is a stationary point of the original optimization problem. Numerical experiments on building graph Fourier basis are conducted to demonstrate the efficiency of the proposed algorithms. [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]