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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]

A time-dependent vibrational electronic coupled-cluster (VECC) approach is proposed to simulate photo-electron/UV-VIS absorption spectra as well as time-dependent properties for non-adiabatic vibronic models, going beyond the Born–Oppenheimer approximation. A detailed derivation of the equations of motion and a motivation for the ansatz are presented. The VECC method employs second-quantized bosonic construction operators and a mixed linear and exponential ansatz to form a compact representation of the time-dependent wave-function. Importantly, the method does not require a basis set, has only a few user-defined inputs, and has a classical (polynomial) scaling with respect to the number of degrees of freedom (of the vibronic model), resulting in a favorable computational cost. In benchmark applications to small models and molecules, the VECC method provides accurate results compared to multi-configurational time-dependent Hartree calculations when predicting short-time dynamical properties (i.e., photo-electron/UV–VIS absorption spectra) for non-adiabatic vibronic models. To illustrate the capabilities, the VECC method is also successfully applied to a large vibronic model for hexahelicene with 14 electronic states and 63 normal modes, developed in the group by Aranda and Santoro [J. Chem. Theory Comput. 17, 1691, (2021)]. [ABSTRACT FROM AUTHOR]

We continue the study of level numbers and level vectors of a bounded linear operator between normed linear spaces. We prove that every linear operator on a finite-dimensional polyhedral Banach space possesses only finitely many level numbers. We also explore the level numbers of weighted permutation operators between $ \ell _p $ ℓ p spaces. Furthermore, we introduce the concept of approximate level numbers and study its properties and relation with level numbers. In particular, our results indicate that the concept of level numbers can be potentially applied to the spectral theory of operators, in the setting of normed linear spaces. [ABSTRACT FROM AUTHOR]

Numerical range is the subject of study for over a century. Its extension to matricial range is defined in terms of completely positive maps. A linear map ϕ between matrix algebras is completely positive if and only if its Choi matrix $ C(\phi) $ C (ϕ) is positive semidefinite. In this paper, we extend the study to some joint matricial ranges defined by those ϕ's where $ C(\phi) $ C (ϕ) is Hermitian with specified spectrum. We prove some convexity theorems which extend previous results on generalized joint numerical ranges and matricial ranges. We also extend Bohnenblust's result on joint positive definiteness of Hermitian matrices and Friedland and Loewy's result on the existence of a nonzero matrix with multiple first eigenvalue in subspaces of Hermitian matrices. [ABSTRACT FROM AUTHOR]

In this article, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method, we find explict periodic waves and we also present a characterization for all positive and periodic solutions for the model using the monotonicity of the period map in terms of the energy levels. The monotonicity of the period map is also useful to obtain the quantity and multiplicity of non-positive eigenvalues for the associated linearized operator and to do so, we use tools of the Floquet theory. Finally, we prove the spectral stability by analyzing the difference between the number of negative eigenvalues of a convenient linear operator restricted to the space constituted by zero-mean periodic functions and the number of negative eigenvalues of the matrix formed by the tangent space associated to the low order conserved quantities of the evolution model. [ABSTRACT FROM AUTHOR]

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]

We introduce and study the crossing map, a closed linear map acting on operators on the tensor square of a given Hilbert space that is inspired by the crossing property of quantum field theory. This map turns out to be closely connected to Tomita–Takesaki modular theory. In particular, crossing symmetric operators, namely those operators that are mapped to their adjoints by the crossing map, define endomorphisms of standard subspaces. Conversely, such endomorphisms can be integrated to crossing symmetric operators. We also investigate the relation between crossing symmetry and natural compatibility conditions with respect to unitary representations of certain symmetry groups, and furthermore introduce a generalized crossing map defined by a real object in an abstract C∗-tensor category, not necessarily consisting of Hilbert spaces and linear maps. This latter crossing map turns out to be closely related to the (unshaded, finite-index) subfactor theoretical Fourier transform. Lastly, we provide families of solutions of the crossing symmetry equation, solving in addition the categorical Yang–Baxter equation, associated with an arbitrary Q-system. [ABSTRACT FROM AUTHOR]

This article establishes the boundary Hölder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions n ≤ 9 n\leq 9 , for C 1 , 1 C^{1\smash{,}1} domains. We consider equations − L ⁢ u = f ⁢ ( u ) -Lu=f(u) in a bounded C 1 , 1 C^{1\smash{,}1} domain Ω ⊂ R n \Omega\subset\mathbb{R}^{n} , with u = 0 u=0 on ∂ Ω \partial\Omega , where 퐿 is a linear elliptic operator with variable coefficients and f ∈ C 1 f\in C^{1} is nonnegative, nondecreasing, and convex. The stability of 푢 amounts to the nonnegativity of the principal eigenvalue of the linearized equation − L − f ′ ⁢ ( u ) -L-f^{\prime}(u) . Our result is new even for the Laplacian, for which [X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math. 224 (2020), 2, 187–252] proved the Hölder continuity in C 3 C^{3} domains. [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]

A bounded linear operator T on a Banach space X is called a convex-cyclic operator if there exists a vector $ x \in X $ x ∈ X such that the convex hull of $ Orb(T, x) $ O r b (T , x) is dense in X. In this paper, for given an aperiodic element g in a locally compact group G, we give some sufficient conditions for a weighted translation operator $ T_{g,w}: f \mapsto w\cdot f*\delta _g $ T g , w : f ↦ w ⋅ f ∗ δ g on $ \mathfrak {L}^{p}(G) $ L p (G) to be convex-cyclic. A necessary condition is also studied. At the end, to explain the obtained results, some examples are given. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give expressions for modular linear differential operators (MLDOs) of any order. In particular, we show that they can all be described in terms of Rankin-Cohen brackets and a modified Rankin-Cohen bracket found by Kaneko and Koike. We also give more uniform descriptions of MLDOs in terms of canonically defined higher Serre derivatives and an extension of Rankin-Cohen brackets, as well as in terms of quasimodular forms and almost holomorphic modular forms. The last of these descriptions involves the holomorphic projection map. The paper also includes some general results on the theory of quasimodular forms on both cocompact and non-cocompact subgroups of SL_2(\mathbb {R}), as well as a slight sharpening of a theorem of Martin and Royer on Rankin-Cohen brackets of quasimodular forms. [ABSTRACT FROM AUTHOR]

Multiplication by a given modular form can be viewed as a linear map on the space of modular forms. By computing its adjoint operator, one can obtain certain cusp forms whose Fourier coefficients are special values of Dirichlet series of Rankin-Selberg type associated to modular forms. We generalize this idea to the space of almost holomorphic modular forms with some cuspidal conditions. We prove that the generating function of special values of the Dirichlet series at certain points is a quasi-modular form. [ABSTRACT FROM AUTHOR]

A problem of completing a linear map on C^*-algebras to a completely positive map is analyzed. It is shown that whenever such a completion is feasible there exists a unique minimal completion. This theorem is used to show that under some very general conditions a completely positive map almost everywhere equivalent to a quasi-pure map is actually equal to that map. [ABSTRACT FROM AUTHOR]

We consider Kantorovich modifications of linking operators and Stancu modifications of the classical Bernstein operators. For the modified operators we determine the limits of iterates and the invariant measures. In order to find the limits we have to solve systems of linear equations and to this end we use a suitable iterative algorithm. [ABSTRACT FROM AUTHOR]

In the present work, the authors introduce a new linear differential operator associated with $ q $-calculus which uses the $ q $-derivative operator. For the new class involving the linear differential operator, interesting properties like coefficient inequality, growth, distortion, convolution and radii of starlikeness are established. Various connections of the present class with that of the earlier works are pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new sequence of Riemann-Liouville type fractional Bernstein-Kantorovich operators $ K_{n}^{\alpha}(f;x) $ depending on a parameter $ \alpha $. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first and second order modulus of continuity of these operators. Moreover, we introduce a new operator that preserves affine functions from Riemann-Liouville type fractional Bernstein-Kantorovich operators. Further, we define the bivariate case of Riemann-Liouville type fractional Bernstein-Kantorovich operators and investigate the order of convergence. Some numerical results are given to illustrate the convergence of these operators and its comparison with the classical case of these operators. [ABSTRACT FROM AUTHOR]

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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]

In distribution networks with distributed generators (DGs), power generation and load demand exhibit increased randomness and volatility, and the line parameters also suffer more frequent fluctuations, which may result in significant state shifts. Existing model-driven methods face challenges in efficiently solving uncertain power flow, especially as the size of the system increases, making it difficult to meet the demand for rapid power flow analysis. To address these issues, this paper proposes an SVR-based interval power flow (IPF) prediction method for distribution networks with DGs integration. The method utilizes intervals to describe system uncertainty and employs Support Vector Regression (SVR) for model training. The input feature vector consists of the intervals of active power generation, load demand, and line parameters, while the output feature vector represents the intervals of voltage or line transmission power. Ultimately, the SVR-based IPF prediction model is established, capturing the linear mapping relationship between input data and output IPF variables. Simulation results demonstrate that the proposed method exhibits high prediction accuracy, strong adaptability, and optimal computation efficiency, meeting the requirements for rapid and real-time power flow analysis while considering the uncertainty in distribution networks with DGs integration. [ABSTRACT FROM AUTHOR]

Differential geometry of a kind of helical hypersurface family that depends on six parameters within the seven‐dimensional Euclidean space is explored. The curvatures of these hypersurfaces are determined, and their minimality is examined, along with illustrative examples being provided. Lastly, the Laplace–Beltrami operator for that kind of hypersurfaces is calculated. [ABSTRACT FROM AUTHOR]

The concept of an involution or anti‐involution is a self‐inverse linear mapping that plays a prominent role in the theory of algebraic structures, particularly rings, hyperrings, ordered semigroups, and ordered semihypergroups. Nowadays, the study of involutions in ordered hyperstructures is a particular area of research in the field of hyperstructure theory, especially in ordered hyperstructures. In this research, we apply the concept of fuzzy sets to anti‐involution hyperideals of ordered anti‐involution LA‐semihypergroups. We shall propose the mathematical formulation of fuzzy anti‐involution hyperideals in ordered anti‐involution LA‐semihypergroups and provide some examples. This concept is a novel extension of fuzzy involution ordered semigroups. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we construct a nonlinear regularized Hermitian and skew-Hermitian splitting (NRHSS) iteration method for a class of nonlinear saddle-point systems. The NRHSS iteration method can be seen as a generalization of the Peaceman-Rachford splitting method with monotone plus skew-symmetric splitting (PRSM/MSS). Besides, if the nonlinear operator reduces to the linear case, the NRHSS iteration method reduces to the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration method. Hence, the NRHSS iteration method can also be seen as a generalization of the RHSS iteration method. To overcome the heavy workload when direct solution methods are used, we also propose an inexact version of the NRHSS iteration method. Detailed theoretical analysis regarding the convergence properties is given. Numerical experiments on image deblurring problems are carried out to illustrate the feasibility and efficiency of the NRHSS iteration method. [ABSTRACT FROM AUTHOR]

In this paper, the basic reproduction number (BRN) is explored in terms of a degenerate reaction–diffusion host–pathogen model. The expressions of BRN R0$$ {R}_0 $$ and local basic reproduction number (LBRN) R¯0(x)$$ {\overline{R}}_0(x) $$ for the model are calculated. Some basic properties of R0$$ {R}_0 $$ with respect to the general diffusion rates and the asymptotic profiles of R0$$ {R}_0 $$ with respect to large diffusion rates and small diffusion rates are established by introducing two compact operators H1$$ {H}_1 $$ and H2$$ {H}_2 $$. Our results reveal that R0$$ {R}_0 $$ is the spectral radius of a sum of the product of additive operators and strongly positive compact linear operators with spectral radii one. [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]

An improved implicit direct-forcing immersed boundary method (DF-IBM) is presented for simulating incompressible flows around complex rigid structures undergoing arbitrary motion. The current approach harnesses the pressure implicit with splitting of operators algorithm to handle the fluid–solid system's dual constraints in a segregated manner. As a result, the divergence-free condition is preserved throughout the Eulerian domain, and the no-slip velocity boundary condition is exactly enforced on the immersed boundary. A new pressure Poisson equation (PPE) is derived, incorporating the boundary force where the no-slip condition is already met, enabling the use of fast iterative PPE solvers without modifications. The improvement involves integrating Lagrangian weight methods having better reciprocity over the IBM-related linear operators with the implicit formulation. An additional force initialization scheme is introduced to further boost the algorithm's performance. The method's accuracy, efficiency, and capability are verified through various stationary and moving immersed boundary cases. The results are validated against experimental and numerical data from the literature. The proposed improvements seamlessly integrate into existing incompressible fluid solvers with minimal adjustments to the original system equations, highlighting their ease of implementation. [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]

To address the problem of dynamic degradation over a finite-precision platform of chaotic maps and the reversibility of linear chaotic maps, we propose an improved model over GF( 2 n ) that is called the nondegenerate m-Dimensional (m ≥ 2) Integer-Domain Chaotic Maps (mD-IDCMs). This model incorporates modular exponentiation operation, and is capable of constructing nondegenerate IDCMs of any dimension. Moreover, we prove the irreversibility of mD-IDCM and analyze its chaotic behaviors in terms of positive Lyapunov Exponents (LEs). The results of theoretical analysis show that the proposed mD-IDCM model can obtain the desired positive LEs by appropriately configuring its coefficient matrix. Then, we present two instances, and analyze their LEs, Kolmogorov entropy, Sample entropy, Correlation dimension, and the dynamic analysis indicates that the chaotic map constructed by mD-IDCM has ergodicity within a sufficiently large chaotic range. Finally, we design a Pseudo-Random Number Generator (PRNG) with a key to verify the practicability of the mD-IDCM. [ABSTRACT FROM AUTHOR]

In order to improve the walking stability of a biped robot in multiple scenarios and reduce the complexity of the Central Pattern Generator (CPG) model, a new CPG walking controller based on multivariate linear mapping was proposed. At first, in order to establish a dynamics model, the lower limb mechanical structure of the biped robot was designed. According to the Lagrange and angular momentum conservation method, the hybrid dynamic model of the biped robot was established. The initial value of the robot's passive walking was found by means of Poincaré mapping and cell mapping methods. Then, a multivariate linear mapping model was established to form a new lightweight CPG model based on a Hopf oscillator. According to the parameter distribution of the new CPG model, a preliminary parameter-tuning idea was proposed. At last, the joint simulation of MATLAB and V-REP shows that the biped robot based on the new CPG control has a stable periodic gait in flat and uphill scenes. The proposed method could improve the stability and versatility of bipedal walking in various environments and can provide general CPG generation and a tuning method reference for robotics scholars. [ABSTRACT FROM AUTHOR]

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring R = F p + w 1 F p + w 2 F p + w 2 2 F p + w 1 w 2 F p + w 1 w 2 2 F p , where w 1 2 − α 2 = 0 , w 1 w 2 = w 2 w 1 , w 2 3 − β 2 w 2 = 0 , and α , β ∈ F p ∖ { 0 } , for a prime p. We begin by introducing a Gray map defined over R , which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over R using Calderbank–Shore–Steane (CSS) construction. [ABSTRACT FROM AUTHOR]

This work examines subordination conclusions for a specific subclass of p-valent meromorphic functions on the punctured unit disc of the complex plane where the function has a pole of order p. A new linear operator is used to define the subclass that is being studied. Furthermore, we present several corollaries with intriguing specific situations of the 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]

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions. [ABSTRACT FROM AUTHOR]

Let H be a separable complex Hilbert space, B (H) the algebra of bounded linear operators on H , μ a finite Borel measure on R + with the finite (n + 1) -th moment, f (z) : = ∫ R + e − t z d μ (t) for all ℜ z ⩾ 0 , C Ψ (H) , and | | · | | Ψ the ideal of compact operators and the norm associated to a symmetrically norming function Ψ , respectively. If A , D ∈ B (H) are accretive, then the Laplace transformer on B (H) , X ↦ ∫ R + e − t A X e − t D d μ (t) is well defined for any X ∈ B (H) as is the newly introduced Taylor remainder transformer R n (f ; D , A) X : = f (A) X − ∑ k = 0 n 1 k ! ∑ i = 0 k (− 1) i k i A k − i X D i f (k) (D). If A , D * are also (n + 1) -accretive, ∑ k = 0 n + 1 (− 1) k n + 1 k A n + 1 − k X D k ∈ C Ψ (H) and | | · | | Ψ is Q* norm, then | | · | | Ψ norm estimates for ∑ k = 0 n + 1 n + 1 k A k A n + 1 − k 1 2 R n (f ; D , A) X ∑ k = 0 n + 1 n + 1 k D n + 1 − k D * k 1 2 are obtained as the spacial cases of the presented estimates for (also newly introduced) Taylor remainder transformers related to a pair of Laplace transformers, defined by a subclass of accretive operators. [ABSTRACT FROM AUTHOR]

In this study, the dynamics of a reaction–diffusion equation on a bounded interval at the first dynamic transition point is investigated. The main difference of this study from the existing ones in the dynamic transition literature is that in this work, no specific form of the nonlinear operator is assumed except that it is an analytic function of u$u$ and ux$u_x$. The linear operator is also assumed to be a general operator with sinusoidal eigenvectors. Moreover, we assume that there is a first transition as a real simple eigenvalue changes sign. With this general framework, we make a rigorous analysis of the dependence of the first transition dynamics on the coefficients of the Taylor expansion of the nonlinear operator. The main tool we use in this study is the center manifold reduction combined with the classification of the dynamic transition theory. This study is a generalization of a recent work where the nonlinear operator contains only low‐order (quadratic and cubic) nonlinearities. The current generalization requires certain technical difficulties such as the validity of the genericity conditions on the Taylor coefficients of the nonlinear operator and a bootstrapping argument using these genericity conditions. The results of this work can be generalized in various directions, which are discussed in the conclusions section. [ABSTRACT FROM AUTHOR]

We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the Hausdorff-, Cesàro-, integration operator, and their adjoints, as well as some combinations of those. For the composition of the Hausdorff- and the Cesàro-operator, we give estimates of the decay of the corresponding singular values. As a curiosity, this provides also an example of two practically relevant non-compact operators, for which their composition is compact. Furthermore, we characterize those operators for which a composition with a non-compact operator gives a compact one. [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]

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]

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]

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 dynamics of the piecewise planar rotations $ F_{\lambda }(z)=\lambda (z-H(z)), $ F λ (z) = λ (z − H (z)) , with $ z\in {\mathbb C} $ z ∈ C , $ H(z)=1 $ H (z) = 1 if $ \mathrm {Im}(z)\ge 0, $ Im (z) ≥ 0 , $ H(z)=-1 $ H (z) = − 1 if $ \mathrm {Im}(z) \lt 0, $ Im (z) < 0 , and $ \lambda =\mathrm {e}^{i \alpha } \in {\mathbb C} $ λ = e iα ∈ C , being α a rational multiple of π. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set formed by the preimages of the discontinuity line. We prove that any connected component of this set is open, bounded and periodic under the action of $ F_\lambda $ F λ , with a period ℓ, that depends on the connected component. Furthermore, $ F_\lambda ^\ell $ F λ ℓ restricted to each component acts as a rotation with a period which also depends on the connected component. As a consequence, any point in the regular set is periodic. Among other results, we also prove that for any connected component of the regular set, its boundary is a convex polygon with certain maximum number of sides. [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]

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]

For a bounded analytic function $\varphi $ on the unit disk $\mathbb {D}$ with $\|\varphi \|_\infty \le 1$ , we consider the defect operators $D_\varphi $ and $D_{\overline \varphi }$ of the Toeplitz operators $T_{\overline \varphi }$ and $T_\varphi $ , respectively, on the weighted Bergman space $A^2_\alpha $. The ranges of $D_\varphi $ and $D_{\overline \varphi }$ , written as $H(\varphi)$ and $H(\overline \varphi)$ and equipped with appropriate inner products, are called sub-Bergman spaces. We prove the following three results in the paper: for $-1 , the space $H(\varphi)$ has a complete Nevanlinna–Pick kernel if and only if $\varphi $ is a Möbius map; for $\alpha>-1$ , we have $H(\varphi)=H(\overline \varphi)=A^2_{\alpha -1}$ if and only if the defect operators $D_\varphi $ and $D_{\overline \varphi }$ are compact; and for $\alpha>-1$ , we have $D^2_\varphi (A^2_\alpha)= D^2_{\overline \varphi }(A^2_\alpha)=A^2_{\alpha -2}$ if and only if $\varphi $ is a finite Blaschke product. In some sense, our restrictions on $\alpha $ here are best possible. [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]

Let G be a generalized matrix algebra. We show that under certain conditions, each generalized Lie n -derivation associated with a linear map on G is a sum of a generalized derivation and a central map vanishing on all (n − 1) -th commutators and is also a sum of a generalized inner derivation and a Lie n -derivation. As an application, generalized Lie n -derivations on von Neumann algebras are characterized. [ABSTRACT FROM AUTHOR]

The main result is that the only non‐trivial closed ideal in the Banach algebra L(Lp)$L(L^p)$ of bounded linear operators on Lp(0,1)$L^p(0,1)$, 1⩽p<∞$1\leqslant p < \infty$, that has a left approximate identity is the ideal of compact operators. The algebra L(L1)$L(L^1)$ has at least one non‐trivial closed ideal that has a contractive right approximate identity as well as many, including the unique maximal ideal, that do not have a right approximate identity. [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]