Early Improvement Predicts Treatment Response in Depression: An Ecological Momentary Assessment Study in an Inpatient and Day Clinic Setting.

A decision-making framework with Complex Linear Diophantine Fuzzy Aczel Alsina aggregation operator for real-world problem

Kernels for composition of positive linear operators

While over a hundred articles discuss second-order differential inequalities and subordinations in the complex plane, very few address the relatively unexplored classes of third-order fuzzy differential subordination and superordination. This paper builds upon the recently proposed concepts of third-order fuzzy differential subordination and superordination, which are developed using a linear operator and a meromorphic function. By applying techniques based on the fundamental notion of admissible functions, we begin by defining the appropriate class of such functions necessary for deriving new results in third-order fuzzy differential subordination. The study reveals the establishment of sandwich-type theorems, linking these new findings with established methods in third-order fuzzy differentiation and superordination theory.

The main goal of this paper is to provide a unified analysis of de nominals in French and noncanonical genitive nominals in Russian. Unlike previous literature that relies on partitivity, quantification and case assignment, we argue that a unified formal account can be substantiated by guaranteeing a uniform syntactic analysis of weak indefinite expressions not based on quantification, a semantic dependency of de nominals and non-canonical genitives to non-veridicality, and different linearization operations at Spell-out that guarantee concatenation of words or subwords (French de*n, Russian n⊕gen).

This paper introduces a novel and versatile framework for numerical radius inequalities within complex Hilbert spaces, building upon the generalized real and imaginary parts of an operator defined by Kittaneh and Stojiljković [Kittaneh F, Stojiljković V. New generalized numerical radius inequalities for Hilbert space operators. J Inequal Appl. 2026: 25. doi:10.1186/s13660-026-03438-3]. We define a new generalized numerical radius, w h , g Re ( A ) , and show its properties as a norm on the C ∗ -algebra of bounded linear operators, B ( H ) , under specified conditions. The proposed framework encompasses existing definitions and generalizations, yielding new identities and refined bounds for w h , g Re ( A ) . The adaptability of w h , g Re ( A ) through the functions h and g allows it to reduce to well-known inequalities already established in the literature, including those by Sheikhhosseini et al. [Sheikhhosseini A, Khosravi M, Sababheh M. The weighted numerical radius. Ann Funct Anal. 2022;13:3. doi:10.1007/s43034-021-00148-3] and Kittaneh [Kittaneh F. Numerical radius inequalities for Hilbert space operators. Studia Math. 2005;168:73–80. doi: 10.4064/sm168-1-5]. The work further explores various inequalities, including those involving powers of operators and operator matrices, providing extensions and refinements to previous results in the field.

Abstract In this article, a generalized Schur complement defined by the Moore-Penrose inverse and *congruence to a Hermitian block matrix are studied for a generalization of Haynsworth’s theorem. As an application, an alternative proof of Albert’s characterization of positivity is obtained [A. Albert, SIAM J. Appl. Math. (1969)]. Some of our results of matrices are extended to bounded linear operators.

Abstract We consider a class of bounded linear operators between Banach spaces, which we call operators with the Kato property, that includes the family of strictly singular operators between those spaces. We show that if $$T:E\rightarrow F$$ T : E → F is a dense-range operator with that property and E has a separable quotient, then for each proper dense operator range $$R\subset E$$ R ⊂ E there exists a closed subspace $$X\subset E$$ X ⊂ E such that E / X is separable, T ( X ) is dense in F and $$R+X$$ R + X is infinite-codimensional. If $$E^*$$ E ∗ is weak*-separable, the subspace X can be built so that, in addition to the former properties, $$R\cap X = \{0\}$$ R ∩ X = { 0 } . Some applications to the geometry of Banach spaces are given. In particular, we provide the next extensions of well-known results of Johnson and Plichko: if X and Y are quasicomplemented but not complemented subspaces of a Banach space E and X has a separable quotient, then X contains a closed subspace $$X_1$$ X 1 such that $$\dim (X/X_1)= \infty $$ dim ( X / X 1 ) = ∞ and $$X_1$$ X 1 is a quasicomplement of Y , and if $$T:E\rightarrow F$$ T : E → F is an operator with non-closed range and E has a separable quotient, then there exists a weak*-closed subspace $$Z\subset E^*$$ Z ⊂ E ∗ such that $$T^*(F^*)\cap Z = \{0\}$$ T ∗ ( F ∗ ) ∩ Z = { 0 } . Some refinements of these results, in the case that $$E^*$$ E ∗ is weak*-separable, are also given. Finally, we show that if E is a Banach space with a separable quotient, then $$E^*$$ E ∗ is weak*-separable if, and only if, for every closed subspace $$X\subset E$$ X ⊂ E and every proper dense operator range $$R\subset E$$ R ⊂ E containing X there exists a quasicomplement Y of X in E such that $$Y\cap R = \{0\}$$ Y ∩ R = { 0 } .

Let ( u ( n ) ) n ∈ N be an arithmetic progression of natural integers in base b ∈ N ∖ { 0 , 1 } . We consider the following sequences: s ( n ) = u ( 0 ) u ( 1 ) ⋯ u ( n ) ¯ b formed by concatenating the first n + 1 terms of ( u ( n ) ) n ∈ N in base b from the right; s g ( n ) = u ( n ) u ( n − 1 ) ⋯ u ( 0 ) ¯ b ; and ( s ∗ ( n ) ) n ∈ N , given by s ∗ ( 0 ) = u ( 0 ) , s ∗ ( n ) = s ( n ) s g ( n − 1 ) ¯ b , n ≥ 1 . We construct explicit formulae for these sequences and use basic concepts of linear difference operators to prove they are not P-recursive (holonomic). We also present an alternative proof that follows directly from their definitions. We implemented ( s ( n ) ) n ∈ N and ( s g ( n ) ) n ∈ N in the decimal base when ( u ( n ) ) n ∈ N = N ∖ { 0 } .

Let B ( H ) be the algebra of all bounded linear operators acting on a complex Hilbert space H . The polar decomposition theorem asserts that every operator T ∈ B ( H ) can be uniquely written as T = V T | T | , the product of a partial isometry V T ∈ B ( H ) that has the same kernel as that of T and the modulus | T | := ( T ∗ T ) 1 / 2 of T. In this paper, we obtain the form of all bijective linear maps Φ on B ( H ) for which V Φ ( T ) and V Φ ( S ) are unitary similar whenever T , S ∈ B ( H ) are two operators unitary similar. We also obtain the form of all bijective linear maps Φ on B ( H ) for which Φ ( V T ) = V Φ ( T ) for all T ∈ B ( H ) . Furthermore, a number of related results and consequences is obtained.

We study heat and wave type equations on a separable Hilbert space H by considering non-local operators in time with any positive densely defined linear operator with discrete spectrum. We show the explicit representation of the solution and analyse the time-decay rate in a scale of suitable Sobolev space. We perform similar analysis on multi-term heat and multi-wave type equations. The main tool here is the Fourier analysis which can be developed in a separable Hilbert space based on the linear operator involved. As an application, the same Cauchy problems are considered and analysed in the setting of a graded Lie group. In this case our analysis relies on the group Fourier analysis. An extra ingredient in this framework allows, in the case of heat type equations, to establish L p - L q estimates for 1 ⩽ p ⩽ 2 ⩽ q < + ∞ for the solutions on graded Lie group groups. Examples and applications of the developed theory are given, either in terms of self-adjoint operators on compact or non-compact manifolds, or in the case of particular settings of graded Lie groups. The results of this paper significantly extend in different directions the results of Part I [de Moraes WAA, Restrepo JE, Ruzhansky M. Heat and wave type equations with non-local operators, I Compact Lie groups. Int Math Res Not IMRN. 2024;2:1299–1328. doi: 10.1093/imrn/rnad017], where operators on compact Lie groups were considered. We note that the results obtained in this paper are also new already in the Euclidean setting of R n .

Abstract Let and be the algebra of all bounded linear operators on a complex Hilbert space and the Jordan algebra of all self‐adjoint operators in , respectively. In this paper, we give characterizations of rank one operators by the pseudospectrum on ‐Lie product of bounded linear operators and discuss some properties about the pseudospectrum. As applications, we obtain the structures of all surjective maps preserving the pseudospectrum of ‐Lie product on and , respectively.

&lt;p&gt;A mapping of the set of undirected simple (loopless) graphs to itself is a linear operator if it maps the edgeless graph to the edgeless graph and maps the union of graphs to the union of their images. A linear operator preserves a set if it maps that set to itself. We study linear operators that map sets defined by the restriction of their chromatic number. For example the set of all graphs whose chromatic number is at least &lt;span class="math inline"&gt;\(k\)&lt;/span&gt; for some fixed &lt;span class="math inline"&gt;\(3\leq k\leq n\)&lt;/span&gt;. We show these linear operators must be vertex permutations.&lt;/p&gt;

The rapid evolution of machine learning model training has outpaced the development of corresponding measurement and testing tools, leading to two significant challenges. Firstly, developers of deep learning frameworks struggle to identify operators that fail to meet precision criteria, as these issues may only manifest in a few data points. Secondly, model trainers lack effective methods to estimate precision loss caused by operators during training. To address these issues, we introduce a Pythonic framework inspired by common network layer definitions in deep learning. Our framework includes two new layers: the Fuzz Layer and the Check Layer, designed to aid in measurement and testing. The Fuzz Layer introduces minor perturbations to tensor inputs for any deterministic layer under testing (LUT). The Check Layer then measures precision by analyzing the differences before and after the perturbation. This approach estimates a lower bound of the maximal relative error and alerts developers or trainers of potential bugs if the difference exceeds a pre-defined tolerance. Additionally, Check Layers can be used independently to conduct precision tests for specific layers, ensuring the precision of operators during runtime. Despite the additional memory and time requirements, this runtime testing ensures proper training of the original model. We demonstrate the utility of our framework, FuMi, through two experiments. First, we tested 21 torch operators across nine popular machine learning models using PyTorch for various tasks, finding that the conv2d and linear operators often fail to meet precision requirements. Second, to showcase the generalizability of our framework, we tested the ATTENTION operator. By comparing different implementations of state-of-the-art ATTENTION operators, we found that the maximum relative error of the ATTENTION operator is not less than 1%, which is 13 times larger than that measured by Predoo (a unit test tool). This framework provides a robust solution for identifying precision issues in deep learning models, ensuring reliable and accurate model training.

Multi-composite activated neural network operators can be understood as positive linear operators, allowing them to be analyzed using standard, established theory. Formed by composing multiple general activation functions, these operators act upon continuous real-valued functions defined on a compact interval. This work presents a quantitative analysis of how quickly these operators converge to the unit operator. Utilizing general inequalities based on the modulus of continuity—applicable to either the function itself or its derivative—this study establishes both uniform and Lp approximation results. Furthermore, the analysis incorporates the convexity of functions to produce related, specific results.

The present work deals with the existence and periodicity of solutions to non-autonomous partial differential equations of retarded, infinite, and neutral type in the framework of fading memory space, which is defined axiomatically. In our strategy, we rely on Sadovskii's fixed-point theorem. On the other hand, the family of linear operators $A(\cdot)$ is assumed to be non-densely defined and verified by the Acquistapace-Terreni conditions. Finally, we propose an application to illustrate our results.

The present paper deals with the stability analysis for the geodesic flow of a step-two nilpotent Lie group equipped with a left-invariant pseudo-Riemannian metric. The Lie-Poisson equation is formulated using the so-called j-mapping, a linear operator associated to the step-two nilpotent Lie algebras equipped with the induced scalar product. The stability of equilibrium points for the Hamilton equation is determined in terms of their Williamson types.

Modeling forced nonlinear multivariable dynamical systems remains challenging, particularly when first-principles models are unavailable or strong nonlinear couplings are present. In recent years, data-driven approaches grounded in the Koopman operator theory have gained attention for their ability to represent nonlinear dynamics via linear evolution in appropriately lifted spaces. This work presents a data-driven modeling framework for forced nonlinear multiple-input multiple-output (MIMO) systems based on Hankel Dynamic Mode Decomposition with control and lifting functions (HDMDc+Lift). The proposed methodology exploits Hankel matrices to encode temporal correlations and employs lifting functions to approximate the Koopman operator’s action on observable functions. As a result, an augmented-order linear state-space model is identified exclusively from input–output data, without relying on explicit knowledge of the system’s governing equations. The effectiveness of the proposed approach is demonstrated using operational data from a real multivariable tank system that was not used during the identification stage. The identified model achieves a coefficient of determination exceeding 0.87 in multi-step prediction tasks. Furthermore, spectral analysis of the resulting linear operator reveals that the dominant dynamical modes of the physical system are accurately captured. At the same time, additional modes associated with nonlinear interactions are also identified. These results highlight the HDMDc+Lift framework’s ability to provide accurate and interpretable linear representations of forced nonlinear MIMO dynamics.

We present a novel microlocal analysis of a non-linear ray transform, , arising in Compton scattering tomography (CST). Due to attenuation effects in CST, the integral weights depend on the reconstruction target, f, which has singularities. Thus, standard linear Fourier integral operator (FIO) theory does not apply as the weights are non-smooth. The V-line (or broken ray) transform, , can be used to model the attenuation of incoming and outgoing rays. Through novel analysis of , we characterize the location and strength of the singularities of the ray transform weights. In conjunction, we provide new results which quantify the strength of the singularities of distributional products based on the Sobolev order of the individual components. By combining this new theory, our analysis of , and classical linear FIO theory, we determine the Sobolev order of the singularities of . The strongest (lowest Sobolev order) singularities of are shown to correspond to the wavefront set elements of the classical Radon transform applied to f, and we use this idea and known results on the Radon transform to prove injectivity results for . In addition, we present novel reconstruction methods based on our theory, and we validate our results using simulated image reconstructions.

This study proposes a new formalized approach to the stabilization of linear transformations in artificial neural networks, based on discrete algebraic properties. In contrast to existing stability methods that rely on spectral norms, regularization techniques, or empirical heuristics, this work introduces the concept of algebraic stabilization—stability that arises from the structural properties of the matrices defining linear operators. The central object of investigation is the class of integer-valued matrices for which exponentiation to a form of the type Wk=I+μD is possible, where D∈Zn×n,μ∈Z&gt;1. A well-known problem in group algebra is considered that guarantees the existence of such an exponent under the condition that μ is coprime with the determinant of W. Within this framework, modular arithmetic, reduction modulo μ, and the group structure of GLnZμ are employed, thereby linking the proposed method to the theory of finite groups and linear automata. The advantages of the approach are discussed, including formal control over the iterative behavior of transformations, compatibility with quantized and finitely representable networks, the possibility of embedding stabilizing conditions directly into the network architecture, and the potential to improve model interpretability and reliability. At the same time, limitations are identified, particularly those related to constructive implementation, the selection of suitable hyperparameters, and generalization to broader classes of transformations.