Linear Operators Research Articles

A New Presentation for Specht Modules with Distinct Parts

We obtain a new presentation for Specht modules whose conjugate shapes have strictly decreasing parts by introducing a linear operator on the space generated by column tabloids. The generators of the presentation are column tabloids and the relations form a proper subset of the Garnir relations of Fulton. The results in this paper generalize earlier results of the authors and Stanley on Specht modules of staircase shape.

Theory on New Fractional Operators Using Normalization and Probability Tools

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel class of linear operators with memory effects to extend the L-fractional and the ordinary derivatives, using probability tools. A Mittag–Leffler-type function is introduced to solve linear problems, and nonlinear equations are addressed with power series, illustrating the methods for the SIR epidemic model. The inverse operator is constructed, and a fundamental theorem of calculus and an existence-and-uniqueness result for differintegral equations are proven. A conjecture on deconvolution is raised, which would permit completing the proposed theory.

Sodium-chloride difference is not strongly correlated with base excess in chronic kidney disease: an anion gap problem.

The prevalence of metabolic acidosis is high in patients with chronic kidney disease (CKD). For the diagnosis, a blood gas analysis is necessary, but not always available. The aim of the study was to evaluate the base excess (BE) of the sodium-chloride difference (BENa-Cl = Na+-Cl--34mmol/l) as a screening parameter for hyperchloremic metabolic acidosis. We retrospectively performed acid-base analyses of 168non-dialysedpatientswith CKDaccording to the physiologic and to the Stewart's approach. We performed linear regression analysis, Bland-Altman plot and receiver operating characteristics (ROC) analysis of BENa-Cland BE to evaluate the accuracy of BENa-Clpredicting the BE. We further investigated possible confounding factors. The corrected R2for the correlation of BENa-Cland BE was 0.60 (p < 0.001). The Bland-Altman plot showed a good overall agreement. The bias was negligible, but the 95%-limits of agreement showed a wide interval (10.4mmol/l). For BE ≤ 2mmol/l, the ROC analysis yielded an AUC of 0.89 and moderate sensitivity (0.75) and specificity (0.86) for the optimal BENa-Clthreshold (≤ 2mmol/l). Subgroup analysis showed similar results. The main factor for the imprecision of BENa-Clpredicting the BE across all stages of CKD is the variability of the serum anion gap (SAG). The BENa-Clis not an adequate parameter for screening of hyperchloremic acidosis because of the high variability of the SAG. Only, if the BENa-Clis ≤ 5mmol/l, a hyperchloremic acidosis should be suspected. Therefore, a complete blood gas analysis is necessary for the correct diagnosis of acid-base disorders in patients with chronic kidney disease.

New a Priori estimate for stochastic 2D Navier–Stokes equations with applications to invariant measure

AbstractThe paper deals with the stochastic two-dimensional Navier–Stokes equations for homogeneous and incompressible fluids, set in a bounded domain with Dirichlet boundary conditions. We consider additive noise in the form $$G\,\textrm{d}W$$ G d W , where W is a cylindrical Wiener process and G a bounded linear operator with range dense in the domain of $$A^\gamma $$ A γ , A being the Stokes operator. While it is known that existence of invariant measure holds for $$\gamma &gt;1/4$$ γ &gt; 1 / 4 , previous results show its uniqueness only for $$\gamma &gt; 3/8$$ γ &gt; 3 / 8 . We fill this gap and prove uniqueness and strong mixing property in the range $$\gamma \in (1/4, 3/8]$$ γ ∈ ( 1 / 4 , 3 / 8 ] by adapting the so-called Sobolevskiĭ-Kato-Fujita approach to the stochastic N–S equations. This method provides new a priori estimates, which entail both better regularity in space for the solution and strong Feller and irreducibility properties for the associated Markov semigroup.

Boundary Hölder continuity of stable solutions to semilinear elliptic problems in 𝐶1,1 domains

Abstract 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 𝐿 is a linear elliptic operator with variable coefficients and f ∈ C 1 f\in C^{1} is nonnegative, nondecreasing, and convex. The stability of 𝑢 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.

A note on Lin’s and numerical radii inequalities and application in portfolio optimization

We consider Lin's generalizations of Reid's and Halmos' inequalities via polar decomposition approach and establish several numerical radii inequalities for the product of two bounded linear operators on a Hilbert space. The established inequalities are a significant extension of Lin's inequalities. We then explore the applications of the inequalities in finance where when constructing an investment portfolio, investors often need to allocate capital across different assets to achieve a desired risk-return profile. Lin's inequalities can be used to ensure that the portfolio weights satisfy certain constraints, such as minimum and maximum exposure to particular asset classes or sectors.

Property (Baw1) and Weyl type theorems

In this paper, we study property (Baw1) for bounded linear operators defined by Zariouh and Lombarkia in [12] as a variant of Weyl type theorem. We discuss property (Baw1) for operators satisfying single valued extension property (SVEP). We explore conditions related to the direct summands to ensure the extension of the property (Baw1) from and to their direct sum .

One-way reflection waveform inversion with depth-dependent gradient preconditioning

Summary Reflection Waveform Inversion (RWI) is a technique that uses pure reflection data to estimate subsurface background velocity, relying on evolving seismic images. Conventional RWI operates in a cyclic workflow, with two key components in each cycleâ&amp;#x80;&amp;#x94;migration and reflection tomography. Conventional RWI may result in suboptimal background velocity estimation, partly due to limited or unresolved resolution within each component in each cycle. While gradient preconditioning with the reciprocal of Hessian information helps resolve this issue in both components of RWI, it becomes impractical for a large number of model parameters. One-way reflection waveform inversion (ORWI) is a reflection waveform inversion technique in which the forward modeling scheme operates in one direction (downward and then upward) via virtual parallel depth levels within the medium. Leveraging the ORWI framework, we decompose and reduce the linear Hessian operator (also known as the approximate Hessian or Gauss-Newton Hessian) into multiple smaller sub-operators. In particular, the diagonal blocks of the mono-frequency approximate Hessian operators, each corresponding to a single depth level within the medium, are extracted and inverted to precondition the corresponding mono-frequency gradients in both the migration and reflection tomography components of ORWI. This depth-dependent gradient preconditioning transforms standard ORWI into a high-resolution, yet computationally feasible version aimed at addressing suboptimal velocity estimation, referred to as high-resolution ORWI (HR-ORWI). The effectiveness of the proposed approach is demonstrated through successful applications to synthetic data examples.

Mathematical Analysis for Honeybee Dynamics Under the Influence of Seasonality

In this paper, we studied a mathematical model for honeybee population diseases under the influence of seasonal environments on the long-term dynamics of the disease. The model describes the dynamics of two different beehives sharing a common space. We computed the basic reproduction number of the system as the spectral radius of either the next generation matrix for the autonomous system or as the spectral radius of a linear integral operator for the non-autonomous system, and we deduced that if the reproduction number is less than unity, then the disease dies out in the honeybee population. However, if the basic reproduction number is greater than unity, then the disease persists. Finally, we provide several numerical tests that confirm the theoretical findings.

On the pseudo-similarity of bounded linear operators

On the pseudo-similarity of bounded linear operators

On Some Topological Properties of Ch(X) and of the Dual Operator

The hypo-topology on the algebra C(X) of real-valued continuous functions defined on a Tychonoff space 2X f ∈ C(X) with its hypograph, hypof = {(x,t) ∈ X x R:f(x) ≥ t}. This topology is very useful in the calculus of variations and in optimization theory (e.g. Maximization problems). We denote C(X) with the hypo-topology by Ch(X). Our study deals with fundamental properties of these function spaces, and with the linear operators on them and as well as the characterization of the topological properties of Ch(X) in terms of topological properties of the base space X. We are studying the linear operator between the functional algebras Ch(X) and the Ch(Y). We are primarily concerned with the continuity of the evaluation functional, the general evaluation and the continuity of the characters of Ch(X) before the investigation of the properties of the dual operator f* of a continuous function f:X→Y. This operator is defined by f*:Ch (Y)→Ch (X), where f*(g)=gof for all g ∈ C(Y). The continuity of f* enables us to characterize the continuity of an algebra homomorphism of the type φ:Ch (Y)→Ch (X) for a realcompact space Y. For such a space, we present a type of Riesz theorm with states that an algebra homomorphism φ: Ch(Y)→Ch(X) is continuous if and only if there exists a unique hypo-function f: X→Y such that φ = f*. There after, we give the equivalence between the properties of f and those of f*. The study of the continuous linear functional on Ch(X) helps us to compute the topological dual space of this algebra. We show here that this dual space is useful only when the set of isolated points of X is dense.

A New Subclass of Univalent Functions Defined by Raducanu-Orhan Linear Differential Operator

A New Subclass of Univalent Functions Defined by Raducanu-Orhan Linear Differential Operator

Stability and receptivity analyses of the heated flat-plate boundary layer with variable viscosity

This article presents the modal, non-modal, and resolvent analyses of the boundary layer developed over a heated flat-plate with temperature-dependent viscosity using linear stability theory. The governing equations are derived in the normal velocity–vorticity form by imposing small infinitesimal disturbances on the base flow with the Oberbeck-Boussinesq (OB) approximation. The spectral method is employed to discretize the governing stability equations in the wall-normal direction. The base flow comes from the similarity solution using R-K4th order method along with shooting techniques. The effects of viscosity stratification, inertia, shearing, and buoyancy on the stability of the boundary layer are investigated by varying the sensitivity parameter (ϵ), Reynolds (Re), Prandtl (Pr), and Richardson numbers (Ri). The modal analysis shows the time-asymptotic behavior of the disturbances and onset of instability is mainly caused by amplification of Tollmien–Schlichting (T-S) waves similar to as observe in the traditional Blasius case. The modal stability increases with increase in sensitivity parameter and stabilizing effects become more pronounced for liquid than gas. However, the thermal effects lead to destabilize the flow and strong destabilizing effects produced for higher Prandtl number. On the other hand, non-modal analysis displays an early transient growth of disturbances and an existence of these non-normality effects identified from the pseudospectra via. resolvent analysis. The non-modal growth increases with increase in ϵ even though T-S modes shift towards the damped region, which indicates the continuous modes in the eigenspectrum are more dominated than the discrete T-S modes due to the thermal effects. To clarify this qualitative change, a component-wise input–output analysis is performed to measure the receptivity to particular external disturbances. The results show the thermal energy of the disturbances is converted into kinetic energy due to thermal effects, resulting in strong receptivity amplification at the continuous mode due to the non-normality of the linear operator. Thus, the boundary layer under the influence of viscosity stratification, heating, and shearing effects is vulnerable to free-stream disturbances that could significantly affect bypass transition

Metal–Organic Framework-Derived CeO2/Gold Nanospheres in a Highly Sensitive Electrochemical Sensor for Uric Acid Quantification in Milk

In this work, CeBTC (a cerium(III) 1,3,5-benzene-tricarboxylate), was used as a precursor for obtaining CeO2 nanoparticles (nanoceria) with better sensor performances than CeO2 nanoparticles synthesized by the solvothermal method. Metal–organic framework-derived nanoceria (MOFdNC) were functionalized with spheric gold nanoparticles (AuNPs) to further improve non-enzymatic electrode material for highly sensitive detection of prominent biocompound uric acid (UA) at this modified carbon paste electrode (MOFdNC/AuNPs&amp;CPE). X-ray powder diffraction (XRPD) and transmission electron microscopy (TEM) analysis were used for morphological structure characterization of the obtained nanostructures. Cyclic voltammetry and electrochemical impedance spectroscopy, both in an [Fe(CN)6]3−/4− redox system and uric acid standard solutions, were used for the characterization of material electrocatalytic performances, the selection of optimal electrode modifier, and the estimation of nature and kinetic parameters of the electrode process. Square-wave voltammetry (SWV) was chosen, and the optimal parameters of technique and experimental conditions were established for determining uric acid over MOFdNC/AuNPs&amp;CPE. Together with the development of the sensor, the detection procedure was optimized with the following analytical parameters: linear operating ranges of 0.05 to 1 µM and 1 to 50 µM and a detection limit of 0.011 µM, with outstanding repeatability, reproducibility, and stability of the sensor surface. Anti-interference experiments yielded a stable and nearly unchanged current response with negligible or no change in peak potential. After minor sample pretreatment, the proposed electrode was successfully applied for the quantification of UA in milk.

Eccentric p-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs

The extension of the concept of p-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space M afforded by the associated Arens–Eells space, along with the duality between M and the metric dual space M# defined by the real-valued Lipschitz functions on M. However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of p-summability) exist. One approach involves considering specific subsets of the unit ball of M# for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference in the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, in the last part of this paper, we use the theoretical tools obtained in the first part to analyze metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry.

Chaos for endomorphisms of completely metrizable groups and linear operators on Fréchet spaces

Using some techniques from topological dynamics, we give a uniform treatment of Li-Yorke chaos, mean Li-Yorke chaos and distributional chaos for continuous endomorphisms of completely metrizable groups, and characterize three kinds of chaos (resp. extreme chaos) in terms of the existence of the so-called semi-irregular points (resp. irregular points). We exhibit some examples of inner automorphisms of Polish groups to illustrate the results. We also apply our results to the chaos theory of continuous linear operators on Fréchet spaces, which improves some results in the literature.

Circuit and graver walks and linear and integer programming

We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.

An inexact Matrix-Newton method for solving NEPv

In this paper, an inexact Newton method for solving real-valued nonlinear eigenvalue problems with eigenvector dependency (NEPv) is introduced that is able to solve the problem on a matrix level. Our main contribution is to derive a variant of Newton's method that uses global Krylov methods such as global GMRES to solve the linear operator equation necessary to compute the Newton correction in a matrix-free way. The advantages that this second order method has over the well-established SCF algorithm are explained and visualized by a variety of numerical experiments.

Ultrametric Fredholm operators and approximate pseudospectrum

PurposeThe paper deals with ultrametric bounded Fredholm operators and approximate pseudospectra of closed and densely defined (resp. bounded) linear operators on ultrametric Banach spaces.Design/methodology/approachThe author used the notions of ultrametric bounded Fredholm operators and approximate pseudospectra of operators.FindingsThe author established some results on ultrametric bounded Fredholm operators and approximate pseudospectra of closed and densely defined (resp. bounded) linear operators on ultrametric Banach spaces.Originality/valueThe results of the present manuscript are original.

Diffraction‐Driven Parallel Convolution Processing with Integrated Photonics

AbstractTraditional electronic processors often struggle with bandwidth limitations and high power consumption when executing extensive linear operations for deep learning tasks. Optical computing has emerged as a promising alternative, offering parallel and energy‐efficient computation capabilities. Yet, the development of high‐density optical computing architectures on integrated photonic platforms remains limited, hindered by constraints in neuron scalability and control engineering complexities. Addressing these challenges, this work presents a diffraction‐driven multi‐kernel optical convolution unit (MOCU) that enables on‐chip parallel convolution processing. By utilizing cascaded silica 1D metalines as pre‐trained large‐scale weights and employing spatial multiplexing at the output, MOCU allows simultaneous passive computation of diverse convolutions within a single unit. This architecture facilitates the construction of optical convolutional neural networks (OCNNs), enabling efficient machine vision processing with a streamlined design. To mitigate errors in MOCU‐embedded OCNNs, a lightweight electronic neural network operates concurrently to calibrate systematic deviations via a low‐rank adaptation (LoRA) algorithm, with minimal overhead. The fabricated MOCU chip demonstrates the highest independent 8‐kernel convolutions in parallel, each with a kernel size and occupying just 0.06 . This architecture effectively merges photonic and electronic technologies, offering a scalable design pathway for energy‐efficient, high‐density deep learning hardware.