Continuous Operators Research Articles

Mayer-Homology Learning Prediction of Protein-Ligand Binding Affinities

Artificial intelligence-assisted drug design is revolutionizing the pharmaceutical industry. Effective molecular features are crucial for accurate machine learning predictions, and advanced mathematics plays a key role in designing these features. Persistent homology theory, which equips topological invariants with persistence, provides valuable insights into molecular structures. The standard homology theory is based on a differential rule for the boundary operator that satisfies [Formula: see text] = 0. Our recent work has extended this rule by employing Mayer homology with generalized differentials that satisfy [Formula: see text] = 0 for [Formula: see text] 2, leading to the development of persistent Mayer homology (PMH) theory and richer topological information across various scales. In this study, we utilize PMH to create a novel multiscale topological vectorization for molecular representation, offering valuable tools for descriptive and predictive analyses in molecular data and machine learning prediction. Specifically, benchmark tests on established protein-ligand datasets, including PDBbind-v2007, PDBbind-v2013, and PDBbind-v2016, demonstrate the superior performance of our Mayer homology models in predicting protein-ligand binding affinities.

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.

Anomalous and linear holographic hard wall models for light unflavored mesons

In this work we consider anomalous and linear holographic hard wall (HW) models for light unflavored mesons inspired by the AdS/CFT correspondence. The anomalous dimensions depend on the logarithm of the spin S of the meson state and come from a semiclassical analysis of gauge/string duality. The anomalous HW model produces very good masses and good Regge trajectories for mesons compared with PDG data. Inspired by this anomalous HW model we also propose a phenomenological modification of the dimension of the boundary operators such that the model produces asymptotic linear Regge trajectories. Both anomalous HW models considered here present mass percentage deviations of less than 3%, when compared with PDG data. Published by the American Physical Society 2024

Towards complexity in de Sitter space from the doubled-scaled Sachdev-Ye-Kitaev model

How can we define complexity in dS space from microscopic principles? Based on recent developments pointing towards a correspondence between a pair of double-scaled Sachdev-Ye-Kitaev (DSSYK) models/ 2D Liouville-de Sitter (LdS2) field theory/ 3D Schwarzschild de Sitter (SdS3) space in [1–3], we study concrete complexity proposals in the microscopic models and their dual descriptions. First, we examine the spread complexity of the maximal entropy state of the doubled DSSYK model. We show that it counts the number of entangled chord states in its doubled Hilbert space. We interpret spread complexity in terms of a time difference between antipodal observers in SdS3 space, and a boundary time difference of the dual LdS2 CFTs. This provides a new connection between entanglement and geometry in dS space. Second, Krylov complexity, which describes operator growth, is computed for physical operators on all sides of the correspondence. Their late time evolution behaves as expected for chaotic systems. Later, we define the query complexity in the LdS2 model as the number of steps in an algorithm computing n-point correlation functions of boundary operators of the corresponding antipodal points in SdS3 space. We interpret query complexity as the number of matter operator chord insertions in a cylinder amplitude in the DSSYK, and the number of junctions of Wilson lines between antipodal static patch observers in SdS3 space. Finally, we evaluate a specific proposal of Nielsen complexity for the DSSYK model and comment on its possible dual manifestations.

Mathematical Modelling of Spatially Inhomogeneous Non-Stationary Interaction of Pests with Transgenic and Non-Modified Crops Considering Taxis

Introduction. This paper addresses a unified spatially inhomogeneous, non-stationary model of interaction between genetically modified crop resources (corn) and the corn borer pest, which is also present on a relatively small section of non-modified corn. The model assumes that insect pests influence both types of crops and are capable of independent movement (taxis) towards the gradient of plant resources. It also considers diffusion processes in the dynamics of all components of the unified model, biomass growth, genetic characteristics of both types of plant resources, processes of crop consumption, phenomena of growth and degradation, diffusion, and mutation of pests. The model allows for predictive calculations aimed at reducing crop losses and increasing the resistance of transgenic crops to pests by slowing down the natural mutation rate of the pest. Materials and Methods. The mathematical model is an extension of Kostitsin’s model and is formulated as an initial-boundary value problem for a nonlinear system of convection-diffusion equations. These equations describe the spatiotemporal dynamics of biomass density changes in two types of crops — transgenic and non-modified — as well as the specific populations (densities) of three genotypes of pests (the corn borer) resulting from mutations. The authors linearized the convection-diffusion equations by applying a time-lag method on the time grid, with nonlinear terms from eachequation taken from the previous time layer. The terms describing taxis are presented in a symmetric form, ensuring the skew-symmetry of the corresponding continuous operator and, in the case of spatial grid approximation, the finite-difference operator. Results. A stable monotonic finite-difference scheme is developed, approximating the original problem with second-order accuracy on a uniform 2D spatial grid. Numerical solutions of model problems are provided, qualitatively corresponding to observed processes. Solutions are obtained for various ratios of modified and non-modified sections of the field. Discussion and Conclusion. The obtained results regarding pest behavior, depending on the type of taxis, could significantly extend the time for pests to acquire Bt resistance. The concentration dynamics of pests moving in the direction of the food gradient differs markedly from the concentration of pests moving towards a mate for reproduction.

Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries

We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.

Chaotic phenomena in continuous linear operators on a Fréchet space

Chaotic phenomena in continuous linear operators on a Fréchet space

Rayleigh Quotient and Surjectivity of Nonlinear Operators in Hilbert space

We consider continuous operators acting in a real Hilbert space and indicate conditions ensuring their continuous invertibility and/or surjectivity. In the case of bounded linear operators, these facts are well-known from basic Functional Analysis. The objective of this work is to indicate how similar properties can be proved also when the operators are not necessarily linear, using as a main tool their Rayleigh quotient and especially its lower and upper bound. In particular, we focus our attention on gradient operators and show a quantitative criterion that ensures their surjectivity through the positivity of an additional constant related to the measure of noncompactness.

On fusing matrices associated with conformal boundary conditions

In the context of rational conformal field theories (RCFT) we look at the fusing matrices that arise when a topological defect is attached to a conformal boundary condition. We call such junctions open topological defects. One type of fusing matrices arises when two open defects fuse while another arises when an open defect passes through a boundary operator. We use the topological field theory approach to RCFTs based on Frobenius algebra objects in modular tensor categories to describe the general structure associated with such matrices and how to compute them from a given Frobenius algebra object and its representation theory. We illustrate the computational process on the rational free boson theories. Applications to boundary renormalisation group flows are briefly discussed.

Block diagonal Calderón preconditioning for scattering at multi-screens

A preconditioner is proposed for Laplace exterior boundary value problems on multi-screens. To achieve this, the quotient-space boundary element method and operator preconditioning are combined. For a fairly general subclass of multi-screens, it is shown that this approach paves the way for block diagonal Calderón preconditioners which achieve a spectral condition number that grows only logarithmically with decreasing mesh size, just as in the case of simple screens. Since the resulting scheme contains many more degrees of freedom than strictly required, strategies are presented to remove almost all redundancy without significant loss of effectiveness of the preconditioner. The performance of this method is verified by providing representative numerical results. Further numerical experiments suggest that these results can be extended to a much wider class of multi-screens that cover essentially all geometries encountered in practice, leading to a significantly reduced simulation cost.

Lie product type formulas for continuous multilinear operators

In this paper we prove various Lie product type formulas for continuous multilinear operators. A sample result: Let k≥2 be a natural number, X1,..., Xk, Y be Banach algebras with unit and T:X1×⋅⋅⋅×Xk→Y a continuous k-linear operator such that T(1,...,1)=1. Let also (an)n∈N be a sequence of natural numbers such that limn→∞⁡an=∞. Then for all x1∈X1,..., xk∈Xk we havelimn→∞⁡[T(ex1an,...,exkan)]an=eT(x1,1,...,1)+T(1,x2,1,...,1)+⋅⋅⋅+T(1,...,1,xk).

Development of productivity model of continuous miner operators working in hazardous underground mine environmental conditions

Engaging in underground coal mining involves inherent risks, as miners confront hazardous conditions and unpredictable surroundings. Despite the increasing prevalence of renewable energy, coal continues to be a significant source of power. Extended work hours and exposure to underground conditions can have detrimental effects on miners’ well-being, safety, and efficiency. The competence of continuous miner (CM) operators is pivotal for the overall effectiveness of mining operations. This study undertakes an examination of how the underground environment influences the health and productivity of CM operators through the measurement of real-time data. Moreover, it investigates the economic implications stemming from environmental and physiological factors. Even slight adjustments in conditions have the potential to bolster annual revenue by up to INR 2.66 crores. Additionally, prolonged work hours and exposure may lead to a notable 5.5% decline in productivity levels.

Order Continuous Operators

The order continuous operators consider one of important topic in functional analysis and its applications, the affiliations among order continuous operators and the other classes of operators such as -order are continuous, order bounded, and singular operators, have been studied and investigated, we proved that if an order bounded operator concerning two Riesz space with Dedekind complete is continuous and ordered, then is order continuous, and this paper shows that if is space that is countable, now is not -order continuous, while is uncountable, then is necessarily -order continuous, by giving an example we showed that null ideal for the operator is band when is bounded ordered, further, it is ordered and continuous. Finally, we concluded the operator that is a positively and orderly continuous map on ordered dense with memorizing Riesz subspace of a Riesz space with its range is Dedekind complete, it has only unique ordered continuous expansion all of space.

Nikishin’s theorem and factorization through Marcinkiewicz spaces

Consider L 0 L^0 , the F F -space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin’s classical result on the factorization of sublinear continuous operators from a Banach space to L 0 L^0 , we prove a theorem that characterizes those maps from any quasi-metric space into L 0 L^0 that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces.

Novel Deep CNNs Explore Regions, Boundaries, and Residual Learning for COVID-19 Infection Analysis in Lung CT.

COVID-19 poses a global health crisis, necessitating precise diagnostic methods for timely containment. However, accurately delineating COVID-19-affected regions in lung CT scans is challenging due to contrast variations and significant texture diversity. In this regard, this study introduces a novel two-stage classification and segmentation CNN approach for COVID-19 lung radiological pattern analysis. A novel Residual-BRNet is developed to integrate boundary and regional operations with residual learning, capturing key COVID-19 radiological homogeneous regions, texture variations, and structural contrast patterns in the classification stage. Subsequently, infectious CT images undergo lesion segmentation using the newly proposed RESeg segmentation CNN in the second stage. The RESeg leverages both average and max-pooling implementations to simultaneously learn region homogeneity and boundary-related patterns. Furthermore, novel pixel attention (PA) blocks are integrated into RESeg to effectively address mildly COVID-19-infected regions. The evaluation of the proposed Residual-BRNet CNN in the classification stage demonstrates promising performance metrics, achieving an accuracy of 97.97%, F1-score of 98.01%, sensitivity of 98.42%, and MCC of 96.81%. Meanwhile, PA-RESeg in the segmentation phase achieves an optimal segmentation performance with an IoU score of 98.43% and a dice similarity score of 95.96% of the lesion region. The framework's effectiveness in detecting and segmenting COVID-19 lesions highlights its potential for clinical applications.

Global solutions of approximation problems in Krein spaces

Three approximation problems in Krein spaces are studied, namely the indefinite weighted least squares problem and the related problems of indefinite abstract splines and smoothing. In each case, we analyse the existence of a linear and continuous operator that maps each data point to its solution and when the associated operator problem considering the J-trace has a solution.

On diagnostics of the properties of functional-gradient composites considering rheology

Abstract The solution of inverse problems on determining the variable rheological properties of functional-gradient beams using additional information about the angle of rotation of the end section is presented within the framework of the most used models: Euler–Bernoulli and Timoshenko. Within the framework of the concept of complex modules, nonlinear operator equations are obtained that connect the given and sought functions (instantaneous and long-term module, relaxation time). The solution is based on the linearization method and the formulation of an iterative process, at each step of which it is necessary to solve complex-valued boundary value problems and invert completely continuous operators with complex-valued kernels. Functions reflecting the laws of change in long-term and instantaneous modules have been restored. The results of computational experiments in some special cases are presented.

Pole-skipping for massive fields and the Stueckelberg formalism

Pole-skipping refers to the special phenomenon that the pole and the zero of a retarded two-point Green’s function coincide at certain points in momentum space. We study the pole-skipping phenomenon in holographic Green’s functions of boundary operators that are dual to massive p-form fields and the dRGT massive gravitational fields in the AdS black hole background. Pole-skipping points for these systems are computed using the near horizon method. The relation between the pole-skipping points of massive fields and their massless counterparts is revealed. In particular, as the field mass m is varied from zero to non-zero, the pole-skipping phenomenon undergoes an abrupt change with doubled pole-skipping points found in the massive case. This arises from the breaking of gauge invariance due to the mass term and the consequent appearance of more degrees of freedom. We recover the gauge invariance using the Stueckelberg formalism by introducing auxiliary dynamical fields. The extra pole-skipping points are identified to be associated with the Stueckelberg fields. We also observe that, as the mass varies, some pole-skipping points of the wave number q may move from a non-physical region with complex q to a physical region with real q.

Renormalization group flows in AdS and the bootstrap program

We study correlation functions of the bulk stress tensor and boundary operators in Quantum Field Theories (QFT) in Anti-de Sitter (AdS) space. In particular, we derive new sum rules from the two-point function of the stress tensor and its three-point function with two boundary operators. In AdS2, this leads to a bootstrap setup that involves the central charge of the UV limit of the bulk QFT and may allow to follow a Renormalization Group (RG) flow non-perturbatively by continuously varying the AdS radius. Along the way, we establish the convergence properties of the newly discovered local block decomposition of the three-point function.

Difference operators on fuzzy sets

Based on the properties of the difference operator on crisp sets, a fuzzy difference operator in fuzzy logic is defined as a continuous binary operator on the closed unit interval with some boundary conditions. In this paper, the structures and properties of fuzzy difference operators are studied. The main results are: (1) Using the axiomatic approach, some generalizations of classical tautologies for fuzzy difference operators are obtained. (2) Based on the model theoretic approach, the fuzzy difference operator constructed by a nilpotent t-norm and a strong negation is characterized. (3) the paper discusses the relationship between the fuzzy difference operator and symmetric difference operator which was raised in [3].