Continuous Operators Research Articles

Low Energy Limit for the Resolvent of Some Fibered Boundary Operators

For certain Dirac operators $\eth_{\phi}$ associated to a fibered boundary metric $g_{\phi}$, we provide a pseudodifferential characterization of the limiting behavior of $(\eth_{\phi}+k\gamma)^{-1}$ as $k\searrow 0$, where $\gamma$ is a self-adjoint operator anti-commuting with $\eth_{\phi}$ and whose square is the identity. This yields in particular a pseudodifferential characterization of the low energy limit of the resolvent of $\eth_{\phi}^2$, generalizing a result of Guillarmou and Sher about the low energy limit of the resolvent of the Hodge Laplacian of an asymptotically conical metric. As an application, we use our result to give a pseudodifferential characterization of the inverse of some suspended version of the operator $\eth_{\phi}$. One important ingredient in the proof of our main theorem is that the Dirac operator $\eth_{\phi}$ is Fredholm when acting on suitable weighted Sobolev spaces. This result has been known to experts for some time and we take this as an occasion to provide a complete explicit proof.

The norm of continuous linear operator between two fuzzy quasi-normed spaces

<abstract> <p>In this paper, firstly, we introduce the concepts of continuity and boundedness of linear operators between two fuzzy quasi-normed spaces with general continuous <italic>t</italic>-norms, prove the equivalence of them, and point out that the set of all continuous linear operators forms a convex cone. Secondly, we establish the family of star quasi-seminorms on the cone of continuous linear operators, and construct a fuzzy quasi-norm of a continuous linear operator.</p> </abstract>

Hahn-Banach-Type Theorems and Subdifferentials for Invariant and Equivariant Order Continuous Vector Lattice-Valued Operators with Applications to Optimization

Abstract We give some versions of Hahn-Banach, sandwich, duality, Moreau--Rockafellar-type theorems, optimality conditions and a formula for the subdifferential of composite functions for order continuous vector lattice-valued operators, invariant or equivariant with respect to a fixed group G of homomorphisms. As applications to optimization problems with both convex and linear constraints, we present some Farkas and Kuhn-Tucker-type results.

On maps preserving skew symmetric operators

Given a conjugation C on a separable complex Hilbert space H, a bounded linear operator T on H is said to be C-skew symmetric if CTC = -T*. This paper describes the maps, on the algebra of all bounded linear operators acting on H, that preserve the difference of C-skew symmetric operators for every conjugation C on H.

On unbounded order continuous operators

Let $U$ and $V$ be two Archimedean Riesz spaces. An operator $S:U\rightarrow V$ is said to be unbounded order continuous ($uo$-continuous), if $r_{\alpha }\overset{uo}{\rightarrow }0$ in $U$ implies $Sr_{\alpha }\overset{uo}{% \rightarrow }0$ in $V$. In this paper, we give some properties of the $uo$% -continuous dual $U_{uo}^{\sim }$ of $U$. We show that a nonzero linear functional $f$ on $U$ is $uo$-continuous if and only if $f$ is a linear combination of finitely many order continuous lattice homomorphisms. The result allows us to characterize the $uo$-continuous dual $U_{uo}^{\sim }.$ In general, by giving an example that the $uo$-continuous dual $U_{uo}^{\sim }$ is not a band in $U^{\sim }$, we obtain the conditions for the $uo$% -continuous dual of a Banach lattice $U$ to be a band in $U^{\sim }$. Then, we examine the properties of $uo$-continuous operators. We show that $S$ is an order continuous operator if and only if $S$ is an unbounded order continuous operator when $S$ is a lattice homomorphism between two Riesz spaces $U$ and $V$. Finally, we proved that if an order bounded operator $S:U\rightarrow V$ between Archimedean Riesz space $U$ and atomic Dedekind complete Riesz space $V$ is $uo$-continuous, then $\left\vert S\right\vert $ is $uo$-continuous.

A Tale of HodgeRank and Spectral Method: Target Attack Against Rank Aggregation is the Fixed Point of Adversarial Game.

Rank aggregation with pairwise comparisons has shown promising results in elections, sports competitions, recommendations, and information retrieval. However, little attention has been paid to the security issue of such algorithms, in contrast to numerous research work on the computational and statistical characteristics. Driven by huge profit, the potential adversary has strong motivation and incentives to manipulate the ranking list. Meanwhile, the intrinsic vulnerability of the rank aggregation methods is not well studied in the literature. To fully understand the possible risks, we focus on the purposeful adversary who desires to designate the aggregated results by modifying the pairwise data in this paper. From the perspective of the dynamical system, the attack behavior with a target ranking list is a fixed point belonging to the composition of the adversary and the victim. To perform the targeted attack, we formulate the interaction between the adversary and the victim as a game-theoretic framework consisting of two continuous operators while Nash equilibrium is established. Then two procedures against HodgeRank and RankCentrality are constructed to produce the modification of the original data. Furthermore, we prove that the victims will produce the target ranking list once the adversary masters the complete information. It is noteworthy that the proposed methods allow the adversary only to hold incomplete information or imperfect feedback and perform the purposeful attack. The effectiveness of the suggested target attack strategies is demonstrated by a series of toy simulations and several real-world data experiments. These experimental results show that the proposed methods could achieve the attacker's goal in the sense that the leading candidate of the perturbed ranking list is the designated one by the adversary.

PSYCOLOGICAL PROVISION OF PROCESSES MODERNIZATION OF METALLURGICAL PRODUCTION MANAGEMENT

The creation of an ergonomic methodical approach to the processes modernization of metallurgical production management, which involves a human factor at the difficult man-machine system development and exploitation and the realization degree estimation of ergonomic requirements on the different stages of operator`s activity planning. The purpose of the research is to define the influence features of ergonomic and psychological factors on the activity efficiency of continuous broadband rolling mill operators in the conditions of metallurgical enterprises modernization of the two countries. An analytical model of the organization of the research works devoted to the ergonomic modernization of man-machine systems was developed. Searching and purpose-oriented investigations on the different stages of man-machine system development and exploitation were modelled from the sketchy projection till the exploitation of the system. Theoretical, system analytical and experimental methods were used. The results of processes ergonomic modernization of metallurgical production management were discussed in two countries under new economic (market) conditions. The factors, which define the tension in rolling-mill operator`s activity, were exposed. Some recommendations on modernization of the existing technology and labor organization were formulated. The scientific novelty consists in the development and adaptation of the methodical approach that describes adequately the cognitive complexity and psychophysiological tension in operator`s activity as the modernization factors of modern metallurgical production in two countries. The practical value is the creation of an ergonomic ensuring system of development, exploitation and modernization of difficult men-machine systems and metallurgical production management processes.

A new approach for direct discretization of fractional order operator in delta domain

The fractional order system (FOS) comprises fractional order operator. In order to obtain the discretized version of the fractional order system, the first step is to discretize the fractional order operator, commonly expressed as s?m, 0 < m < 1. The fractional order operator can be used as fractional order differentiator or integrator, depending upon the values of . In general, there are two approaches for discretization of fractional order operator, one is indirect method of discretization and another is direct method of discretization. The direct discretization method capitalizes the method of formation of generating function where fractional order operator s?m is expressed as a function of Z in the shift operator parameterization and continued fraction expansion (CFE) method is then utilized to get the corresponding discrete domain rational transfer function. There is an inherent problem with this discretization method using shift operator parameterization (discrete Z-domain). At fast sampling time, the discretized version of the continuous time operator or system should resemble that of the continuous time counterpart if the sampling theorem is satisfied. At very high sampling rate, the shift operator parameterized system fails to provide meaningful information due to its numerical ill conditioning. To overcome this problem, Delta operator parameterization for discretization is considered in this paper, where at fast sampling rate, the continuous time results can be obtained from the discrete time experiments and therefore a unified framework can be developed to get the discrete time results and continuous time results hand to hand. In this paper a new generating function is proposed to discretize the fractional order operator using the Gauss-Legendre 2-point quadrature rule. Additionally, the function has been expanded using the CFE in order to obtain rational approximation of the fractional order operator. The detailed mathematical formulations along with the simulation results in MATLAB, with different fractional order systems are considered, in order to prove the newness of this formulation for discretization of the FOS in complex Delta domain.

Continuous Mapping Convolution for Large-Scale Point Clouds Semantic Segmentation

In this letter, we introduce MappingConvSeg, a continuous convolution network for semantic segmentation of large-scale point clouds. In particular, a conceptually simple, end-to-end learnable, and continuous convolution operator is proposed for learning spatial correlation of unstructured 3-D point clouds. For each local point set, the unstructured point features are first mapped onto a series of learned kernel points based on the spatial relationship, and the continuous convolution is then applied to capture specific local geometrical patterns. Taking the proposed mapping convolution operation as the building block, a hierarchical network is then built for large-scale point cloud semantic segmentation. Experimental results conducted on two public benchmarks, including Toronto-3D and Stanford large-scale 3-D Indoor Spaces (S3DIS) dataset, demonstrate the superiority of the proposed method.

Perturbation properties of fractional strongly continuous cosine and sine family operators

<abstract><p>Perturbation theory has long been a very useful tool in the hands of mathematicians and physicists. The purpose of this paper is to prove some perturbation results for infinitesimal generators of fractional strongly continuous cosine families. That is, we impose sufficient conditions such that $ A $ is the infinitesimal generator of a fractional strongly continuous cosine family in a Banach space $ X $, and $ B $ is a bounded linear operator in $ X $, then $ A+B $ is also the infinitesimal generator of a fractional strongly continuous cosine family in $ X $. Our results coincide with the classical ones when $ \alpha = 2 $. Furthermore, depending on commutativity condition of linear bounded operators, we propose the elegant closed-form formulas for uniformly continuous perturbed fractional operator cosine and sine functions. Finally, we present an example in the context of one-dimensional perturbed fractional wave equation to demonstrate the applicability of our theoretical results and we give some comparisons with the existing literature.</p></abstract>

On the left (right) condition pseudospectrum of linear operators

The aim of this paper is twofold: On the one hand, we suggest a characterization for the left (right) condition pseudospectrum of bounded linear operators. On the other hand, we give an analogue of the spectral mapping theorem for left (right) condition pseudospectrum.

Generalized Drazin-1-meromorphic invertible operators and generalized Kato-1-meromorphic decomposition

In this paper we generalize the concept of Koliha-Drazin invertible operators by introducing generalized Drazin-1-meromorphic invertible operators. A bounded linear operator T on a Banach space X is said to be 1-meromorphic if every non-zero point of its spectrum is an isolated point. For T we say that it is generalized Drazin-1-meromorphic invertible if there exists a bounded linear operator S acting on X such that TS = ST, STS = S, TST ? T is 1-meromorphic, while T admits a generalized Kato-1-meromorphic decomposition if there exists a pair of T-invariant closed subspaces (M,N) such that X = M?N, the reduction TM is Kato and TN is 1-meromorphic.

Dirichlet problem with measurable data in rectifiable domains

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk D is due to the dissertation of Luzin. The paper [11] was devoted to the Dirichlet problem with continuous boundary data for quasilinear Poisson equations in smooth (C1) domains. The present paper is devoted to the Dirichlet problem with arbitrary measurable (over natural parameter) boundary data for the quasilinear Poisson equations in any Jordan domains with rectifiable boundaries. For this purpose, it is constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations ?U = G with the sources G ? Lp, p > 1. The latter makes it possible to apply the Leray-Schauder approach to the proof of theorems on the existence of regular nonclassical solutions of the measurable Dirichlet problem for quasilinear Poisson equations of the form ?U(z) = H(z)?Q(U(z)) for multipliers H ? Lp with p > 1 and continuous functions Q : R ? R with Q(t)/t ? 0 as t ? ?. Here the boundary values are interpreted in the sense of angular (along nontangential paths) limits that are a traditional tool of the geometric function theory in comparison with variational interpretations in PDE. As consequences, we give applications to some concrete semi-linear equations of mathematical physics, arising under modelling various physical processes such as diffusion with absorption, plasma states, stationary burning etc.

A problem with a non-local condition for a fourth-order equation with multiple characteristics

In this article, we consider a non-local problem with an integral condition for a fourth-order equation. The unique solvability of the problem is proved. The proof of the uniqueness of a solution is based on the a priori estimates derived in the paper. To prove the existence of a solution, the problem is reduced to two Goursat problems for second-order equations, and the equivalence of the stated problem and the resulting system of Goursat problems is proved. One of the problems of the system is the classical Goursat problem. The second problem is a characteristic problem for an integro-differential equation with a non-local integral condition on one of the characteristics. It is impossible to apply the well-known methods of substantiating the solvability of problems with conditions on characteristics to the study of this problem. The introduction of a new unknown function made it possible to reduce the second problem to an equation with a completely continuous operator, to verify, on the basis of the uniqueness theorem, that it is solvable and, by virtue of the proven equivalence of the problems, that the problem posed is solvable.

On the weak compactness of limited completely continuous operators

On the weak compactness of limited completely continuous operators

An efficient spectral-Galerkin method for a new Steklov eigenvalue problem in inverse scattering

<abstract><p>An efficient spectral method is proposed for a new Steklov eigenvalue problem in inverse scattering. Firstly, we establish the weak form and the associated discrete scheme by introducing an appropriate Sobolev space and a corresponding approximation space. Then, according to the Fredholm Alternative, the corresponding operator forms of weak formulation and discrete formulation are derived. After that, the error estimates of approximated eigenvalues and eigenfunctions are proved by using the spectral approximation results of completely continuous operators and the approximation properties of orthogonal projection operators. We also construct an appropriate set of basis functions in the approximation space and derive the matrix form of the discrete scheme based on the tensor product. In addition, we extend the algorithm to the circular domain. Finally, we present plenty of numerical experiments and compare them with some existing numerical methods, which validate that our algorithm is effective and high accuracy.</p></abstract>

Real-Time Power Management for Microgrids With Dynamic Boundaries and Multiple Source Locations

The dynamic boundary concept enables more flexible and efficient operation of microgrids with distributed energy resources (DER) that are intermittent in nature. As the integration of renewables continues to accelerate, an adaptive power management module that enables dynamic boundary operations in microgrids with an increasing number of source locations is essential for the fast and low-cost deployment of microgrid controllers. The power management module introduced in this paper is capable of handling the increased complexity in topological variations and transitions stemming from the dynamic boundaries and multiple source locations. This includes real-time operation of multiple islands with dynamic boundaries, initiation of topological transitions (merging and separation of islands), and automatic source coordination for power sharing and frequency regulation. All functions in the power management module are designed to be automatically adaptable to arbitrary microgrids with non-meshed topologies so that the deployment of the controller at new microgrid sites can be expedited with a reduced cost. The module has been implemented on NI’s CompactRIO system as an essential part of an MG controller and tested on a converter-based hardware testbed (HTB). Testing results validated the effectiveness of the algorithms under various operating conditions.

Modified inertial projection and contraction algorithms for solving variational inequality problems with non-Lipschitz continuous operators

Modified inertial projection and contraction algorithms for solving variational inequality problems with non-Lipschitz continuous operators

Higher genera for proper actions of Lie groups, II: The case of manifolds with boundary

Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the Rapid Decay condition and is such that G/K has nonpositive sectional curvature, we define higher Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and to a generalized G-equivariant Dirac operator D on M with L^2-invertible boundary operator D_\partial. We then establish a higher index formula for these C^*-indices and use it in order to introduce higher genera for M, thus generalizing to manifolds with boundary the results that we have established in Part 1. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.

Ultradifferentiable extension theorems: A survey

We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney’s classical extension theorem, with special emphasis on the existence of continuous linear extension operators. The focus is on Denjoy–Carleman classes for which we develop the theory from scratch and discuss important related concepts such as (non-)quasianalyticity. It allows us to give an efficient and, to a fair extent, elementary introduction to Braun–Meise–Taylor classes based on their representation as intersections and unions of Denjoy–Carleman classes.