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A square matrix A has the usual Jordan canonical form that describes the structure of A via eigenvalues and the corresponding Jordan blocks. If A is a linear relation in a finite-dimensional linear space \mathfrak {H} (i.e., A is a linear subspace of \mathfrak {H} \times \mathfrak {H} and can be considered as a multivalued linear operator), then there is a richer structure. In addition to the classical Jordan chains (interpreted in the Cartesian product \mathfrak {H} \times \mathfrak {H}), there occur three more classes of chains: chains starting at zero (the chains for the eigenvalue infinity), chains starting at zero and also ending at zero (the singular chains), and chains with linearly independent entries (the shift chains). These four types of chains give rise to a direct sum decomposition (a Jordan-like decomposition) of the linear relation A. In this decomposition there is a completely singular part that has the extended complex plane as eigenvalues; a usual Jordan part that corresponds to the finite proper eigenvalues; a Jordan part that corresponds to the eigenvalue \infty; and a multishift, i.e., a part that has no eigenvalues at all. Furthermore, the Jordan-like decomposition exhibits a certain uniqueness, closing a gap in earlier results. The presentation is purely algebraic, only the structure of linear spaces is used. Moreover, the presentation has a uniform character: each of the above types is constructed via an appropriately chosen sequence of quotient spaces. The dimensions of the spaces are the Weyr characteristics, which uniquely determine the Jordan-like decomposition of the linear relation. [ABSTRACT FROM AUTHOR]

The main and auxiliary radiating elements are linked by linear mathematical relation and coated with nanocomposite (NC) materials to enhance radiation characteristics. Nanocomposite-based electric material (NE) and nanocomposite-based magnetic material (NM) coated on auxiliary radiating elements to enhance the electromagnetic radiation of the proposed antenna. The radiating elements coated with NE graphene oxide (GO) and NM barium ferrite (BF) have higher electric field and magnetic field distribution respectively. The linear relation of main and auxiliary elements is verified by two substrates FR4, Teflon with correlated results. This proposed NC-coated antenna produces a bandwidth of 4.15 GHz, 4.02 dBi average gain and radiation efficiency of 95.17%, being superior to the antenna without NC coating. Fabricated antenna has 1.56–5.71 GHz bandwidth that includes 2.4/5.2 GHz WLAN, 2.5/3.5 GHz WiMAX, 2.4 GHz ISM, 5G SUB-6 GHz, 5G NR 78 and 2.4/5 GHz Wi-Fi bands. [ABSTRACT FROM AUTHOR]

In this paper, we analyse properties like nullity, defect, ascent and descent of the powers of regular linear relations. We improve some results related to closure and regularity of powers of linear relations in normed spaces. Further, the obtained results are applied to investigate the descent and essential descent spectrum and to give some stability results. [ABSTRACT FROM AUTHOR]

In its original form the peak model for rank one supersingular perturbations of class ɦ--4 or higher of a nonnegative self-adjoint operator requires that the Gram matrix of the model should be diagonal. Here we remove the restriction on the Gram matrix. In particular we explain the origin of the Krein Q-function associated with the Gram matrix. [ABSTRACT FROM AUTHOR]

In this paper, we study some local spectral properties of 3 x 3-block matrix linear relations. More precisely, we give a necessary and sufficient conditions for linear relations matrix to have the single valued property in terms of the surjective spectrum and the analytic residuum of its diagonal entries. On the following, by means of it, we illustrate a new characterization of spectra and local spectra of linear relations matrix. [ABSTRACT FROM AUTHOR]

Let (L,Γ)$(\mathfrak {L},\Gamma)$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space H$\mathfrak {H}$. Let MΓ$M_\Gamma$ be the Weyl family corresponding to (L,Γ)$(\mathfrak {L},\Gamma)$. We cope with two main topics. First, since MΓ$M_\Gamma$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation MΓ(z)$M_\Gamma (z)$, for some z∈C∖R$z\in \mathbb {C}\setminus \mathbb {R}$, becomes a nontrivial task. Regarding MΓ(z)$M_\Gamma (z)$ as the (Shmul'yan) transform of zI$zI$ induced by Γ, we give conditions for the equality in MΓ(z)¯⊆MΓ¯(z)¯$\overline{M_\Gamma (z)}\subseteq \overline{M_{\overline{\Gamma }}(z)}$ to hold and we compute the adjoint MΓ¯(z)∗$M_{\overline{\Gamma }}(z)^*$. As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for T+$T^+$ is nonempty. Based on the criterion for the closeness of MΓ(z)$M_\Gamma (z)$, we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family MΓ$M_\Gamma$ corresponding to a boundary relation Γ for T+$T^+$ is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair (L′,Γ′)$(\mathfrak {L}^\prime ,\Gamma ^\prime)$ with its Weyl family MΓ′$M_{\Gamma ^\prime }$. The transformation scheme is either Γ′=ΓV−1$\Gamma ^\prime =\Gamma V^{-1}$ or Γ′=VΓ$\Gamma ^\prime =V\Gamma$ with suitable linear relations V. Results in this direction include but are not limited to: a 1‐1 correspondence between (L,Γ)$(\mathfrak {L},\Gamma)$ and (L′,Γ′)$(\mathfrak {L}^\prime ,\Gamma ^\prime)$; the formula for MΓ′−MΓ$M_{\Gamma ^\prime }-M_\Gamma$, for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple (L,Γ0,Γ1)$(\mathfrak {L},\Gamma _0,\Gamma _1)$ with kerΓ=T$\ker \Gamma =T$ and T0=T0∗$T_0=T^*_0$ (second scheme, Hilbert space case). [ABSTRACT FROM AUTHOR]

It is a classical result that the Weyl function of a simple symmetric operator in a Hilbert space determines a boundary triple uniquely up to unitary equivalence. We generalize this result to a simple symmetric operator in a Pontryagin space, where unitary equivalence is replaced by the similarity realized via a standard unitary operator. [ABSTRACT FROM AUTHOR]

A potted experiment was conducted to explore the accumulation of arsenic (As) and predict the uptake of As by a wheat-maize rotation system in calcareous soil with different rates of sewage sludge (SS) amendment over two consecutive years. The SS amendment decreased the pH value of calcareous soil but increased the cation exchange capacity (CEC), calcium carbonate (CC), organic carbon (OC) and As accumulation in soil and crops with increasing SS addition. The As bioconcentration factor (BCF) of wheat and maize had a significant negative correlation with pH, CC and a significant positive correlation with OC. Soil CEC had a significant positive correlation only with the As BCF of wheat. Regression analysis showed that soil As, pH, OC, CC and CEC were good predictors of the As concentration in wheat/maize. The regression model for each part of the wheat/maize plants had a high model efficiency value and explained 67~88% of the variability. The R2 values of the wheat and maize grain prediction models were 79% and 76%, respectively. Thus, these models contribute to the study of As risk assessment for sewage sludge utilisation in calcareous soil-wheat/maize rotation systems.

For a closed linear relation everywhere defined on a Hilbert space the concepts of isometry, co-isometry, partial isometry, and generalized inverse are introduced and studied. Part of the results proved in this paper improve and generalize some results known for these concepts. In particular, we extend those of [Acta Sci. Math. (Szeged), 70 (2004), 767–781] and [Studia Math. 205 (2011), no. 1, 71–82]. [ABSTRACT FROM AUTHOR]

Weighted mean temperature of the atmosphere, T m , is a key parameter for retrieving the precipitable water vapor from Global Navigation Satellite System observations. It is commonly estimated by a linear model that relates to surface temperature T s . However, the linear relationship between T m and T s is associated with geographic regions and affected by the weather. To better estimate the T m over China, we analyzed the region-specific and weather-dependent characteristics of this linear relationship using 860,054 radiosonde profiles from 88 Chinese stations between 2005 and 2018. The slope coefficients of site-specific linear models are 0.35~0.95, which generally reduce from northeast to southwest. Over southwest China, the slope coefficient changes drastically, while over the northwest, it shows little variation. We developed a T s ∼ T m linear model using the data from rainless days as well as a model using the data from rainy days for each station. At half the stations, mostly located in west and north China, the differences between the rainy-day and rainless-day T m models are significant and larger than 0.5% (1%) in mean (maximal) relative bias. The regression precisions of the rainy-day models are higher than that of the rainless-day models averagely by 28% for the stations. Radiosonde data satisfying T m − T s > 10   K and T s − T m > 30   K most deviate from linear regression models. Results suggest that the former situation is related to low surface temperature (<270 K), as well as striking temperature and humidity inversions below 800 hPa, while the latter situation is related to high surface temperature (>280 K) and a distinct humidity inversion above 600 hPa. [ABSTRACT FROM AUTHOR]

In terms of spaces of boundary values, we formulate and prove criteria of maximal θ-accretivity and maximal nonnegativity for the proper extension of a closed nonnegative linear relation in a Hilbert space. In the case of differential operators, this directly leads to boundary conditions. [ABSTRACT FROM AUTHOR]

A bounded linear relation T on a complex Banach space X is called a relatively regular linear relation, if T B T = T and T (0) ⊆ N (B) for some bounded operator B on X. This paper contains some properties, characterizations and decomposably results of relatively regular linear relation. [ABSTRACT FROM AUTHOR]

We study the continuity of linear relations defined on asymmetric normed spaces with values in normed spaces. We give some geometric charactirization of these mappings. As an application, we prove the Banach-Steinhaus theorem in the framework of asymmetric normed spaces.

We parametrize the space of double cosets of the group G L (n , k) with respect to two subgroups T−, T+ of block strictly triangular matrices. In the appendix, we consider the quasi-regular representation of G L (n , C) in L2 on T − ∖ G L (n , C) , observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free. [ABSTRACT FROM AUTHOR]

Given two range space relations A and B in Hilbert spaces, we characterize the existence of a range space operator T such that A = BT, respectively A = TB. [ABSTRACT FROM AUTHOR]

In this note we prove that positive multi-valued operators on Riesz spaces which are everywhere defined are always continuous. Further, we prove that the sum of two positive relations having the same null part is itself positive. This result allows us to study the domination problem in the class of positive relations. [ABSTRACT FROM AUTHOR]

Background and Objectives: Quantifying the acid value of stored wheat is of great significance for timely treatment and avoiding waste. For the low‐fat content in wheat, the aim of this study is to propose a microchemical way for the acid value analysis. Findings: Twenty‐four wheat samples are taken from the grain reserve depot and the acid values of them are analyzed by the spectral response of one phthalocyanine dye, tetra‐α‐octoxy substituted zinc phthalocyanine, to a fatty acid. Under the experimental conditions, a linear relation exists between the absorbance of this dye at 749 nm (A749 nm) and the concentration logarithm (lncb) of fatty acid (palmitic acid) in chloroform, R2 = 1.000. Conclusions: By the aforesaid linearity, 1 µl of extracted wheat oil is enough to determine the acid values of wheat samples, which can avoid the interference from the color of the sample in traditional titration. Significance and Novelty: The UV‐visible method has great potential in the actual analysis of the acid value of wheat samples and has important reference value for the analysis of other related agricultural products, food and drugs, and so on. [ABSTRACT FROM AUTHOR]

• Surface tension is observed to be quasi-universally linked to DCP for saturated simple liquids. • A new DCP based modification strategy for deviations from universality was proposed. • The validation AADs of the model from the experimental data is within 2.04 %. In this work, the interfacial property of surface tension has been observed to be quasi-universally linked to a dimensionless calorimetric parameter (DCP) for saturated simple liquids. This parameter, derived from thermodynamic calorimetric properties and exclusively influenced by molecular interactions, holds promise for predicting interfacial property and displaying universal behavior. Similar to the excess entropy scaling, empirical evidence indicates linear relations between interfacial properties with the DCP, when appropriate scaling is applied. Based on this observation, formulas for calculating these properties through DCP has been be developed. For fluids with strong molecular interactions, we have also identified their deviations from universality and have proposed a new DCP based modification strategy. These new DCP-based models were evaluated and validated by comparing them with experimental data for 19 pure fluids from different molecular structure catalogues. The average absolute deviations (AADs) were within 2.04 %. [ABSTRACT FROM AUTHOR]

This paper is concerned with stability of deficiency indices of Hermitian subspaces (i.e. linear relations) under relatively bounded perturbations in Hilbert spaces. Several results about invariance of deficiency indices of Hermitian subspaces under relatively bounded perturbations are established. As a consequence, invariance of self-adjointness of Hermitian subspaces under relatively bounded perturbations is obtained. In addition, it is shown that the deficiency indices may shrink in the special case that the relative bound is equal to 1. The results obtained in the present paper generalize the corresponding results for symmetric operators to more general Hermitian subspaces, some of which relax or improve certain conditions of the related results in existing literatures. [ABSTRACT FROM AUTHOR]

In this paper, we detail some results of the essential spectra of closed linear operators pencils defined on a Banach space X. This is done by enacting a relation between the essential spectrum of closed linear operator pencils and that of linear relations. We identify some essential spectra of an unbounded 2 × 2 block operator matrices pencils with non diagonal domain acting in Banach spaces. [ABSTRACT FROM AUTHOR]

Aiming at the problems of poor accuracy of data feature extraction and large classification error in library archives data classification methods, an automatic classification method of library archives data based on data mining is designed. Firstly, the linear relationship between the characteristic variables of library archives data is determined, and the linear coefficient of archives data characteristics is calculated; Then, the characteristic states of library archives data are divided into three states, the characteristic data are normalized, and the adaptive differential evolution algorithm is used to remove the noise in the characteristics of library archives data; Finally, the mapping relation training model in data mining is used to input the data feature training set, and the file data features are labeled according to different weights; Establish automatic data classification model. The experimental results show that the highest accuracy of this method is about 97%. [ABSTRACT FROM AUTHOR]

We identify 13 isomorphism classes of indecomposable coisotropic relations between Poisson vector spaces and show that every coisotropic relation between finite-dimensional Poisson vector spaces may be decomposed as a direct sum of multiples of these indecomposables. We also find a list of 13 invariants, each of which is the dimension of a space constructed from the relation, such that the 13-vector of multiplicities and the 13-vector of invariants are related by an invertible matrix over Z. It turns out to be simpler to do the analysis above for isotropic relations between presymplectic vector spaces. The coisotropic/Poisson case then follows by a simple duality argument.

We investigate in this paper the isolated points of the approximate point spectrum of a closed linear relation acting on a complex Banach space by using the concepts of quasinilpotent part and the analytic core of a linear relation. [ABSTRACT FROM AUTHOR]

Weighted mean temperature of the atmosphere, Tm, is a key parameter for retrieving the precipitable water vapor from Global Navigation Satellite System observations. It is commonly estimated by a linear model that relates to surface temperature Ts. However, the linear relationship between Tm and Ts is associated with geographic regions and affected by the weather. To better estimate the Tm over China, we analyzed the region-specific and weather-dependent characteristics of this linear relationship using 860,054 radiosonde profiles from 88 Chinese stations between 2005 and 2018. The slope coefficients of site-specific linear models are 0.35~0.95, which generally reduce from northeast to southwest. Over southwest China, the slope coefficient changes drastically, while over the northwest, it shows little variation. We developed a Ts∼Tm linear model using the data from rainless days as well as a model using the data from rainy days for each station. At half the stations, mostly located in west and north China, the differences between the rainy-day and rainless-day Tm models are significant and larger than 0.5% (1%) in mean (maximal) relative bias. The regression precisions of the rainy-day models are higher than that of the rainless-day models averagely by 28% for the stations. Radiosonde data satisfying Tm−Ts>10 K and Ts−Tm>30 K most deviate from linear regression models. Results suggest that the former situation is related to low surface temperature (280 K) and a distinct humidity inversion above 600 hPa.

The Toeplitz-Hausdorff Theorem which was established in 1918 asserts that the numerical range of an operator is always convex. We prove this theorem for the numerical range of a linear relation relative to another linear relation. We also show that the closure of this numerical range contains the relative spectrum for these two linear relations. The classical results for a single linear relation can be deduced from the results obtained here. [ABSTRACT FROM AUTHOR]

We investigate in this paper the isolated points of the approximate point spectrum of a closed linear relation acting on a complex Banach space by using the concepts of quasinilpotent part and the analytic core of a linear relation.

Let A and B be two closed linear relations acting between two Banach spaces X and Y and let λ be a complex number. We study the behavior of the nullity and deficiency of A when perturbed by λB. In particular, we show the existence of a constant ρ > 0 for which both the nullity and deficiency of A do not remain constant when A is perturbed by λB for all λ inside the disk |λ| < ρ. It turns out, however, that these quantities do not depend on λ in the specified disk, that is, both the nullity and deficiency of A - λB are uniform on the specified disk. [ABSTRACT FROM AUTHOR]

Motivated by Aiena, Trapani and Triolo’s work (Filomat 28(2), 263–273, 2014), we introduced some concepts of the local spectral theory and the single-valued extension property abbreviated SVEP of the closed linear relations on a Banach space. After that, we analyzed basic proprieties of these notions and established a relationship between the analytic spectral subspace and the analytic core (see Theorem 3.5). [ABSTRACT FROM AUTHOR]

This paper studies trace class perturbation of closed linear relations in Hilbert spaces. The concept of trace class perturbation of closed relations is introduced by orthogonal projections. Equivalent characterizations of compact and trace class block operator matrices are first given in terms of their elements, separately. By using them, several equivalent and sufficient characterizations of trace class perturbation of closed linear relations are obtained. [ABSTRACT FROM AUTHOR]

In the paper, a minimal relation L0 generated by an integral equation with operator measures is defined and a description of the adjoint relation LQ is given. For this minimal relation, we construct a space of boundary values (a boundary triplet) satisfying the abstract "Green formula" and get a description of maximal dissipative (accumulative) and also self-adjoint extensions of the minimal relation. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce and study the S-pseudospectra and the essential S-pseudospectra of linear relations. We start by giving the definition and we inves- tigate the characterization and some properties of these S-pseudospectra. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the new notions of the ε -pseudospectra and the essential ε -pseudospectra of linear relations and investigate their characterizations. Beside that, we establish the relationship between the pseudospectra (resp. essential ε -pseudospectra) of a sequence of closed linear relations and its limit in a Banach space. We finish by applying some obtained results of the essential ε -pseudospectra of transport equation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the stock network construction problem under simultaneous consideration of linear and nonlinear relations between stocks. A novel method based on the conditional probability is proposed to describe the nonlinear relation between stocks. Furthermore, by considering both the linear and nonlinear relations between stocks, a multilayer network is constructed to characterize stock market, in which Pearson correlation network, Granger causality network, and our proposed nonlinear relation network are combined. Finally, several experiments are conducted to illustrate the effectiveness of the proposed approaches. The results show that the proposed multilayer network not only covers more nodes than the Pearson correlation network, but also better balances the relation between prediction accuracy and the number of predictable nodes. [ABSTRACT FROM AUTHOR]

An accurate, safe, environmentally friendly, fast and sensitive electrochemical biosensors were developed to detect xanthine in serum. The metal‐organic framework ZIF‐8 was synthesised and elemental gold was supported on the surface of ZIF‐8 by reduction method to synthesise Ag‐ZIF‐8. The mesoporous carbon material and the synthesised Ag‐ZIF‐8 were, respectively, applied to a glassy carbon electrode to construct biosensors. The constructed biosensor has a good linear relation in the range of 1–280 μmol l−1 of xanthine and the detection limit is 0.167 μmol l−1. The relative standard deviation value in serum samples was <5%, and the recoveries were 96–106%, indicating the good selectivity, stability and reproducibility of this electrochemical biosensor. [ABSTRACT FROM AUTHOR]

We characterize the isolated points of the spectrum, the poles of the resolvent and the Riesz points of a bounded and closed linear relation on a Banach space. The found results include and generalize some known characterizations in the frame of bounded operators. [ABSTRACT FROM AUTHOR]

This paper explores continuous pressure wave characteristics and flow features in a periodic transient flow induced by a rotary pressure wave generator. A three-dimensional CFD model based on the realizable k-ϵ turbulence model is established to conduct the simulations. The results show that the turbulent kinetic energy evolution in the periodic transient flow is similar to that in an instantaneous transient flow. In addition, there exists an optimal frequency corresponding to the maximum wave amplitude, which is closely related to the fluid properties and system parameters. When the disturbance frequency is less than the optimal frequency, the pressure and velocity derivative satisfy a linear relation in all zones of the upstream pipe. In contrast, a linear zone away from the rotor is locally formed at regions where the linear relation is approximately valid. Correspondingly, a nonlinear zone is near the rotor where the linear relation is inapplicable. The disturbance frequency significantly affects the axial wave amplitude distribution. [ABSTRACT FROM AUTHOR]

Given a closed linear relation T between two Hilbert spaces H and K , the corresponding first and second coordinate projections P T and Q T are both linear contractions from T to H , and to K , respectively. In this paper we investigate the features of these graph contractions. We show among other things that P T P T ∗ = (I + T ∗ T) - 1 , and that Q T Q T ∗ = I - (I + T T ∗) - 1 . The ranges ran P T ∗ and ran Q T ∗ are proved to be closely related to the so called ‘regular part’ of T. The connection of the graph projections to Stone’s decomposition of a closed linear relation is also discussed. [ABSTRACT FROM AUTHOR]

We define a minimal operator L0 generated by an integral equation with an operator measure and prove necessary and sufficient conditions for the operator L0 to be densely defined. In general, L ∗ 0 is a linear relation. We give a description of L ∗ 0 and establish that there exists a one-to-one correspondence between relations Lb with the property L0 ⊂Lb⊂L ∗ 0 and relations θ entering in boundary conditions. In this case we denote Lb = Lθ. We establish conditions under which linear relations Lθ and θ together have the following properties: a linear relation (l.r) is self-adjoint; l.r is closed; l.r is invertible, i.e., the inverse relation is an operator; l.r has the finitedimensional kernel; l.r is well-defined; the range of l.r is closed; the range of l.r is a closed subspace of the finite codimension; the range of l.r coincides with the space wholly; l.r is continuously invertible. We describe the spectrum of Lθ and prove that families of linear relations Lθ(λ) and θ(λ) are holomorphic together. [ABSTRACT FROM AUTHOR]

The behavior of the domain, the range, the kernel and the multi-valued part of a rational function in a linear relation is analyzed, respectively. We give some basic properties of such linear relations, and we prove that the rational form of the spectral mapping theorem holds in terms of ascent, essential ascent, descent and essential descent. [ABSTRACT FROM AUTHOR]

Let T be a bounded linear relation (multivalued linear operator) in a Banach space X. We say that T is a Saphar type linear relation if its null space N(T) is contained in its generalized range ⋂ n = 1 ∞ T n (X) and if there is a bounded linear operator S such that T S T = T and T (0) ⊂ N (S) . In this paper we show that many characterization results of Saphar type operators remain valid in the context of multivalued linear operators. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to give some perturbation results and some relations between the essential pseudospectra of the sum of two multivalued linear operator and the essential pseudospectra of each of these multivalued linear operator. [ABSTRACT FROM AUTHOR]

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In this paper, we investigate the computational behavior of the exterior point simplex algorithm. Up until now, there has been a major difference observed between the theoretical worst case complexity and practical performance of simplex-type algorithms. Computational tests have been carried out on randomly generated sparse linear problems and on a small set of benchmark problems. Specifically, 6780 linear problems were randomly generated, in order to formulate a respectable amount of experiments. Our study consists of the measurement of the number of iterations that the exterior point simplex algorithm needs for the solution of the above mentioned problems and benchmark dataset. Our purpose is to formulate representative regression models for these measurements, which would play a significant role for the evaluation of an algorithm's efficiency. For this examination, specific characteristics, such as the number of constraints and variables, the sparsity and bit length, and the condition of matrix A, of each linear problem, were taken into account. What drew our attention was that the formulated model for the randomly generated problems reveal a linear relation among these characteristics. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of quotient of linear operators to linear relations. We introduce for two given linear relations A and B , the linear relation quotient B / A and we give a detailed treatment of some basic algebraic and topological properties of this new notion. [ABSTRACT FROM AUTHOR]

This paper is concerned with properties and stability of linear relations, including closed, compact, and Fredholm linear relations. Stability of closedness and compactness of linear relations under relatively bounded perturbations and relatively compact perturbations is studied. In particular, results on the stability of index of Fredholm linear relation and its iterates are discussed. The results obtained in the present paper generalize the corresponding results for single-valued linear operators to multi-valued linear operators, and some improve or relax certain assumptions of the related existing results. [ABSTRACT FROM AUTHOR]