Class Of Inclusions Research Articles

A Bott–Borel–Weil Theorem for Diagonal Ind-groups

AbstractA diagonal ind-group is a direct limit of classical affine algebraic groups of growing rank under a class of inclusions that contains the inclusionas a typical special case. If G is a diagonal ind-group and B ⊂ G is a Borel ind-subgroup, we consider the ind-variety G/B and compute the cohomology H𝓁(G/B,𝒪−λ) of any G-equivariant line bundle 𝒪−λ on G/B. It has been known that, for a generic λ, all cohomology groups of 𝒪−λ vanish, and that a non-generic equivariant line bundle 𝒪−λ has at most one nonzero cohomology group. The new result of this paper is a precise description of when Hj (G/B,𝒪−λ) is nonzero and the proof of the fact that, whenever nonzero, Hj (G/B,𝒪−λ) is a G-module dual to a highest weight module. The main difficulty is in defining an appropriate analog WB of the Weyl group, so that the action of WB on weights of G is compatible with the analog of the Demazure “action” of the Weyl group on the cohomology of line bundles. The highest weight corresponding to Hj (G/B,𝒪−λ) is then computed by a procedure similar to that in the classical Bott–Borel–Weil theorem.

Fuzzy Rule Based Classification and Quantification of Graphite Inclusions from Microstructure Images of Cast Iron

The quantification of three classes of graphite inclusions in cast iron, namely, nodular, flake, and irregular, is the most important process in the foundry industry. This classification is based on the ISO 945 proposed morphology of graphite inclusions. This work presents a novel solution for automatic quantitative analysis of graphite inclusions into the three mentioned classes. The proposed work comprises three stages, namely, preprocessing of micrographs, classification of graphite inclusions, and then quantification of inclusions in each class. An effort has been made in this work to propose a minimum set of features to represent graphite inclusion morphology. The method employs just two geometric shape descriptors: the diameter ratio and the area ratio. A fuzzy rule based classifier is built using known feature values that are efficient in the classification of the three classes of graphite inclusions. The proposed method is automatic, fast, and provides the basis for determining many more morphological parameters that can be determined with the least effort. The results obtained by the proposed method are compared with the manual method. It is observed that the results obtained from the proposed method are useful in the optimization of cast iron manufacturing in the foundry industry.

Selective tau tyrosine nitration in non-AD tauopathies

Previously, we reported the characterization of two novel antibodies that react with tau nitrated at tyrosine 197 (Tau-nY197) and tyrosine 394 (Tau-nY394) in Alzheimer's disease (AD). In this report, we examined whether tau nitration at these sites also occurs in corticobasal degeneration (CBD), progressive supranuclear palsy (PSP) and Pick's disease (PiD), three neurodegenerative tauopathies that contain abundant tau deposits within glial and neuronal cell types but lack amyloid deposition. The reactivity of these antibodies was also compared to two previously characterized antibodies Tau-nY18 and Tau-nY29, specific for tau nitrated at tyrosine 18 and tyrosine 29, respectively. In the present experiments, Tau-nY18 did not label the classical pathological lesions of CBD or PSP but did label the neuronal lesions associated with PiD to a limited extent. In contrast, Tau-nY29 revealed some, but not all classes of tau inclusions associated with both CBD and PSP but did label numerous Pick body inclusions in PiD. Tau-nY197 was restricted to the neuropil threads in both CBD and PSP; however, similar to Tau-nY29, extensive Pick body pathology was clearly labeled. Tau-nY394 did not detect any of the lesions associated with these disorders. In contrast, extensive neuronal and glial tau pathology within these diseases was labeled by Tau-Y197, a monoclonal antibody that reacts within the Y-197-containing proline-rich region of the molecule. Based on our Western and IHC experiments, it appears that nitration of tau at tyrosine 29 is a pathological modification that might be associated with neurodegeneration. Collectively, our data suggest that site-specific tau tyrosine nitration events occur in a disease and lesion-specific manner, indicating that nitration appears to be a highly controlled modification in AD and non-AD tauopathies.

A Monotone+Skew Splitting Model for Composite Monotone Inclusions in Duality

The principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew-adjoint operator. An algorithmic framework is developed for solving this generic problem in a Hilbert space setting. New primal-dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators, and convergence results are established. These algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration. Comparisons with existing methods are made and applications to composite variational problems are demonstrated.

Controllability and strong controllability of differential inclusions

In this paper, we prove sufficient conditions for controllability and strong controllability in terms of the Mordukhovich subdifferential for two classes of differential inclusions. The first one is the class of sub-Lipschitz multivalued functions introduced by Loewen–Rockafellar (1994) [10]. The second one, introduced recently by Clarke (2005) [18], is the class of multivalued functions which are pseudo-Lipschitz and satisfy the so-called tempered growth condition. To do this, we establish an error bound result in terms of the Mordukhovich subdifferential outside Asplund spaces.

New approximation-solvability of general nonlinear operator inclusion couples involving ( A, η, m)-resolvent operators and relaxed cocoercive type operators

In this paper, we consider and study a class of general nonlinear operator inclusion couples involving ( A, η, m)-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to ( A, η, m)-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures.

On positive solutions of one class of evolutionary inclusions of the subdifferential type

We obtain sufficient conditions for the existence of nonnegative solutions of the evolutionary inclusions of subdifferential type with multivalued non-Lipschitz perturbations and, under the additional condition of dissipativity, prove the existence of a global attractor in the class of nonnegative functions.

Existence results for a class of pseudomonotone elliptic-parabolic inclusions

This paper deals with the initial-value problem of nonlinear evolution inclusions of the form d B( u)/ dt+ A( u) ∈ f, v 0 ∈ B( u)(0), where the operator B is induced by a subgradient and A is pseudomonotone. Existence theorem is established via the time discretization technique and the regularization method. In contrast to the previous results, here we impose a weaker coerciveness condition on A and remove the strong monotonicity from B.

On the existence of solutions for a class of second-order differential inclusions and applications

In this paper, by extending the method of guiding functions, we give sufficient conditions for the existence of solutions to the problem (1.1) u ″ ∈ Q ( u ) , u ( 0 ) = u ( 1 ) = 0 , where Q is a multivalued map whose values are not necessarily convex. It is shown how abstract results can be applied to study feedback control systems and to describe a motion of a particle in a 1-D potential.

History-dependent subdifferential inclusions and hemivariational inequalities in contact mechanics

We consider a class of subdifferential inclusions involving a history-dependent term for which we provide an existence and uniqueness result. The proof is based on arguments on pseudomonotone operators and fixed point. Then we specialize this result in the study of a class of history-dependent hemivariational inequalities. Such kind of problems arises in a large number of mathematical models which describe quasistatic processes of contact between a deformable body and an obstacle, the so-called foundation. To provide an example we consider a viscoelastic problem in which the frictional contact is modeled with subdifferential boundary conditions. We prove that this problem leads to a history-dependent hemivariational inequality in which the unknown is the velocity field. Then we apply our abstract result in order to prove the unique weak solvability of the corresponding contact problem.

Existence results for impulsive neutral stochastic evolution inclusions in Hilbert space

This paper is concerned with the existence of mild solutions of a class of impulsive neutral stochastic evolution inclusions in Hilbert space in the case where the right hand side is convex or nonconvex-valued. The results are obtained by using two fixed point theorems for multivalued mappings and evolution system theory.

Boundary value problems of a class of nonlinear partial differential inclusions

This paper deals with the boundary value problems of nonlinear partial differential inclusions, driven by a negative Laplacian, and with the multivalued term which contains the gradient. It is proved the existence of solutions for the inclusions with the convex and nonconvex valued perturbations. The existence of extremal solutions and a strong relaxation theorem are also obtained.

Graph convergence for the H(·, ·)-accretive operator in Banach spaces with an application

In this paper, a concept of graph convergence concerned with the H(·, ·)-accretive operator is introduced in Banach spaces and some equivalence theorems between of graph-convergence and resolvent operator convergence for the H(·, ·)-accretive operator sequence are proved. As an application, a perturbed algorithm for solving a class of variational inclusions involving the H(·, ·)-accretive operator is constructed. Under some suitable conditions, the existence of the solution for the variational inclusions and the convergence of iterative sequence generated by the perturbed algorithm are also given.

On certain classes of functional inclusions with causal operators in Banach spaces

We introduce the notion of a multivalued causal operator and consider an abstract Cauchy problem in a Banach space for various classes of functional inclusions with causal operators. The methods of the topological degree theory for condensing maps are applied to obtain local and global existence results for this problem and to study the continuous dependence of a solution set on initial data. As application we generalize some existence results for semilinear functional differential inclusions and Volterra integro-differential inclusions with delay.

Set-valued variational-like inclusions with H -η-accretive operators in Banach spaces

In this work, a new class of set-valued variational-like inclusions involving H -η-accretive operators is introduced and studied. A new iterative procedure for computing approximate solutions for the class of set-valued variational-like inclusions and convergence results are established.

Some existence results on a class of inclusions

In this paper‎, ‎we introduce the generalized system nonlinear variational inclusions‎ ‎and prove the existence of its solution in normed spaces‎. ‎We provide‎ ‎examples of applications related to a system nonlinear variational‎ ‎inclusions in the sense of Verma‎, ‎a coupled fixed point problem‎, ‎considered by Bhaskar and Lakshmikantham‎, ‎a coupled coincidence‎ ‎point considered by Lakshmikantham and Ćirić‎. ‎Also‎, ‎we‎ ‎generalized coupled best approximations theorem‎.

On controllability for a class of second-order differential inclusions

By using a suitable fixed point theorem a sufficient condition for controllability is obtained for a Sturm-Liouville type differential inclusion in the case when the right hand side has convex values.

On a nonlocal problem for fractional integrodifferential inclusions in Banach spaces

This paper investigates a class of fractional functional integrodifferential inclusions with nonlocal conditions in Banach spaces. The existence of mild solutions of these inclusions is determined under mixed continuity and Carathéodory conditions by usin

Numerical approximation of a class of discontinuous systems of fractional order

In this paper we investigate the possibility to formulate an implicit multistep numerical method for fractional differential equations, as a discrete dynamical system to model a class of discontinuous dynamical systems of fractional order. For this purpose, the problem is continuously transformed into a set-valued problem, to which the approximate selection theorem for a class of differential inclusions applies. Next, following the way presented in the book of Stewart and Humphries (Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996) for the case of continuous differential equations, we prove that a variant of Adams–Bashforth–Moulton method for fractional differential equations can be considered as defining a discrete dynamical system, approximating the underlying discontinuous fractional system. For this purpose, the existence and uniqueness of solutions are investigated. One example is presented.

Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay

This paper deals with the controllability of a class of impulsive neutral stochastic functional differential inclusions with infinite delay in an abstract space. Sufficient conditions for the controllability are derived with the help of the fixed point theorem for discontinuous multi-valued operators due to Dhage. An example is provided to illustrate the obtained theory.