Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property

We initiate a study of structural properties of the quotient algebra $\mathcal K(X)/\mathcal A(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_0$ into $\mathcal K(Z)/\mathcal A(Z)$, where $Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a non-separable space $c_0(\Gamma)$ into $\mathcal K(Z_{FJ})/\mathcal A(Z_{FJ})$, where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel.Comment: 21 page

Generalization of Szász operators involving multiple Sheffer polynomials

The present work deals with the mathematical investigation of some generalizations of the Szász operators. In this work, the multiple Sheffer polynomials are introduced. The generalization of Szász operators involving multiple Sheffer polynomials are considered. Convergence properties of these operators are verified with the help of the universal Korovkin-type result and the order of approximation is calculated by using classical modulus of continuity. Further, the convergence of these operators are also discussed in weighted spaces of functions on the positive semi-axis and estimate the approximation with the help of weighted modulus of continuity. The theoretical results are exemplified choosing the special cases of multiple Sheffer polynomials

Error estimates for DeepOnets: A deep learning framework in infinite dimensions

DeepOnets have recently been proposed as a framework for learning nonlinear operators mapping between infinite dimensional Banach spaces. We analyze DeepOnets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepOnets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ODE, an elliptic PDE with variable coefficients and nonlinear parabolic and hyperbolic PDEs. In all these examples, we prove that DeepOnets break the curse of dimensionality, thus demonstrating the efficient approximation of infinite-dimensional operators with this machine learning framework

The quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property

We initiate a study of structural properties of the quotient algebra K (X)/A(X) of the compact-by-approximable operators on Banach spaces X failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from c(0) into K (Z)/A(Z), where Z belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space c(0)(Gamma) into K (Z(FJ))/A(Z(FJ)), where Z(FJ) is a universal compact factorisation space arising from the work of Johnson and Figiel.Peer reviewe

Covering rough sets based on neighborhoods: An approach without using neighborhoods

Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the coverings themselves, we introduce unary and composition operations on coverings. A notion of homomorphismis provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin

Three-loop universal anomalous dimension of the Wilson operators in N=4 SUSY Yang-Mills model

We present results for the three-loop universal anomalous dimension of Wilson twist-2 operators in the N=4 Supersymmetric Yang-Mills model. These results are obtained by extracting the most complicated contributions from the three loop non-singlet anomalous dimensions in QCD which were calculated recently. Their singularities at j=1 agree with the predictions obtained from the BFKL equation for N=4 SYM in the next-to-leading order. The asymptotics of universal anomalous dimension at large j is in an agreement with the expectations based on an interpolation between weak and strong coupling regimes in the framework of the AdS/CFT correspondence.Comment: LaTeX file, 13 pages, no figures. Some corrections, additional remarks and references. In the last version the analysis of anomalous dimension at j -> 2 was improve