Factorization Of Operators Research Articles

APPROXIMATE APPROACH FOR THE MATHEMATICAL MODEL OF DISPLACEMENT-TRACTION WITH DEAD LOADS

This paper is concerned with the mathematical modeling of the displacement-traction with elastic bases, under specially selected boundary conditions for place and traction. The approximate approach is based on the theoretical results of the iterative factorization of operators given sufficiently smooth data, and smooth solutions. The problem resulting from the discretization of the original problem using the approximate approach occurs twice at each step of the proposed iterative process. The efficiency of the numerical method of iterative factorization explains that it is suitable for practical implementation in the computer.

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.

Factorization of two-Lipschitz integral operators

Factorization of two-Lipschitz integral operators

Two-loop non-planar four-point topology with massive internal loop

We study a set of two-loop non-planar master integrals needed for the NNLO QCD corrections to diphoton and dijet production at hadron colliders. The top-sector topology contains an internal massive fermion loop and is known to contain elliptic curves. Leveraging the method of differential equations, we provide a comprehensive discussion for deriving an ϵ-factorized differential equation related to the most intricate sector within the Feynman integral family. Despite the dependence on multiple scales and the presence of two elliptic sectors, we demonstrate how to leverage the properties of their maximal cuts and the factorization of the Picard-Fuchs operator to deal with the complexity of the analytic computation. In particular, we construct a transformation matrix that brings the differential equations into a format enabling the convenient expression of analytic results in terms of Chen’s iterated integrals.

On rotated CMV operators and orthogonal polynomials on the unit circle

Split-step quantum walk operators can be expressed as a generalized version of CMV operators with complex transmission coefficients, which we call rotated CMV operators. Following the idea of Cantero, Moral and Velazquez's original construction of the original CMV operators from the theory of orthogonal polynomials on the unit circle (OPUC), we show that rotated CMV operators can be constructed similarly via a rotated version of OPUCs with respect to the same measure, and admit an analogous LM -factorization as the original CMV operators. We also develop the rotated second kind polynomials corresponding to the rotated OPUCs. We then use the LM -factorization of rotated alternate CMV operators to compute the Gesztesy–Zinchenko transfer matrices for rotated CMV operators.

Factorization of abstract operators into two second degree operators and its applications to integro-differential equations

Boundary value problem B1x = f with an abstract linear operator B1, corresponding to an Fredholm integro-differential equation with ordinary or partial differential operator is researched. An exact solution to B1x = f in the case when a bijective operator B1 has a factorization of the form B1 =BB0 where B and B0 are two linear more simple than B1 second degree abstract operators, received. Conditions for factorization of the operator B1 and a criterion for bijectivity of B1 are found.

Lipschitz Transformations and Maurey-Type Non-Homogeneous Integral Inequalities for Operators on Banach Function Spaces

We introduce a method based on Lipschitz pointwise transformations to define a distance on a Banach function space from its norm. We show how some specific lattice geometric properties (p-convexity, p-concavity, p-regularity) or, equivalently, some types of summability conditions (for example, when the terms of the terms in the sums in the range of the operator are restricted to the interval [−1,1]) can be studied by adapting the classical analytical techniques of the summability of operators on Banach lattices, which recalls the work of Maurey. We show a technique to prove new integral dominations (equivalently, operator factorizations), which involve non-homogeneous expressions constructed by pointwise composition with Lipschitz maps. As an example, we prove a new family of integral bounds for certain operators on Lorentz spaces.

P-SV-wave propagation in heterogeneous media: Velocity-stress distributional finite-difference method

We have developed a 2D distributional-finite-difference (DFD) algorithm for modeling the propagation of seismic waves in heterogeneous media in the time domain. We revisit the classic staggered finite-difference algorithm by substituting the standard finite-difference operators with the recently introduced DFD operators with the methodological differences we underline. We find that the DFD operators improve the simulation accuracy while maintaining the simple structure of the finite-difference algorithm. Thanks to its weak formalism, the newly developed algorithm accurately and naturally accounts for the free surface, which is a substantial improvement for finite-difference approaches. Moreover, we develop an efficient factorization for the DFD operators. It limits the computational cost of the proposed algorithm to twice that of the finite-difference algorithm. Numerical examples demonstrate that the extra computational burden is well compensated by the superior accuracy of the DFD operators.

Engineering isospectrality in multidimensional photonic systems: Multidimensional quasi-isospectrality in photonics.

Selective manipulation of energy levels plays an essential role in realizing multichannel wave devices. One of the representative examples is to utilize the concept of quasi-isospectrality: a family of wave systems with an almost identical spectrum except for a part of energy levels. Most approaches toward quasi-isospectrality have employed analytical methods based on symmetry or tridiagonalization, such as supersymmetry, Householder, or Lanczos transformations. Although such analytical approaches provide deterministic and stable designs based on operator factorizations, the mathematical strictness in the factorizations, at the same time, hinders isospectral engineering in a given multidimension. Here we develop the semi-analytical method for engineering isospectrality in multidimensional photonic systems. The method provides the systematic perturbation for the target energy level shifts by decomposing the allowed form of system changes into the perturbation basis. We demonstrate the isospectrality of lower-, higher-, and random-order states while imposing the designed shifts on the other states. The stability analysis shows that the accuracy of the method is determined by the ranges of isospectral state numbers and perturbation strength. The systematic, free-form, and multidimensional natures of the proposed method show great potential for the platform-transparent design of multichannel devices.

Factorization of Hilbert operators

In this research, we introduce some factorization for Hilbert operators of order n based on two important classes of Hausdorff operators, Cesàro and gamma. These factorizations lead us to some new inequalities, one is a generalized version of Hilbert’s inequality. Moreover, as an application of our factorization, we compute the norm of Hilbert operators on some matrix domains.

Numerical Factorization of Propagation Operator for Hyperbolic Equations and Application to One-way, True Amplitude One-way Equations and Bremmer Series

Numerical Factorization of Propagation Operator for Hyperbolic Equations and Application to One-way, True Amplitude One-way Equations and Bremmer Series

Cartesian Operator Factorization Method for Hydrogen

We generalize Schrödinger’s factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach is the fact that the Hamiltonian is represented as a sum over factorizations in terms of coupled operators that depend on the coordinates and momenta in each Cartesian direction. We determine the eigenstates and energies, the wavefunctions in both coordinate and momentum space, and we also illustrate how this technique can be employed to develop the conventional confluent hypergeometric equation approach. The methodology developed here could potentially be employed for other Hamiltonians that can be represented as the sum over coupled Schrödinger factorizations.

Direct sampling methods for isotropic and anisotropic scatterers with point source measurements

<p style='text-indent:20px;'>In this paper, we consider the inverse scattering problem for recovering either an isotropic or anisotropic scatterer from the measured scattered field initiated by a point source. We propose two new imaging functionals for solving the inverse problem. The first one employs a 'far-field' transform to the data which we then use to derive and provide an explicit decay rate for the imaging functional. In order to analyze the behavior of this imaging functional we use the factorization of the near field operator as well as the Funk-Hecke integral identity. For the second imaging functional the Cauchy data is used to define the functional and its behavior is analyzed using the Green's identities. Numerical experiments are given in two dimensions for both isotropic and anisotropic scatterers.</p>

On tensor products of nuclear operators in Banach spaces

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

Factorization of Hermite subdivision operators from polynomial over-reproduction

We study the case of Hermite subdivision operators satisfying a spectral condition of order greater than their size. We show that this can be characterized by operator factorizations involving Taylor operators and difference factorizations of a rank one vector scheme. Giving explicit expressions for the factorization operators, we put into evidence that the factorization only depends on the order of the spectral condition but not on the polynomials that define it. We further show that the derivation of these operators is based on an interplay between Stirling numbers and p-Cauchy numbers (or generalized Gregory coefficients).

$$\ell$$-weights and factorization of transfer operators

We analyze the $\ell$--weights of the evaluation and $q$-oscillator representations of the quantum loop algebras $\mathrm U_q(\mathcal L(\mathfrak{sl}_{\, l + 1}))$ for $l = 1$ and $l = 2$ and prove the factorization relations for the transfer operators of the associated quantum integrable systems.

$\ell$-weights and factorization of transfer operators

Дан анализ $\ell$-весов представлений вычисления и $q$-осцилляторных представлений алгебр петель $\mathrm U_q(\mathcal L(\mathfrak{sl}_{l+1}))$ для $l=1$ и $l=2$, доказаны соотношения факторизации для трансфер-операторов соответствующих квантовых интегрируемых систем.

A sage package for the symbolic-numeric factorization of linear differential operators

We present a SageMath implementation of the symbolic-numeric algorithm introduced by van der Hoeven in 2007 for factoring linear differential operators whose coefficients are rational functions.

A calculation of the Weyl anomaly for 6D conformal higher spins

In this work we continue the study of the one-loop partition function for higher derivative conformal higher spin (CHS) fields in six dimensions and its holographic counterpart given by massless higher spin Fronsdal fields in seven dimensions.In going beyond the conformal class of the boundary round 6-sphere, we start by considering a Ricci-flat, but not conformally flat, boundary and the corresponding Poincaré-Einstein space-filling metric. Here we are able to match the UV logarithmic divergence of the boundary with the IR logarithmic divergence of the bulk, very much like in the known 4D/5D setting, under the assumptions of factorization of the higher derivative CHS kinetic operator and WKB-exactness of the heat kernel of the dual bulk field. A key technical ingredient in this construction is the determination of the fourth heat kernel coefficient b6 for Lichnerowicz Laplacians on both 6D and 7D Einstein manifolds. These results allow to obtain, in addition to the already known type-A Weyl anomaly, two of the three independent type-B anomaly coefficients in terms of the third, say c3 for instance.In order to gain access to c3, and thus determine the four central charges independently, we further consider a generic non Ricci-flat Einstein boundary. However, in this case we find a mismatch between boundary and bulk computations for spins higher than two. We close by discussing the nature of this discrepancy and perspectives for a possible amendment.

On Factorization of Functional Operators with Reflection on the Real Axis

Problems of factorization of matrix functions are closely connected with the solution of matrix Riemann boundary value problems and with the solution of vector singular integral equations. In this article, we study functional operators with orientation-reversing shift reflection on the real axes. We introduce the concept of multiplicative representation of functional operators with shift and its partial indices. Based on the classical notion of matrix factorization, the correctness of the definitions is shown. A theorem on the relationship between factorization of functional operators with reflection and factorization of the corresponding matrix functions is proven.