Orthogonal Complement Research Articles

Scattering for Schrödinger operators with potentials concentrated near a subspace

We study the scattering properties of Schrödinger operators with bounded potentials concentrated near a subspace of R d \mathbb {R}^d . For such operators, we show the existence of scattering states and characterize their orthogonal complement as a set of surface states, which consists of states that are confined to the subspace (such as pure point states) and states that escape it at a sublinear rate, in a suitable sense. We provide examples of surface states for different systems including those that propagate along the subspace and those that escape the subspace arbitrarily slowly. Our proof uses a novel interpretation of the Enss method [Comm. Math. Phys. 61 (1978), pp. 285–291] in order to obtain a dynamical characterization of the orthogonal complement of the scattering states.

Bridging Mathematics and Cellulose Studies: Investigating Closed Subspaces Hilbert Base Construction

This research focuses on mathematically analyzing specific aspects of cellulose, particularly its molecular sequences. The goal is to employ Hilbert base construction to quantify and understand the structural and mechanical characteristics of cellulose at the molecular level. This is crucial for advancing our knowledge in plant biology and engineering cellulose-based materials. The study rigorously investigates mathematical properties, such as the equivalence between complete subspaces and closed subspaces, the interplay between closed subspaces and their orthogonal complements, and the existence of the closest vector within the Hilbert space to closed subspaces. The outcomes of this research can be adapted and extended to contribute to the construction and manipulation of Hilbert bases specifically for understanding and characterizing the polymer sequences in cellulose. Moreover, the application of these mathematical concepts extends to both structural and functional analysis of cellulose, encompassing mechanical behavior, chemical interactions, and functional attributes. This rigorous mathematical approach offers a more nuanced understanding of cellulose beyond its physical structure, paving the way for groundbreaking advancements in cellulose research and applications.

Notes on totally geodesic foliations of a complete semi-Riemannian manifold

In this paper, we prove that the orthogonal complement $\mathcal{F}^{\perp}$ of a totally geodesic foliation $\mathcal{F}$ on a complete semi-Riemannian manifold $(M,g)$ satisfying a certain inequality between mixed sectional curvatures and the integrability tensor of $\mathcal{F}^{\perp}$ is totally geodesic. We also obtain conditions for the existence of totally geodesic foliations on a complete semi-Riemannian manifold $(M,g)$ with bundle-like metric $g$.

Projection-Based Multiple Notch Filtering

A method for multi-frequency notch filtering is described. The approach is based on quadratic programming, which leads to projecting the signal vector onto the orthogonal complement of the subspace spanned by a set of sinusoidal basis vectors, each corresponding to a notch frequency. The projection matrix uses a weighted inner product whose weights are samples of a standard spectral analysis window. An overlap-add method suitable for block processing is also described. The projection-based notch filter (PNF) is compared with linear-phase FIR and IIR multiple notch filters and is found to have improved amplitude and phase distortion.

A Data-Driven State Estimation Framework for Natural Gas Networks With Measurement Noise

The large-scale coverage of natural gas makes the composition structure and operation mode of natural gas network more complex, higher requirements are put forward for the effectiveness and accuracy of state estimation. The existing methods for state estimation of natural gas network with noise are all modeled after processing the data with noise, leading to the real data being distorted to a certain extent. With that in mind, a data-driven method is presented in this paper. While solving the problem of state estimation for natural gas network with measurement noise in the input data, filtering and denoising are unnecessary during state estimation, retaining the complete information of real data. It avoids destruction of real data induced by separating noise from measured data owing to different methods and intensities of noise processing. According to the gas flow characteristic equation of natural gas system, the original problem is converted into a weighted low-rank approximation problem, the search space is shrunk to an orthogonal complement space. The selection of initial values is not merely unrestricted but there will be no accumulation and transmission of iteration error. The effectiveness of the proposed method is demonstrated through simulating 10-node natural gas network. Compared with the Newton’s method, the data-driven method has superior performance, the RMSE achieves 0.2268 and the MAPE achieves 1.63%.

High‐dimensional regression coefficient estimation by nuclear norm plus l1 norm penalization

We propose a new estimator of the regression coefficients for a high‐dimensional linear regression model, which is derived by replacing the sample predictor covariance matrix in the ordinary least square (OLS) estimator with a different predictor covariance matrix estimate obtained by a nuclear norm plus norm penalization. We call the estimator ALgebraic Covariance Estimator‐regression (ALCE‐reg). We make a direct theoretical comparison of the expected mean square error of ALCE‐reg with OLS and RIDGE. We show in a simulation study that ALCE‐reg is particularly effective when both the dimension and the sample size are large, due to its ability to find a good compromise between the large bias of shrinkage estimators (like RIDGE and least absolute shrinkage and selection operator [LASSO]) and the large variance of estimators conditioned by the sample predictor covariance matrix (like OLS and principal orthogonal complement thresholding [POET]).

Semi-Structured Distributional Regression

Combining additive models and neural networks allows to broaden the scope of statistical regression and extend deep learning-based approaches by interpretable structured additive predictors at the same time. Existing attempts uniting the two modeling approaches are, however, limited to very specific combinations and, more importantly, involve an identifiability issue. As a consequence, interpretability and stable estimation are typically lost. We propose a general framework to combine structured regression models and deep neural networks into a unifying network architecture. To overcome the inherent identifiability issues between different model parts, we construct an orthogonalization cell that projects the deep neural network into the orthogonal complement of the statistical model predictor. This enables proper estimation of structured model parts and thereby interpretability. We demonstrate the framework’s efficacy in numerical experiments and illustrate its special merits in benchmarks and real-world applications.

Formalization of Orthogonal Decomposition for Hilbert Spaces

Summary In this article, we formalize the theorems about orthogonal decomposition of Hilbert spaces, using the Mizar system [1], [2]. For any subspace S of a Hilbert space H, any vector can be represented by the sum of a vector in S and a vector orthogonal to S. The formalization of orthogonal complements of Hilbert spaces has been stored in the Mizar Mathematical Library [4]. We referred to [5] and [6] in the formalization.

A Graphical Calculus for Lagrangian Relations

Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.

Bounds on quantum adiabaticity in driven many-body systems from generalized orthogonality catastrophe and quantum speed limit

We provide two inequalities for estimating adiabatic fidelity in terms of two other more handily calculated quantities, i.e., generalized orthogonality catastrophe and quantum speed limit. As a result of considering a two-dimensional subspace spanned by the initial ground state and its orthogonal complement, our method leads to stronger bounds on adiabatic fidelity than those previously obtained. One of the two inequalities is nearly sharp when the system size is large, as illustrated using a driven Rice-Mele model, which represents a broad class of quantum many-body systems whose overlap of different instantaneous ground states exhibits orthogonality catastrophe.

Implementation of a projection and rescaling algorithm for second-order conic feasibility problems

This paper documents a computational implementation of a projection and rescaling algorithm for solving one of the alternative feasibility problems where L is a linear subspace in , is its orthogonal complement, and is the interior of a direct product of second order cones. The gist of the projection and rescaling algorithm is to enhance a low-cost first-order method (a basic procedure) with an adaptive reconditioning transformation (a rescaling step). We give a full description of a Python implementation of this algorithm and present multiple sets of numerical experiments on synthetic problem instances with varied levels of conditioning. Our computational experiments provide promising evidence of the effectiveness of the projection and rescaling algorithm. Our Python code is publicly available. Furthermore, the simplicity of the algorithm makes a computational implementation in other environments completely straightforward.

Heterotic strings on mathbbm{T} 3/ℤ2, Nikulin involutions and M-theory

We first describe the low energy dynamics of ten dimensional heterotic supergravity compactified on the smooth, flat 3-manifold mathbbm{T} 3/ℤ2, without supersymmetry, and explain how it arises from flat heterotic gauge fields. The semi-classical theory has both Coulomb and Higgs branches of non-supersymmetric vacua. We then give an exact worldsheet description as asymmetric orbifolds of mathbbm{T} 3, where the orbifold generator involves a Nikulin non-symplectic involution θ of the even self-dual lattice Γ(19,3). Along the way we briefly compare our findings with M-theory on K3/θ. Our construction gives a novel CFT description of the semi-classical field theory moduli space. In particular, the Wilson line parameters in the lattice I ⊂ Γ(19,3) of signature (19 − s, 1) which is invariant under θ, and in its orthogonal complement N , correspond respectively to Coulomb and Higgs branch moduli. There is a rich pattern of transitions amongst Higgs and Coulomb branches which we describe using the worldsheet theory.

Определение обобщенных реакций в механизмах параллельной структуры

Analysis methods utilising screw theory are the main tool for investigating parallel mechanisms. The paper considers the specifics of computing generalised contact forces in these mechanisms featuring less than six degrees of freedom. We show that, in order to calculate the desired contact force magnitude corresponding to a certain wrench of the constraint imposed onto the output link, it is necessary to project the wrenches of the active kinematic pairs in the chain onto a vector space representing an orthogonal complement to the link space of the mechanism. We use a translational guiding mechanism with three degrees of freedom as an example to apply our approach to.

Relation to a property of the angular-momentum-zero space of states of four fermions in an angular momentum j=9/2 shell unexpectedly found to be stationary for any rotationally invariant two-body interaction

The existence of states with angular momenta $I=4$ and 6 of four fermions in an angular momentum $j=9/2$ shell that are stationary for any rotationally invariant two-body interaction despite the presence of other states with the same angular momentum, the Escuderos-Zamick states, is shown to be equivalent to the invariance to any such interaction of the span of states generated from $I=0$ states by one-body operators. This invariance is verified by exact calculation independently of previous verifications of the equivalent statement. It explains the occurrence of the Escuderos-Zamick states for just $I=4$ and 6. The action of an arbitrary interaction on the invariant space and its orthogonal complement is analyzed, leading to a relation of the Escuderos-Zamick energy levels to levels with $I=10$ and 12. Aspects of the observed spectra of $^{94}\mathrm{Ru}$, $^{96}\mathrm{Pd}$, and $^{74}\mathrm{Ni}$ are discussed in the light of this relation.

Algorithms for fundamental invariants and equivariants of finite groups

For a finite group, we present three algorithms to compute a generating set of invariants simultaneously to generating sets of basic equivariants,i.e.,equivariants for the irreducible representations of the group. The main novelty resides in the exploitation of the orthogonal complement of the ideal generated by invariants. Its symmetry adapted basis delivers the fundamental equivariants.Fundamental equivariants allow to assemble symmetry adapted bases of polynomial spaces of higher degrees, and these are essential ingredients in exploiting and preserving symmetry in computations. They appear within algebraic computation and beyond, in physics, chemistry and engineering.Our first construction applies solely to reflection groups and consists in applying symmetry preserving interpolation, as developed by the same authors, along an orbit in general position. The fundamental invariants can be read off the H-basis of the ideal of the orbit while the fundamental equivariants are obtained from a symmetry adapted basis of an invariant direct complement to this ideal in the polynomial ring.The second algorithm takes as input primary invariants and the output provides not only the secondary invariants but also free bases for the modules of basic equivariants. These are constructed as the components of a symmetry adapted basis of the orthogonal complement, in the polynomial ring, to the ideal generated by primary invariants.The third and main algorithm proceeds degree by degree, determining the fundamental invariants as forming a H-basis of the Hilbert ideal,i.e.,the polynomial ideal generated by the invariants of positive degree. The fundamental equivariants are simultaneously computed degree by degree as the components of a symmetry adapted basis of the orthogonal complement of the Hilbert ideal.

Parabolic quasi-Sturmian approach to proton-impact ionization of helium

A method for calculating ion-atom ionizing collisions is formulated and applied to single ionization of helium induced by energetic proton impact. Within the frozen-core model for the residual helium ion, the four-body problem in the exit channel is recast as an inhomogeneous Schr\"odinger equation for the Coulomb three-body system $({e}^{\ensuremath{-}},{\text{He}}^{+},{p}^{+})$. The asymptotic behavior of its solution contains the transition amplitude. We suggest to solve the driven equation in the representation of so-called parabolic convoluted quasi-Sturmian (CQS) basis functions which are constructed using quasi-Sturmians (QSs) for the $({e}^{\ensuremath{-}},{\text{He}}^{+})$ and $({p}^{+},{\text{He}}^{+})$ subsystems. By applying the corresponding Coulomb Green's function operators, these QSs are generated from orthogonal complements to square-integrable (${L}^{2}$) Sturmian functions in parabolic coordinates. In the proposed parabolic CQS approach explicit asymptotic expressions for the basis functions provide an expansion of the transition amplitude in terms of ``basis amplitudes,'' which we express analytically. The proton-electron interaction is treated as a perturbation and is approximated by a truncated Sturmian basis-set expansion. It is found that, at least in the high-energy limit, the ionization amplitude converges pretty fast as the number of terms in the separable expansion for the proton-electron potential is increased. The calculated fully differential cross sections for singly ionizing 1-MeV $p$-He collisions in several kinematical regimes in the scattering plane are found to be in reasonable agreement with experimental data and with the theoretical results obtained using the first Born approximation, the 3C model, and the wave-packet convergent close-coupling method. The theory-experiment discrepancy over the binary peak position observed by other authors remains unexplained.

A sequential MUSIC algorithm for scatterers detection in SAR tomography enhanced by a robust covariance estimator

Synthetic aperture radar (SAR) tomography (TomoSAR) is an appealing tool for the extraction of height information of urban infrastructures. Due to the widespread applications of the multiple signal classification (MUSIC) algorithm in source localization, it is a suitable solution in TomoSAR when multiple snapshots (looks) are available. While the classical MUSIC algorithm aims to estimate the whole reflectivity profile of scatterers, sequential MUSIC algorithms are suited for the detection of sparse point-like scatterers. In this class of methods, successive cancellation is performed through orthogonal complement projections on the MUSIC power spectrum. In this work, a new sequential MUSIC algorithm named recursive covariance cancelled MUSIC (RCC-MUSIC), is proposed. This method brings higher accuracy in comparison with the previous sequential methods at the cost of a negligible increase in computational cost. Furthermore, to improve the performance of RCC-MUSIC, it is combined with the recent method of covariance matrix estimation called correlation subspace. Utilizing the correlation subspace method results in a denoised covariance matrix which in turn, increases the accuracy of subspace-based methods. Several numerical examples are presented to compare the performance of the proposed method with the relevant state-of-the-art methods. As a subspace method, simulation results demonstrate the efficiency of the proposed method in terms of estimation accuracy and computational load.

Spectral geometry on manifolds with fibered boundary metrics I: Low energy resolvent

We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibered boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibered boundary (aka ϕ-) pseudodifferential operator when the resolvent parameter tends to zero. This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibers are points. The new feature in the case of non-trivial fibers is that the resolvent has different asymptotic behavior on the subspace of forms that are fiberwise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate ‘split’ pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on ϕ-manifolds.

A trust-region augmented Hessian implementation for state-specific and state-averaged CASSCF wave functions.

In this work, we present a one-step second-order converger for state-specific (SS) and state-averaged (SA) complete active space self-consistent field (CASSCF) wave functions. Robust convergence is achieved through step restrictions using a trust-region augmented Hessian (TRAH) algorithm. To avoid numerical instabilities, an exponential parameterization of variational configuration parameters is employed, which works with a nonredundant orthogonal complement basis. This is a common approach for SS-CASSCF and is extended to SA-CASSCF wave functions in this work. Our implementation is integral direct and based on intermediates that are formulated in either the sparse atomic-orbital or small active molecular-orbital basis. Thus, it benefits from a combination with efficient integral decomposition techniques, such as the resolution-of-the-identity or the chain-of-spheres for exchange approximations. This facilitates calculations on large molecules, such as a Ni(II) complex with 231 atoms and 5154 basis functions. The runtime performance of TRAH-CASSCF is competitive with the other state-of-the-art implementations of approximate and full second-order algorithms. In comparison with a sophisticated first-order converger, TRAH-CASSCF calculations usually take more iterations to reach convergence and, thus, have longer runtimes. However, TRAH-CASSCF calculations still converge reliably to a true minimum even if the first-order algorithm fails.

Planes in four-space and four associated CM points

To any two-dimensional rational plane in four-dimensional space one can naturally attach a point in the Grassmannian Gr(2,4) and four shapes of lattices of rank two. Here, the first two lattices originate from the plane and its orthogonal complement, and the second two essentially arise from the accidental local isomorphism between SO(4) and SO(3)×SO(3). As an application of a recent result of Einsiedler and Lindenstrauss on algebraicity of joinings, we prove simultaneous equidistribution of all of these objects under two splitting conditions.