Orthonormal Basis Research Articles

Some properties of a complex symmetric operator

A square matrix with complex entries which is equal to its transpose is called a complex symmetric matrix. Such matrices are found in many areas of pure and applied mathematics. A conjugation on H is defined as a conjugate linear isometric involution C on a Hilbert Space H. A bounded linear operator T : H → H has a complex symmetric matrix with regard to an orthonormal basis if, for every conjugation C on H, T = CT *C. Such an operator is called a Complex symmetric operator. Several characteristics of Complex Symmetric matrices and associated operators are discussed in this article.

Stereodivergent photobiocatalytic radical cyclization through the repurposing and directed evolution of fatty acid photodecarboxylases.

Despite their intriguing photophysical and photochemical activities, naturally occurring photoenzymes have not yet been repurposed for new-to-nature activities. Here we engineered fatty acid photodecarboxylases to catalyse unnatural photoredox radical C-C bond formation by leveraging the strongly oxidizing excited-state flavoquinone cofactor. Through genome mining, rational engineering and directed evolution, we developed a panel of radical photocyclases to facilitate decarboxylative radical cyclization with excellent chemo-, enantio- and diastereoselectivities. Our high-throughput experimental workflow allowed for the directed evolution of fatty acid photodecarboxylases. An orthogonal set of radical photocyclases was engineered to access all four possible stereoisomers of the stereochemical dyad, affording fully diastereo- and enantiodivergent biotransformations in asymmetric radical biocatalysis. Molecular dynamics simulations show that our evolved radical photocyclases allow near-attack conformations to be easily accessed, enabling chemoselective radical cyclization. The development of stereoselective radical photocyclases provides unnatural C-C-bond-forming activities in natural photoenzyme families, which can be used to tame the stereochemistry of free-radical-mediated reactions.

An entropy conserving/stable discontinuous Galerkin solver in entropy variables based on the direct enforcement of entropy balance

Numerical schemes that guarantee the preservation of flow properties as prescribed by physical laws, ensure high-fidelity results and improved computational robustness. This work presents a highly accurate entropy conserving/stable solver for the solution of the compressible Euler equations. The method uses a modal Discontinuous Galerkin approximation and takes advantage of both entropy variables and the Direct Enforcement of Entropy Balance (DEEB), proposed by Abgrall [1] to fulfill the second law of thermodynamics at the discrete level. Thanks to the use of an orthonormal modal basis, the cost of adding the DEEB correction to the spatially discretized equations is negligible. Entropy variables are used directly, as unknowns, or to assemble the discrete operators while solving for the conservative variables. In the latter case, the conservative variables are “converted” into the entropy ones through an L2 “entropy projection”. If this strategy is coupled with DEEB, a very efficient alternative to the direct use of the entropy variables is obtained, with a dramatic improvement in robustness over the direct use of the conservative variables. For the time integration both an explicit Strong Stability Preserving Runge-Kutta scheme and the implicit Generalized Crank-Nicolson method are considered. The latter allows for the construction of a discrete entropy conservative/stable scheme both in space and time to be used as a reference. Convergence and conservation properties are demonstrated together with robustness and computational efficiency on a suite of two-dimensional test cases, showing that by coupling DEEB with entropy projection the solver robustness is improved while not affecting the stability limit.

Exact Diagonalization of SU(N) Fermi-Hubbard Models.

We show how to perform exact diagonalizations of SU(N) Fermi-Hubbard models on L-site clusters separately in each irreducible representation (irrep) of SU(N). Using the representation theory of the unitary group U(L), we demonstrate that a convenient orthonormal basis, on which matrix elements of the Hamiltonian are very simple, is given by the set of semistandard Young tableaux (or, equivalently, the Gelfand-Tsetlin patterns) corresponding to the targeted irrep. As an application of this color factorization, we study the robustness of some SU(N) phases predicted in the Heisenberg limit upon decreasing the on-site interaction U on various lattices of size L≤12 and for 2≤N≤6. In particular, we show that a long-range color ordered phase emerges for intermediate U for N=4 at filling 1/4 on the triangular lattice.

A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

In this paper we consider the numerical solution of fractional differential equations. In particular, we study a step-by-step procedure, defined over a graded mesh, which is based on a truncated expansion of the vector field along the orthonormal Jacobi polynomial basis. Under mild hypotheses, the proposed procedure is capable of getting spectral accuracy. A few numerical examples are reported to confirm the theoretical findings.

Development and application of a power Bonferroni mean operator based on the foreign fiber content grade evaluation method

Cubic q-rung orthogonal fuzzy sets (C q-ROFSs) are a sophisticated mathematical tool used to handle complex evaluation information in multi-attribute decision-making problems. In specific decision-making problems, the power Bonferroni mean (PBM) operator can reflect the correlation between different attributes and mitigate the impact of extreme evaluation information, thereby providing more practical value. This paper focuses on expanding the PBM operator into the C q-ROFS environment and deriving new PBM operators: the cubic q-rung orthogonal power Bonferroni averaging operator and weight cubic q-rung orthogonal PBM operator. The proposed operator shows strong flexibility and stability in the cubic q-rung orthogonal fuzzy environment. In the absence of weight information, there is a dearth of literature addressing the acceptable advantage and decision stability in the C q-ROFS environment; considering the regret behavior of decision information, a VIKOR method based on regret theory is proposed. The proposed method aggregates information using the proposed operator, determines the scheme and weights at two levels of attributes, and constructs a relative proximity decision matrix. Then, the VIKOR method calculates the group utility value and individual regret value based on the regret perception value to rank the alternatives. Finally, the method is applied to evaluate the cotton foreign fiber content, and its stability and effectiveness are verified through sensitivity analysis and comparison with existing methods.

Linear canonical Stockwell transform and the associated multiresolution analysis

In this article, we propose a new class of transform called linear canonical Stockwell transform (LCST) and obtain its basic properties along with the inner product relation and reconstruction formula. We characterize the range of the transform and show that its range is the reproducing kernel Hilbert space. We also develop a multiresolution analysis (MRA) associated with the proposed transform together with the construction of the orthonormal basis for .

Measurement and Policy Optimization of Regional Preschool Education Development Level Based on Generalized Orthogonal Fuzzy Sets and Prospect Theory

Preschool education belongs to non-compulsory enlightenment education, and it is difficult to measure and analyze the development of preschool education in different regions because of its multiple attributes and diversity of influencing factors. In addition, decision makers will be limited by their own cognition when facing multi-attribute factors and uncertain factors, and there is a big gap between the decision results given and the actual situation. Therefore, this chapter introduces generalized orthogonal fuzzy sets and prospect theory into the measurement model of preschool education development level based on technique for order of preference by similarity to ideal solution (TOPSIS) method to improve the decision accuracy of decision makers. The experimental results show that the model can effectively deal with uncertain information and improve the analysis accuracy, and the results are more in line with the actual situation than other models. At the same time, it can intuitively compare and analyze the development of preschool education in different regions, and provide reliable data support for decision makers.

ANOVA approximation with mixed tensor product basis on scattered points

In this paper we consider an orthonormal basis, generated by a tensor product of Fourier basis functions, half period cosine basis functions, and the Chebyshev basis functions. We deal with the approximation problem in high dimensions related to this basis and design a fast algorithm to multiply with the underlying matrix, consisting of rows of the non-equidistant Fourier matrix, the non-equidistant cosine matrix and the non-equidistant Chebyshev matrix, and its transposed. Using this, we derive the ANOVA (analysis of variance) decomposition for functions with partially periodic boundary conditions through using the Fourier basis in some dimensions and the half period cosine basis or the Chebyshev basis in others. We consider sensitivity analysis in this setting, in order to find an adapted basis for the underlying approximation problem. More precisely, we find the underlying index set of the multidimensional series expansion. Additionally, we test this ANOVA approximation with mixed basis at numerical experiments, and refer to the advantage of interpretable results.

Matrix representation of Toeplitz operators on Newton spaces

In this paper, we study several properties of an orthonormal basis {Nn(z)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\{N_{n}(z)\\}$\\end{document} for the Newton space N2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N^{2}({\\mathbb{P}})$\\end{document}. In particular, we investigate the product of Nm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N_{m}$\\end{document} and Nm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N_{m}$\\end{document} and the orthogonal projection P of Nn‾Nm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\overline{N_{n}}N_{m}$\\end{document} that maps from L2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{2}(\\mathbb{P})$\\end{document} onto N2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N^{2}(\\mathbb{P})$\\end{document}. Moreover, we find the matrix representation of Toeplitz operators with respect to such an orthonormal basis on the Newton space N2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N^{2}({\\mathbb{P}})$\\end{document}.

Optimal Design of Group Orthogonal Phase-Coded Waveforms for MIMO Radar

Digital radio frequency memory (DRFM) has emerged as an advanced technique to achieve a range of jamming signals, due to its capability to intercept waveforms within a short time. multiple-input multiple-output (MIMO) radars can transmit agile orthogonal waveform sets for different pulses to combat DRFM-based jamming, where any two groups of waveform sets are also orthogonal. In this article, a group orthogonal waveform optimal design model is formulated in order to combat DRFM-based jamming by flexibly designing waveforms for MIMO radars. Aiming at balancing the intra- and intergroup orthogonal performances, the objective function is defined as the weighted sum of the intra- and intergroup orthogonal performance metrics. To solve the formulated model, in this article, a group orthogonal waveform design algorithm is proposed. Based on a primal-dual-type method and proper relaxations, the proposed algorithm transforms the original problem into a series of simple subproblems. Numerical results demonstrate that the obtained group orthogonal waveforms have the ability to flexibly suppress DRFM-based deceptive jamming, which is not achievable using p-majorization–minimization (p-MM) and primal-dual, two of the most advanced orthogonal waveform design algorithms.

Some aspects of the Floquet theory for the heat equation in a periodic domain

We treat the linear heat equation in a periodic waveguide Π⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi \\subset {{\\mathbb {R}}}^d$$\\end{document}, with a regular enough boundary, by using the Floquet transform methods. Applying the Floquet transform F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\ extsf{F}}}$$\\end{document} to the equation yields a heat equation with mixed boundary conditions on the periodic cell ϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varpi $$\\end{document} of Π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi $$\\end{document}, and we analyse the connection between the solutions of the two problems. The considerations involve a description of the spectral projections onto subspaces HS⊂L2(Π)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {H}}}_S \\subset L^2(\\Pi )$$\\end{document} corresponding certain spectral components. We also show that the translated Wannier functions form an orthonormal basis in HS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {H}}}_S$$\\end{document}.

The sup-norm problem beyond the newform

AbstractIn this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $\mathrm{GL}_2(\mathbb{Z}_p)$ -type V in $\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.

Targeted optimization in small-scale atomic structure calculations: application to Au I

The lack of reliable atomic data can be a severe limitation in astrophysical modelling, in particular of events such as kilonovae that require information on all neutron-capture elements across a wide range of ionization stages. Notably, the presence of non-orthonormalities between electron orbitals representing configurations that are close in energy can introduce significant inaccuracies in computed energies and transition probabilities. Here, we propose an explicit targeted optimization (TO) method that can effectively circumvent this concern while retaining an orthonormal orbital basis set. We illustrate this method within the framework of small-scale atomic structure models of Au I, using the Grasp2018 multiconfigurational Dirac–Hartree–Fock atomic structure code. By comparing to conventional optimization schemes we show how a TO approach improves the energy level positioning and ordering. TO also leads to better agreement with experimental data for the strongest E1 transitions. This illustrates how small-scale models can be significantly improved with minor computational costs if orbital non-orthonormalities are considered carefully. These results should prove useful to multi-element atomic structure calculations in, for example, astrophysical opacity applications involving neutron-capture elements.

Robustness of vortex wave based communications to operational variables

Orbital angular momentum (OAM) based acoustic communications are a promising method of increasing bandwidth in underwater acoustic communications networks as their unique phase patterns form an orthogonal basis set on which communications protocols may be based. The underwater acoustic environment, however, is highly complex and dynamic. These complexities make developing systems capable of long-range communications challenging. Methods for using BELLHOP’s ray tracing software to simulate the time series of vortex-wave based communications signals are presented. Additionally, this study explores the robustness of the inner product deconvolution method of decoding OAM-based communications against various operating conditions, considering both environmental and platform variations. The effects on channel cross-talk are assessed against sound speed profile uncertainty, position errors, Doppler Effect, and turbulence in the water column. These analyses shed light on the operational implementation of OAM-based communications systems.

The Impact of Network Design Interventions on the Security of Interdependent Systems

We study the problem of defending a Cyber-Physical System (CPS) consisting of interdependent components with heterogeneous sensitivity to investments. In addition to the optimal allocation of limited security resources, we analyze the impact of an orthogonal set of defense strategies in the form of network design interventions in the CPS to protect it against the attacker. We first propose an algorithm to simplify the CPS attack graph to an equivalent form which reduces the computational requirements for characterizing the defender's optimal security investments. We then evaluate four types of design interventions in the network in the form of adding nodes in the attack graph, interpreted as introducing additional safeguards, introducing structural redundancies, introducing functional redundancies, and introducing new functionalities. We identify scenarios in which interventions that strengthen internal components of the CPS may be more beneficial than traditional approaches such as perimeter defense. We showcase our proposed approach in two practical use cases: a remote attack on an industrial CPS and a remote attack on an automotive system. We highlight how our results closely match recommendations made by security organizations and discuss the implications of our findings for CPS design.

Efficiency Improvement of Friction Stir Welding Parameters on AZ91 Magnesium Alloy Joints

Abstract: Friction Stir Welding (FSW) is a solid-state welding that permits an intensive formof parts and geometries to be welded with great nature of joints. The use of high-quality Magnesium alloys is extending in shipbuilding industry, particularly for the advancementof naval warships, cruise ships, littoral surface craft and merchant ships. FSW advancementhas been seemed to have various focal points for the exploitation of Magnesium alloys structures, as it is a minimal effort welding process. The objective is to come across the optimum levels of the process parameters in which it yields maximum tensile strength andbetter hardness. A three-factor, three-level design is utilized for optimizing the FSW process parameters and a Taguchi L9 orthogonal array experimental set up is used to anticipate the responses. The system parameters considered are Rotational speed, Transverse speed and the Tool tilt angle. The Friction stir welding is processed for butt joining of Magnesium alloys (AZ91) plates with 6 mm thickness. Tensile testing is attempted on dog-bone kind test specimen for Magnesium alloys. Analysis of Variance (ANOVA) has been used to analyze the effect of different parameters regarding the responses. The microstructural attributes of the welded segments, including base metal, heat affected zone (HAZ), Thermo Mechanically Affected Zone (TMAZ) and Stir Zone (SZ) are examined through Scanning Electron Microscopy (SEM) and Energy Dispersive X-ray (EDX) analysis carried out to quantify the chemical element allocation at the weld interface. The observed optimum condition for Magnesium alloys (AZ91) is 560 rpm, 60 mm/min and 1 degree.

Quantifying coherence via the generalized Wigner–Yanase–Dyson skew information

Quantum coherence is an important feature that distinguishes quantum mechanics from classical physics. In this paper, we study the coherence of a quantum state with respect to a quantum channel, and introduce a coherence measure in terms of the generalized Wigner–Yanase–Dyson skew information. The basic properties and operational significance of the measure are expounded in detail, which shows that it is indeed a veritable quantity and satisfies many desirable conditions intuitively required for the coherence measures. We illustrate the significance of characterizing channels from the coherence perspective through some examples. Finally, we consider the average coherence of a state based on the generalized Wigner–Yanase–Dyson skew information with respect to any complete mutually unbiased bases in prime power dimensional systems, as well as with respect to all orthonormal bases, and show that these two averages are equivalent since they both can be seen as the coherence of the quantum state with respect to the depolarizing quantum channel.

Learnable quantile polynomial chaos expansion: An uncertainty quantification method for interval reliability analysis

Various uncertainties like random input and data noise in aerospace engineering affect flight vehicle safety, making uncertainty quantification important for improving system reliability. Polynomial chaos expansion (PCE) is a versatile method for quantifying stochastic systems’ uncertainty caused by random input, while it cannot quantify data uncertainty caused by noise. To solve this problem, a novel uncertainty quantification method, learnable quantile PCE (LQPCE), is proposed for interval reliability analysis. Unlike the traditional PCE, the multi-dimensional orthonormal basis is constructed for combination input rather than only random variables, and the expansion coefficients are parameterized to learnable weights, ensuring the LQPCE model can capture data noise’s features. The LQPCE Pinball loss function is constructed to learn parameterized expansion coefficients by quantile level iterative sampling, achieving accurate prediction and data uncertainty quantification. The quantified data uncertainty is used for interval reliability analysis, calculating more reliable results about the influence of input uncertainty on stochastic systems’ reliability. Three numerical examples and one engineering application validate the LQPCE method’s effectiveness. The results show that the LQPCE method can quantify data uncertainty accurately and obtain credible interval reliability.

Deciding whether a lattice has an orthonormal basis is in co-NP

AbstractWe show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the lattice isomorphism problem, which is known to be in the complexity class SZK. We achieve this by deploying a result on characteristic vectors by Elkies that gained attention in the context of 4-manifolds and Seiberg-Witten equations, but seems rather unnoticed in the algorithmic lattice community. On the way, we also show that for a given Gram matrix $$G \in \mathbb {Q}^{n \times n}$$ G ∈ Q n × n , we can efficiently find a rational lattice that is embedded in at most four times the initial dimension n, i.e. a rational matrix $$B \in \mathbb {Q}^{4n \times n}$$ B ∈ Q 4 n × n such that $$B^\intercal B = G$$ B ⊺ B = G .