Orthonormal Basis Research Articles

Static Analysis of Anisotropic Doubly-Curved Shell Subjected to Concentrated Loads Employing Higher Order Layer-Wise Theories

In the present manuscript, a Layer-Wise (LW) generalized model is proposed for the linear static analysis of doublycurved shells constrained with general boundary conditions under the influence of concentrated and surface loads. The unknown field variable is modelled employing polynomials of various orders, each of them defined within each layer of the structure. As a particular case of the LW model, an Equivalent Single Layer (ESL) formulation is derived too. Different approaches are outlined for the assessment of external forces, as well as for non-conventional constraints. The doubly-curved shell is composed by superimposed generally anisotropic laminae, each of them characterized by an arbitrary orientation. The fundamental governing equations are derived starting from an orthogonal set of principal coordinates. Furthermore, generalized blending functions account for the distortion of the physical domain. The implementation of the fundamental governing equations is performed by means of the Generalized Differential Quadrature (GDQ) method, whereas the numerical integrations are computed employing the Generalized Integral Quadrature (GIQ) method. In the post-processing phase, an effective procedure is adopted for the reconstruction of stress and strain through-the-thickness distributions based on the exact fulfillment of three-dimensional equilibrium equations. A series of systematic investigations are performed in which the static response of structures with various curvatures and lamination schemes, calculated by the present methodology, have been successfully compared to those ones obtained from refined finite element three-dimensional simulations. Even though the present LW approach accounts for a two-dimensional assessment of the structural problem, it is capable of well predicting the three-dimensional response of structures with different characteristics, taking into account a reduced computational cost and pretending to be a valid alternative to widespread numerical implementations.

Multi-Element UWB Probe Optimization for Medical Microwave Imaging

The need for non-ionizing techniques for medical imaging applications has led to the use of microwave signals. Several systems have been introduced in recent years based on increasing the number of antennas and frequency bandwidth to obtain high resolution and good accuracy in locating objects. A novel microwave imaging system that reduces the number of required antennas for precise target location appropriate for medical applications is presented. The proposed system consists of four UWB extended gap ridge horn (EGRH) antennas covering the frequency band from 0.5 GHz to 1.5 GHz mounted on a cylindrical phantom that mimics the brain in an orthogonal set of two EGRH probes. This configuration has the ability to control both the longitudinal and transversal dimensions of the reconstructed target's image, rather than controlling the spatial resolution, by increasing the frequency band that can be easily affected by medium losses. The system is tested numerically and experimentally by the detection of a cylindrical target within a human brain model.

Alternative approach to quantum imaginary time evolution

There is increasing interest in quantum algorithms (QAs) that are based on the imaginary time evolution (ITE), a successful classical numerical approach to obtain ground states. However, most of the proposals so far require heavy postprocessing computational steps on a classical computer, such as solving linear equations. Here we provide an alternative approach to implement ITE. A key feature in our approach is the use of an orthogonal basis set: the propagated state is efficiently expressed in terms of orthogonal basis states at every step of the evolution. We argue that the number of basis states needed at those steps to achieve an accurate solution can be kept on the order of $n$, the number of qubits, by controlling the precision (number of significant digits) and the imaginary time increment. The number of quantum gates per imaginary time step is estimated to be polynomial in $n$. Additionally, while in many QAs the locality of the Hamiltonian is a key assumption, in our algorithm this restriction is not required. This characteristic of our algorithm renders it useful for studying highly nonlocal systems, such as the occupation-representation nuclear shell model. We illustrate our algorithm through numerical implementation on an IBM quantum simulator.

G-invariant on unary Hermitian lattices over imaginary quadratic fields with class number 2 or 3

In this work, we shall study the unary Hermitian lattices over imaginary quadratic fields. Let E=Q(−d) be an imaginary quadratic field for a square-free positive integer d, and denote by O its ring of integers. For each positive integer m, let Im be the free Hermitian lattice over O with an orthonormal basis, let Sd(1) be the set consisting of all positive definite integral unary Hermitian lattices over O that can be represented by some Im, and let gd(1) be the least positive integer such that all Hermitian lattices in Sd(1) can be uniformly represented by Igd(1). The main results here determine the explicit form of Sd(1) and the exact value of gd(1) for all imaginary quadratic fields E with class number 2 or 3, which naturally generalize the Lagrange's four-square theorem.

LAFITE Reveals the Complexity of Transcript Isoforms in Subcellular Fractions.

Characterization of the subcellular distribution of RNA is essential for understanding the molecular basis of biological processes. Here, the subcellular nanopore direct RNA-sequencing (DRS) of four lung cancer cell lines (A549, H1975, H358, and HCC4006) is performed, coupled with a computational pipeline, Low-abundance Aware Full-length Isoform clusTEr (LAFITE), to comprehensively analyze the full-length cytoplasmic and nuclear transcriptome. Using additional DRS and orthogonal data sets, it is shown that LAFITE outperforms current methods for detecting full-length transcripts, particularly for low-abundance isoforms that are usually overlooked due to poor read coverage. Experimental validation of six novel isoforms exclusively identified by LAFITE further confirms the reliability of this pipeline. By applying LAFITE to subcellular DRS data, the complexity of the nuclear transcriptome is revealed in terms of isoform diversity, 3'-UTR usage, m6A modification patterns, and intron retention. Overall, LAFITE provides enhanced full-length isoform identification and enables a high-resolution view of the RNA landscape at the isoform level.

Progressive pattern orthogonal interleaver set for interleave division multiple access based, non orthogonal multiple access schemes: Beyond 5G perspective

Abstract This communication suggests an orthogonal interleaver set for interleave division multiple access (IDMA) based non orthogonal multiple access (NOMA) schemes from beyond 5G viewpoint to support enormous increase in user count. The method generates an orthogonal interleaver set by providing two mother interleavers as seed to generate other users’ interleaving patterns progressively. The key feature of the proposed scheme is that it reduces implementation complexity and memory requirement at the base station, while implementing iterative multiuser detection (MUD), which most of the interlaever designs suggested in literature do not consider. It provides additional security to the user data due to progressively changing mother interleavers’ pattern along with the conventional purpose of providing unique identity for individual users in the system. The proposed orthogonal interleaver set is tested through simulations under multiple IDMA system configurations. It has been observed that it preserves the bit error rate (BER) performance of the IDMA scheme along with the optimal implementation complexity and minimal information exchange requirement between base station and mobile station to share the interleaver design.

A High-Performance 3-D Imaging Technique Using Simultaneous Azimuth and Elevation Compounding.

A new technique for 3-D imaging with a row-column array (RCA) configuration has been developed. The technique requires an electrostrictive piezoelectric for the active substrate. While the top set of electrodes is connected to RF transmit and receive channels for conventional diverging wave imaging (DWI), the orthogonal bottom set of electrodes is connected to independently controlled variable dc bias channels. By implementing modulated bias patterns compounded across multiple pulses, fine delay control across the bottom elements can be achieved simultaneously with imaging with the top set of electrodes. This resulted in a high-quality two-way focus in both azimuth and elevation. A 20-MHz electrostrictive composite substrate was fabricated, and 64 top ×64 bottom electrodes were patterned and connected to custom beamforming and biasing electronics. The point spread functions were generated in all dimensions, and the -6 dB resolution was measured to be 93 [Formula: see text] axially, [Formula: see text] in the azimuth, and 328 [Formula: see text] in the elevation dimension. This was in good agreement with the simulated resolutions of 80, 273, and 280 [Formula: see text], respectively.

Conserved cell‐type specific signature of resilience to Alzheimer’s disease nominates role for excitatory cortical neurons

AbstractBackgroundAlzheimer’s disease (AD), the leading cause of dementia, affects millions of people worldwide. With no disease‐modifying medication currently available, the human toll and economic costs are rising rapidly. Under current standards, a patient is diagnosed with AD when both cognitive decline and pathology (amyloid plaques and neurofibrillary tangles) are present. Remarkably, some individuals who have AD pathology remain cognitively normal. Uncovering factors that lead to “cognitive resilience” to AD is a promising path to create new targets for therapies. However, technical challenges discovering novel human resilience factors limit testing, validation, and nomination of novel drugs for AD.MethodIn this study, we use single‐nuclear transcriptional profiles of postmortem cortex from human individuals with high AD pathology who were either cognitively normal (resilient) or cognitively impaired (susceptible) at time of death, as well as mouse strains that parallel these differences in cognition with high amyloid.ResultOur cross‐species discovery approach highlights a novel role for excitatory layer 4/5 cortical neurons in promoting cognitive resilience to AD, and nominates several resilience genes that include ATP1A1, GABRB1, PTK2, and ROCK2. Nominated resilience genes were tested for replication in orthogonal data sets and confirmed to be correlated with cognitive resilience. Additionally, we identified several potential mechanisms of resilience, including regulation of membrane potential, axonal and dendritic growth, and general increase of protein cycle, potentially of membrane proteins.ConclusionBecause our discovery of resilience‐associated genes in layer 4/5 cortical neurons originates from an integrated human and mouse transcriptomic space from susceptible and resilient individuals, we are positioned to test causality and perform mechanistic, validation, and pre‐clinical studies in our human‐relevant AD‐BXD mouse panel.

Signatures of Duschinsky Rotation in Femtosecond Coherence Spectra

The motions of nuclei in a molecule can be mathematically described by using normal modes of vibration, which form a complete orthonormal basis. Each normal mode describes oscillatory motion at a frequency determined by the momentum of the nuclei. Near equilibrium, it is common to apply the quantum harmonic-oscillator model, whose eigenfunctions intimately involve combinatorics. Each electronic state has distinct force constants; therefore, each normal-mode basis is distinct. Duschinsky proposed a linearized approximation to the transformation between the normal-mode bases of two electronic states using a rotation matrix. The rotation angles are typically obtained by using quantum-chemical computations or via gas-phase spectroscopy measurements. Quantifying the rotation angles in the condensed phase remains a challenge. Here, we apply a two-dimensional harmonic model that includes a Duschinsky rotation to condensed-phase femtosecond coherence spectra (FCS), which are created in transient–absorption spectroscopy measurements through impulsive excitation of coherent vibrational wavepackets. Using the 2D model, we simulate spectra to identify the signatures of Duschinsky rotation. The results suggest that peak multiplicities and asymmetries may be used to quantify the rotation angle, which is a key advance in condensed-phase molecular spectroscopy.

Jacobi partial waves for a set of 3D noncentral rational scatterers

The common tool of choice for basis expansions for the scattering problem with 3D quantum systems remains the spherical harmonics as eigenfunctions of the Laplace–Beltrami operator on the sphere, with approximations for deviations made around the usually dominant s-wave spherically symmetric state. However, with the growing number of technologically accessible nonspherically symmetric geometries of cold atomic and molecular systems, there is a need to explore as orthonormal bases for partial wave analysis the larger class of weighted Jacobi polynomials, subsuming the spherical harmonics. In particular, the lowest angular state for this bigger class is a toroid instead of a spherical s-orbital. This allows analytic treatment of a wider array of rational angular-dependent potentials which can describe rings and systems with topological constraints such as monopoles. Here, we present exact analytic solutions for the quantum scattering problem by partial wave analysis using the weighted Jacobi polynomials as an expanded basis. We obtain the scattering amplitude, differential and total cross-sections for exactly solvable 3D potentials included in the Smorodinsky-Winternitz noncentral systems with dynamical symmetries. Moreover, this procedure also solves the quantum scattering problem from a novel series of rational trigonometric forms of anisotropic potentials including double ring-shaped configurations.

Optimal pupil basis set for telescope-coronagraph design and perturbation analysis based on the method of moments.

This paper concerns optimization and analysis of telescope-coronagraph systems for direct imaging of exoplanets. In this paper, the coronagraph system, with arbitrary pupil geometry, is theoretically considered, and the governing equation for the pupil design is derived. The method of moments is applied to solve the generalized energy-concentration eigenvalue problem to obtain the optimal pupil apodization and complete sets of orthonormal basis functions for arbitrary pupil geometries. The method yields eigenvalues indicating the fraction of starlight energy encircled in the area of the focal-plane mask (FPM), where starlight can be occulted and/or nulled. In other words, a higher eigenvalue implies less leakage/spillover of light outside of the FPM region and into the planet-discovery zone. Thus, a higher eigenvalue supports better starlight suppression for a given type of coronagraph. This methodology is useful for semi-quantitatively ranking different modes of perturbation with respect to energy spillage in the focal plane independent of coronagraph design details. A model-order-reduction-based sensitivity analysis is conducted to investigate the coupling between different pupil modes induced by aberrations. A pupil mode recovery scheme is presented to offer a theoretically rigorous and computationally efficient approach to reconstruct the optimal pupil mode under an arbitrary phase perturbation. The reconstruction coefficients and recovery-effectiveness factors are derived theoretically and demonstrated numerically. Several numerical examples, including the LUVOIR A and B pupils, are provided to validate and demonstrate the applicability of the proposed methods. The reported methodology enables model-order reduction based on degree of focal-plane energy concentration and reconstruction of optimal pupil apodization vis-á-vis phase aberrations using a precomputed basis set. These features should enhance computational efficiency for coronagraph design and sensitivity analysis.

Light-front wavefunctions of mesons by design

We develop a mechanism to build the light-front wavefunctions (LFWFs) of meson bound states on a small-sized basis function representation. Unlike in a standard Hamiltonian formalism, the Hamiltonian in this method is implicit, and the information of the system is carried directly by the functional form and adjustable parameters of the LFWFs. In this work, we model the LFWFs for four charmonium states, eta _c, J/psi , psi ', and psi (3770) as superpositions of orthonormal basis functions. We choose the basis functions as eigenfunctions of an effective Hamiltonian, which has a longitudinal confining potential in addition to the transverse confining potential from light-front holographic QCD. We determine the basis function parameters and superposition coefficients by employing both guidance from the nonrelativistic description of the meson states and the experimental measurements of the meson decay widths. With the obtained wavefunctions, we study the features of those meson states, including charge radii and parton distribution functions. We use the J/psi LFWF to calculate the meson production in diffractive deep inelastic scattering and ultra-peripheral heavy-ion collisions, and the eta _c LFWF to calculate its diphoton transition form factor. Both results show good agreement with experiments. The obtained LFWFs have simple-functional forms and can be readily used to predict additional experimental observables.

Orthogonal product sets with strong quantum nonlocality on a plane structure

In this paper, we consider the orthogonal product set (OPS) with strong quantum nonlocality. Based on the decomposition of plane geometry, we present a sufficient condition for the triviality of orthogonality-preserving POVM on fixed subsystem and partially answer an open question given by Yuan et al. Phys. Rev. A \textbf{102}, 042228 (2020)}. The connection between the nonlocality and the plane structure of OPS is established. We successfully construct a strongly nonlocal OPS in $\mathcal{C}^{d_{A}}\otimes \mathcal{C}^{d_{B}}\otimes \mathcal{C}^{d_{C}}$ $(d_{A,B,C}\geq 4)$, which contains fewer quantum states, and generalize the structures of known OPSs to any possible three and four-partite systems. In addition, we present several entanglement-assisted protocols for perfectly local discrimination the sets. It is shown that the protocols without teleportation use less entanglement resources on average and these sets can always be discriminated locally with multiple copies of 2-qubit maximally entangled states. These results also exhibit nontrivial signification of maximally entangled states in the local discrimination of quantum states.

A new framework for polynomial approximation to differential equations

In this paper, we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework is based on an expansion of the vector field along an orthonormal basis, and relies on perturbation results for the considered problem. Initially devised for the approximation of ordinary differential equations, it is here further extended and, moreover, generalized to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework.

Spectrality of homogeneous Moran measures on ℝ n

Abstract A Borel probability measure on ℝ n {\mathbb{R}^{n}} is called a spectral measure if the Hilbert space L 2 ⁢ ( μ ) {L^{2}(\mu)} has an orthonormal basis consisting of exponentials. In the present paper we show that, under a mild condition, compatible pairs and lacunary sequences generate homogeneous Moran spectral measures μ on ℝ n {\mathbb{R}^{n}} . As a consequence we show that compatible pairs are necessary and sufficient conditions for certain planar homogeneous Moran measures to become spectral ones.

The minimality of mean square error in chirp approximation using fractional fourier series and fractional fourier transform

Chirps are familiar in nature, have a built-in resistance to noise and interference, and are connected to a wide range of highly oscillatory processes. Detecting chirp oscillating patterns by traditional Fourier series is challenging because the chirp frequencies constantly change over time. Estimating such types of functions considering the partial sums of a Fourier series in Fourier analysis does not permit an approximate solution, which entails more Fourier coefficients required for signal reconstruction. The standard Fourier series, therefore, has a poor convergence rate and is an inadequate approximation. In this study, we use a parameterized orthonormal basis with an adjustable parameter to match the oscillating behavior of the chirp to approximate linear chirps using the partial sums of a generalized Fourier series known as fractional Fourier series, which gives the best approximation with only a small number of fractional Fourier coefficients. We used the fractional Fourier transform to compute the fractional Fourier coefficients at sample points. Additionally, we discover that the fractional parameter has the best value at which fractional Fourier coefficients of zero degrees have the most considerable magnitude, leading to the rapid decline of fractional Fourier coefficients of high degrees. Furthermore, fractional Fourier series approximation with optimal fractional parameters provides the minimum mean square error over the fractional Fourier parameter domain.

Negatively Competitive Incoherent Feedforward Loops Mitigate Winner-Take-All Resource Competition.

The effects of host resource limitations on the function of synthetic gene circuits have gained significant attention over the past years. Hosts, having evolved resource capacities optimal for their own genome, have been repeatedly demonstrated to suffer from the added burden of synthetic genetic programs, which may in return pose deleterious effects on the circuit's function. Three resource controller archetypes have been proposed previously to mitigate resource distribution problems in dynamic circuits: the local controller, the global controller, and a "negatively competitive" regulatory (NCR) controller that utilizes synthetic competition to combat resource competition. The dynamics of negative feedback forms of these controllers have been previously investigated, and here we extend the analysis of these resource allocation strategies to the incoherent feedforward loop (iFFL) topology. We demonstrate that the three iFFL controllers can attenuate Winner-Take-All resource competition between two bistable switches. We uncover that the parameters associated with the synthetic competition in the NCR iFFL controller are paramount to its increased efficacy over the local controller type, while the global controllers demonstrate to be relatively ineffectual. Interestingly, unlike the negative feedback counterpart topologies, iFFL controllers exhibit a unique coupling of switch activation thresholds which we term the "coactivation threshold shift" effect. Finally, we demonstrate that a nearly fully orthogonal set of bistable switches could be achieved by pairing an NCR controller with an appropriate level of controller resource consumption.

Solving inverse non-linear fractional differential equations by generalized Chelyshkov wavelets

The purpose of this research is to employ a method involving Chelyshkov wavelets to construct a numerical solution to the inverse problem of determining the right-hand side function of a non-linear fractional differential equation by utilizing over-measured data. The novelty of this research is that this type of inverse problem is studied by Chelyshkov wavelets. Firstly, the problem is reduced into a system of algebraic equations with an unknown right-hand side by means of the orthonormal base of Chelyshkov wavelets. Secondly, by choosing suitable nodes, this system is transformed into a homogenous system of algebraic equations. The solution of the homogenous system allows us to determine the coefficients of the bases vectors for the solution of the non-linear fractional differential equation. In the final step, the right-hand side is obtained by substituting the constructed solution into a non-linear fractional differential equation. The presented examples illustrate that the numerical solution, obtained by this method, is remarkably close to the exact solution.

Mutually unbiased frames

In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and normalized vectors. It encompasses the already existing notions of unbiasedness for orthonormal bases, regular simplices, equiangular tight frames, positive operator valued measure, and also includes symmetric informationally complete quantum measurements. After introducing the tool, its power is shown by finding the following results about the last mentioned class of constellations: (i) real fiducial states do not exist in any even dimension, and (ii) unknown d-dimensional fiducial states are parameterized, a priori, with roughly 3d/2 real variables only, without loss of generality. Furthermore, multi-parametric families of pure quantum states having minimum uncertainty with regard to several choices of d+1 orthonormal bases are shown, in every dimension d. These last families contain all existing fiducial states in every finite dimension, and the bases include maximal sets of d+1 mutually unbiased bases, when d is a prime number.

Harmonic and anharmonic oscillators on the Heisenberg group

In this article, we present a notion of the harmonic oscillator on the Heisenberg group Hn, which, under a few reasonable assumptions, forms the natural analog of a harmonic oscillator on Rn: a negative sum of squares of operators on Hn, which is essentially self-adjoint on L2(Hn) with purely discrete spectrum and whose eigenvectors are Schwartz functions forming an orthonormal basis of L2(Hn). The differential operator in question is determined by the Dynin–Folland group—a stratified nilpotent Lie group—and its generic unitary irreducible representations, which naturally act on L2(Hn). As in the Euclidean case, our notion of harmonic oscillator on Hn extends to a whole class of so-called anharmonic oscillators, which involve left-invariant derivatives and polynomial potentials of order greater or equal 2. These operators, which enjoy similar properties as the harmonic oscillator, are in one-to-one correspondence with positive Rockland operators on the Dynin–Folland group. The latter part of this article is concerned with spectral multipliers. We obtain useful Lp-Lq-estimates for a large class of spectral multipliers of the sub-Laplacian LHn,2 and, in fact, of generic Rockland operators on graded groups. As a by-product, we obtain explicit hypoelliptic heat semigroup estimates and recover the continuous Sobolev embeddings on graded groups, provided 1 < p ≤ 2 ≤ q < ∞.