Conjugate Pairs Research Articles

AN OPENING OF A NEW HORIZON IN THE THEORY OF QUADRATIC EQUATION : PURE AND PSEUDO QUADRATIC EQUATION – A NEW CONCEPT

In this paper, the author has opened a new horizon in the theory of quadratic equations. The author proved that the value of x which satisfies the quadratic equation cannot be the only criteria to designate as the root or roots of an equation. The author has developed a new mathematical concept of the dimension of a number. By introducing the concept of the dimension of number the author structured the general form of a quadratic equation into two forms: 1) Pure quadratic equation and 2) Pseudo quadratic equation. First of all the author defined the pure and pseudo quadratic equations. In the case of pure quadratic equation ax^2+bx+c=0 , the root of the equation will be a two-dimensional number having one root only while in the case of pseudo quadratic equation ax^2+bx+c=0, the root of the equation will be a one-dimensional number having two roots only. The author proved that all pseudo quadratic equation is factorizable but all factorizable quadratic equation is not a pseudo quadratic equation. The author begs to differ from the conventional theorem: “A quadratic equation has two and only two roots.” By introducing the concept that any quadratic surd is a two-dimensional number, the author developed a new theorem: “In a quadratic equation with rational coefficients, irrational roots cannot occur in conjugate pairs” and proved it. Any form of quadratic equation ax^2+bx+c=0, can be solved by the application of the ‘Theory of Dynamics of Numbers’ even if the discriminant b^2-4ac<0 in real number only without introducing the concept of an imaginary number. Therefore, the question of imaginary roots does not arise in the method of solution of any quadratic equation

Data-Driven Constitutive Modeling via Conjugate Pairs and Response Functions

Response functions completely define the constitutive equations for a hyperelastic material. A strain measure providing an orthogonal stress response, grants response functions directly from experimental curves. One of these strain measures is the Laplace stretch based on QR-decomposition of the deformation gradient. Such a recovery of response functions from experimental data fits the paradigm of data-driven modeling. The set of independent conjugate stress–strain base pairs were proposed as a simple alternative for constitutive modeling and thus might be efficient for data-driven modeling. In the present paper we explore applicability of the conjugate pairs approach for data-driven modeling. The analysis is based on representation of the conjugate pairs in terms of the response functions due to the Laplace stretch. Our analysis shows that one can not guarantee independence of these pairs except in the case of infinitesimal strain.

Tree-level unitarity, causality, and higher-order Lorentz and CPT violation

Higher-order effects of CPT and Lorentz violation within the SME effective framework including Myers-Pospelov dimension-five operator terms are studied. The model is canonically quantized by giving special attention to the arising of indefinite-metric states or ghosts in an indefinite Fock space. As is well-known, without a perturbative treatment that avoids the propagation of ghost modes or any other approximation, one has to face the question of whether unitarity and microcausality are preserved. In this work, we study both possible issues. We found that microcausality is preserved due to the cancellation of residues occurring in pairs or conjugate pairs when they become complex. Also, by using the Lee-Wick prescription, we prove that the $S$ matrix can be defined as perturbatively unitary for tree-level $2\to 2$ processes with an internal fermion line.

Arithmetical functions associated with conjugate pairs of sets under regular convolutions

Two subsets P and Q of the set of positive integers is said to form a conjugate pair if each positive integer n possesses a unique factorization of the form n = ab, a ∈ P, b ∈ Q. In this paper we generalize conjugate pairs of sets to the setting of regular convolutions and study associated arithmetical functions. Particular attention is paid to arithmetical functions associated with k-free integers and k-th powers under regular convolution.

Dual structures for the additive DEA model

We show that the weighted additive DEA score (WA) for the additive DEA model is simultaneously the (dual) support function for a translation of the DEA technology and the gauge of the dual polar set of the translated technology. Those results are used: to show that WA and the indicator function for the translated technology form a dual conjugate pair; to show that WA and the translated technology’s gauge function form a dual polar pair; to develop a simple but exact link between WA and the profit function for the DEA technology; to show that WA and the directional distance function form a dual polar pair; and to develop an exact decomposition of profit inefficiency that extends the Cooper, Pastor, Aparicio, and Borras (2011) and Aparicio et al. (2016) decomposition.

Plasmonic photonic biosensor: in situ detection and quantification of SARS-CoV-2 particles.

We conceptualized and numerically investigated a photonic crystal fiber (PCF)-based surface plasmon resonance (SPR) sensor for rapid detection and quantification of novel coronavirus. The plasmonic gold-based optical sensor permits three different ways to quantify the virus concentrations inside patient's body based on different ligand-analyte conjugate pairs. This photonic biosensor demonstrates viable detections of SARS-CoV-2 spike receptor-binding-domain (RBD), mutated viral single-stranded ribonucleic acid (RNA) and human monoclonal antibody immunoglobulin G (IgG). A marquise-shaped core is introduced to facilitate efficient light-tailoring. Analytes are dissolved in sterile phosphate buffered saline (PBS) and surfaced on the plasmonic metal layer for realizing detection. The 1-pyrene butyric acid n-hydroxy-succinimide ester is numerically used to immobilize the analytes on the sensing interface. Using the finite element method (FEM), the proposed sensor is studied critically and optimized for the refractive index (RI) range from 1.3348-1.3576, since the target analytes RIs fluctuate within this range depending on the severity of the viral infection. The polarization-dependent sensor exhibits dominant sensing attributes for x-polarized mode, where it shows the average wavelength sensitivities of 2,009 nm/RIU, 2,745 nm/RIU and 1,984 nm/RIU for analytes: spike RBD, extracted coronavirus RNA and antibody IgG, respectively. The corresponding median amplitude sensitivities are 135 RIU-1, 196 RIU-1 and 140 RIU-1, respectively. The maximum sensor resolution and figure of merit are found 2.53 × 10-5 RIU and 101 RIU-1, respectively for viral RNA detection. Also, a significant limit of detection (LOD) of 6.42 × 10-9 RIU2/nm is obtained. Considering modern bioassays, the proposed compact photonic sensor will be well-suited for rapid point-of-care COVID testing.

Design of piecewise deformed elliptical gear with closed pitch curve and its conjugate pair

The family of elliptical gears with closed pitch curves includes elliptical gears and deformed elliptical gears. Elliptical gears are not only the most widely used non-circular gears but also the research hotspot in the field of non-uniform transmission. However, adjusting the shape of its pitch curve is difficult, which seriously restricts its popularization. This article proposes a piecewise deformed elliptical gear with a closed pitch curve, which realizes the unity of all kinds of elliptical gears in the mathematical model, makes the shape of the pitch curve and gear ratio easier to adjust, and expands the connotation and application scope of the family of elliptical gears. Moreover, the design approach of transmission with piecewise deformed elliptical gear was studied, and the inspection method of transmission performance was provided. The internal relationship between piecewise deformed elliptical gears and the family of elliptical gears was then analyzed. Finally, the method of computer-aided design (CAD) system developed was also provided. Additionally, a case was provided to verify the above theories. The designed conjugate pair can realize a correct meshing and adjust the gear ratio more flexibly, laying a theoretical foundation to expand the application field of non-uniform transmission.

Superharmonic instability of stokes waves

AbstractA stability of nearly limiting Stokes waves to superharmonic perturbations is considered numerically in approximation of an infinite depth. Investigation of the stability properties can give one an insight into the evolution of the Stokes wave. The new, previously inaccessible branches of superharmonic instability were investigated. Our numerical simulations suggest that eigenvalues of linearized dynamical equations, corresponding to the unstable modes, appear as a result of a collision of a pair of purely imaginary eigenvalues at the origin, and a subsequent appearance of a pair of purely real eigenvalues: a positive and a negative one that are symmetric with respect to zero. Complex conjugate pairs of purely imaginary eigenvalues correspond to stable modes, and as the steepness of the underlying Stokes wave grows, the pairs move toward the origin along the imaginary axis. Moreover, when studying the eigenvalues of linearized dynamical equations we find that as the steepness of the Stokes wave grows, the real eigenvalues follow a universal scaling law, that can be approximated by a power law. The asymptotic power law behavior of this dependence for instability of Stokes waves close to the limiting one is proposed. Surface elevation profiles for several unstable eigenmodes are made available through http://stokeswave.org website.

Mathcal{N} = 1 conformal dualities from unoriented chiral quivers

We study various orientifold projections of 4d mathcal{N} = 1 toric gauge theories, associated with CY singularities known as La,b,a/ℤ2, with a + b even. We obtain superconformal chiral theories that have the same central charge, anomalies and superconformal index, whereas they were different before the orientifold. Some of these projections are implemented by a novel type of orientifold without fixed loci, known as glide orientifold. We claim that these theories flow to the same conformal manifold, and they are connected by quadratic exactly marginal deformations. The latter can be written in terms of conjugate pairs of bifundamental fields of R-charge one, generalizing previous results for unoriented non-chiral theories.

On Quantum Entropy.

Quantum physics, despite its intrinsically probabilistic nature, lacks a definition of entropy fully accounting for the randomness of a quantum state. For example, von Neumann entropy quantifies only the incomplete specification of a quantum state and does not quantify the probabilistic distribution of its observables; it trivially vanishes for pure quantum states. We propose a quantum entropy that quantifies the randomness of a pure quantum state via a conjugate pair of observables/operators forming the quantum phase space. The entropy is dimensionless, it is a relativistic scalar, it is invariant under canonical transformations and under CPT transformations, and its minimum has been established by the entropic uncertainty principle. We expand the entropy to also include mixed states. We show that the entropy is monotonically increasing during a time evolution of coherent states under a Dirac Hamiltonian. However, in a mathematical scenario, when two fermions come closer to each other, each evolving as a coherent state, the total system’s entropy oscillates due to the increasing spatial entanglement. We hypothesize an entropy law governing physical systems whereby the entropy of a closed system never decreases, implying a time arrow for particle physics. We then explore the possibility that as the oscillations of the entropy must by the law be barred in quantum physics, potential entropy oscillations trigger annihilation and creation of particles.

On finite deformation of hyperelastic shell-type structures: Cartesian formulation-based VDQ approach

In this article, a numerical approach is presented for the large deformation analysis of shell-type structures made of Neo-Hookean and Kirchhoff–St Venant materials within the framework of the seven-parameter shell theory. Work conjugate pair of the second Piola–Kirchhoff stress and Green–Lagrange strain tensors are taken for the macroscopic stress and strain measures in this total Lagrangian formulation. By defining displacement vector, deformation gradient and stress tensor in the Cartesian coordinate system, and using the chain rule for taking derivative of tensors, the complications of using the curvilinear coordinate system are bypassed. The variational differential quadrature (VDQ) technique as an effective numerical solution method is applied to obtain the weak form of governing equations. Being locking-free, simple implementation, computational efficiency and fast convergence rate are the main features of the proposed numerical approach. A number of well-known benchmark problems are solved in order to reveal the accuracy and efficiency of the method. It is shown that this approach is able to predict the large deformations of hyperelastic shell-type structures in an efficient way.

Flexible Scanning Method by Integrating Laser Line Sensors with Articulated Arm Coordinate Measuring Machines

Measuring and reconstructing the shape of workpieces have been considered as a fundamental step in both reverse engineering and product quality control. Owing to increasing structural complexity of recent products, measurements from multiple directions are typically required in current scanning techniques. Specifically, the plane structured light can be applied to measure one area of a part at a time, with an additional algorithm required to merge the collected data of each area. Alternatively, the line structured light sensor integrated on CNC machines or CMMs could also realize multi-view measurement. However, the system needs to be repeatedly calibrated at each new direction. This paper presents a flexible scanning method by integrating laser line sensors with articulated arm coordinate measuring machines (AACMM). Since the output of the laser line sensor is 2D raw data in the laser plane, our system model introduces an explicit transformation from the 2D sensor coordinate frame to the 3D base coordinate frame of the AACMM (i.e., the translation and rotation the of the 2D sensor coordinate in the sixth coordinate system of AACMM). To solve the model, the “conjugate pairs” are proposed and identified by measuring a fixed point (e.g., a sphere center). Moreover, a search algorithm is adopted to find the optimal solution, which noticeably boosts the model accuracy. The experimental results show that the error of the system is about 0.2 mm, which is caused by the error of the AACMM, the sensor error and the calibration error. By measuring a complicated part, the proposed system is proved to be flexible and facilitate, with the ability to measure a part expediently from any necessary direction. Furthermore, the proposed calibration method can also be used for robot hand-eye relationship calibration.

Minimal Partitions with a Given s-Core and t-Core

Suppose s and t are coprime positive integers, and let sigma be an s-core partition and tau a t-core partition. In this paper, we consider the set {mathcal {P}}_{sigma ,tau }(n) of partitions of n with s-core sigma and t-core tau . We find the smallest n for which this set is non-empty, and show that for this value of n the partitions in {mathcal {P}}_{sigma ,tau }(n) (which we call (sigma ,tau )-minimal partitions) are in bijection with a certain class of (0, 1)-matrices with s rows and t columns. We then use these results in considering conjugate partitions: we determine exactly when the set {mathcal {P}}_{sigma ,tau }(n) consists of a conjugate pair of partitions, and when {mathcal {P}}_{sigma ,tau }(n) contains a unique self-conjugate partition.

PT-symmetric potentials with imaginary asymptotic saturation

We point out that PT-symmetric potentials \(V_{\mathrm {PT}}(x)\) having imaginary asymptotic saturation, \(V_{\mathrm {PT}}(x=\pm \infty ) =\pm i V_1, V_1 \in {\mathbb {R}}\) are devoid of scattering states and spectral singularity. We show the existence of real (positive and negative) discrete spectrum both with and without complex conjugate pair(s) of eigenvalues (CCPEs). If the eigenstates are arranged in the ascending order of the real part of the discrete eigenvalues, the initial states have few nodes but latter ones oscillate fast. Both real and imaginary parts of \(\psi _n(x)\) vanish asymptotically, and \(|\psi _n(x)|\) are nodeless. For the CCPEs, these are asymmetric and peaking on the left (right) and for real energies these are symmetric and peaking at the origin. For CCPEs \(E_{\pm }\), the eigenstates \(\psi _{\pm }\) follow the interesting property, \(|\psi _+(x)|= N |\psi _-(-x)|, N \in {\mathbb {R}}^+\).

Full and Partial Eigenvalue Placement for Minimum Norm Static Output Feedback Control

The controller design for linear time-invariant state space systems seems to be straightforward and well established. This is not true for static output feedback control, which is still a challenging task. This paper deals with controller design based on eigenvalue assignment. We consider the placement of distinct as well as multiple real eigenvalues or complex conjugate pairs. The desired eigenvalue configurations are characterised in terms of algebraic divisibility of the characteristic polynomial of the closed-loop system. We also consider the problem of partial eigenvalue placement, where not all eigenvalues are fixed by feedback. Degrees of freedom in the controller design are used for the minimization of various matrix norms of the feedback gain matrix.

Quality by design approach for enantioseparation of terbutaline and its sulfate conjugate metabolite for bioanalytical application using supercritical fluid chromatography

Terbutaline is mainly metabolized by sulfoconjugation stereoselectively, favoring its (S)-(+) enantiomer. Reported chiral separations of Terbutaline enantiomers were achieved by various chromatographic methods. However, the simultaneous enantioseparation of Terbutaline and the monosulfate conjugate metabolites was never reported. This study aims at shedding light on the influential factors and interactions leading to successful enantioseparation of Terbutaline and its monosulfate conjugate pairs by Supercritical Fluid Chromatography (SFC) for the first time within a Quality by Design framework using Design of Experiments. The effect of molarity of mobile phase additive, mobile phase flow rate, column temperature and back pressure were evaluated. Compared to previous reports, the response surface interestingly revealed the favorability of high temperature and high flow rate up to 2.25 ml/min for resolution of the two pairs of enantiomers on polysaccharide chiral stationary phase CHIRALPAK IC. In addition, a switch in the elution order of Terbutaline and the sulfate conjugate peak pairs was observed upon elimination of the mobile phase additive where the sulfate conjugate underwent intra-molecular ionic interactions and the change in elution order was only due to TER behavior. The multifactorial interactions would not have been detected with the common one-factor-at-a-time approach during method development, demonstrating the superiority and importance of the Analytical Quality by Design frame in enantioseparation.

Non-Hermitian Spatial Symmetries and Their Stabilized Normal and Exceptional Topological Semimetals.

We study non-Hermitian spatial symmetries-a class of symmetries that have no counterparts in Hermitian systems-and study how normal and exceptional semimetals can be stabilized by these symmetries. Different from internal ones, spatial symmetries act nonlocally in momentum space and enforce global constraints on both band degeneracies and topological quantities at different locations. Inderiving general constraints on band degeneracies and topological invariants, we demonstrate that non-Hermitian spatial symmetries are on an equal footing with, but are essentially different from Hermitian ones. First, we discover the nonlocal Hermitian conjugate pair of exceptional or normal band degeneracies that are enforced by non-Hermitian spatial symmetries. Remarkably, we find that these pairs lead to the symmetry-enforced violation of the Fermion doubling theorem in the long-time limit. Second, with the topological constraints, we unravel that a certain exceptional manifold is only compatible with and stabilized by non-Hermitian spatial symmetries but is intrinsically incompatible with Hermitian spatial symmetries. We illustrate these findings using two three-dimensional models of a non-Hermitian Weyl semimetal and an exceptional unconventional Weyl semimetal. Experimental cold-atom realizations of both models are also proposed.

Four-dimensional computed tomography of the left ventricle, Part I: Motion artifact reduction.

Standard four-dimensional computed tomography (4DCT) cardiac reconstructions typically include spiraling artifacts that depend not only on the motion of the heart but also on the gantry angle range over which the data was acquired. We seek to reduce these motion artifacts and, thereby, improve the accuracy of left ventricular wall positions in 4DCT image series. We use a motion artifact reduction approach (ResyncCT) that is based largely on conjugate pairs of partial angle reconstruction (PAR) images. After identifying the key locations where motion artifacts exist in the uncorrected images, paired subvolumes within the PAR images are analyzed with a modified cross-correlation function in order to estimate 3D velocity and acceleration vectors at these locations. A subsequent motion compensation process (also based on PAR images) includes the creation of a dense motion field, followed by a backproject-and-warp style compensation. The algorithm was tested on a 3D printed phantom, which represents the left ventricle (LV) and on challenging clinical cases corrupted by severe artifacts. The results from our preliminary phantom test as well as from clinical cardiac scans show crisp endocardial edges and resolved double-wall artifacts. When viewed as a temporal series, the corrected images exhibit a much smoother motion of the LV endocardial boundary as compared to the uncorrected images. In addition, quantitative results from our phantom studies show that ResyncCT processing reduces endocardial surface distance errors from 0.9±0.8 to 0.2±0.1mm. The ResyncCT algorithm was shown to be effective in reducing motion artifacts and restoring accurate wall positions. Some perspectives on the use of conjugate-PAR images and on techniques for CT motion artifact reduction more generally are also given.

Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that a conjugate pair of modes loses stability and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing related midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case, we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to \(S^3\). The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.

Efficient world-line-based quantum Monte Carlo method without Hubbard–Stratonovich transformation

By precisely writing down the matrix element of the local Boltzmann operator ({mathrm{e}}^{-tau h}, where h is the Hermitian conjugate pairs of off-diagonal operators), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With the current formula, the Hubbard–Stratonovich transformation is not necessary, accordingly the determinant calculation is not needed, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principles calculations. The current formula also improves the accuracy of the Suzuki–Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be applied to various studies in future.