Conjugate Pairs Research Articles

Decision-Oriented Two-Parameter Fisher Information Sensitivity Using Symplectic Decomposition

The eigenvalues and eigenvectors of the Fisher Information Matrix (FIM) can reveal the most and least sensitive directions of a system and it has wide application across science and engineering. We present a symplectic variant of the eigenvalue decomposition for the FIM and extract the sensitivity information with respect to two-parameter conjugate pairs. The symplectic approach decomposes the FIM onto an even-dimensional symplectic basis. This symplectic structure can reveal additional sensitivity information between two-parameter pairs, otherwise concealed in the orthogonal basis from the standard eigenvalue decomposition. The proposed sensitivity approach can be applied to naturally paired two-parameter distribution parameters, or a decision-oriented pairing via regrouping or re-parameterization of the FIM. It can be used in tandem with the standard eigenvalue decomposition and offer additional insights into the sensitivity analysis at negligible extra cost. Supplementary materials for this article are available online.

Buffer Solutions of known Ionic Strength

pH buffer solutions are those in which minimal pH variations occur when moderate amounts of strong acids or bases are added or diluted. The most common buffers are those used in the intermediate pH zone and are made up of an acid-base conjugate pair (HA/A-), with Ca and Cb as analytical concentrations of acid and base respectively. The buffer capacity of a solution is the measure of its effectiveness in preserving the pH value when adding an acid or a base. Three new programs working under the Windows 10 environment have been developed. The first one, the BUFFER program, allows to prepare buffers of known ionic strength without the need of adding an inert electrolyte, calculating the pH and buffering capacity. On the other hand, the BRÖMSTED method allows calculating the pH of conjugated acid-base systems applying the Newton-Raphson method. In this work two more programs are described, one applying the Brömsted method to monoprotic acids and another new one to diprotic acids.

Experimental study of the nonlinear distortion of non-reciprocal transmission in nonlinear parity-time symmetric LC resonators

At the broken phase of a parity-time (PT)-symmetric dimer where the coupling is weak, the eigenfrequencies are complex conjugate pairs with non-vanishing real parts, leading to oscillation magnitudes with an exponentially growing mode and an exponentially decaying mode. If the large oscillation magnitudes are clamped due to the nonlinear gain of the PT-symmetric dimer, the exponentially growing mode eventually enters the stable oscillations. It was recently demonstrated that such a phenomenon can be utilized for non-reciprocal transmission. The distortion induced by nonlinearity is critical for the non-reciprocal transmission. Here, we experimentally explore the nonlinear distortion in PT-symmetric inductor–capacitor resonators by utilizing discrete components on a printed circuit board. It demonstrates that the IIP3 (the output-referred third-order intercept point) can achieve as high as 38.7 dBm at the frequency of 14.5 MHz corresponding to the maximum forward transmission. The noise figure of the system is measured to be about 11.25 dB.

Efficient Recursive Convolution Schemes for the Accurate FDTD Modeling of Graphene Interband Contribution via Complex Conjugate Pairs

The accurate time-domain numerical modeling of graphene surface conductivity due to the interband contribution is presented in this paper. Initially, the lower frequency limit for the interband conductivity inclusion is studied, highlighting that even in the far-infrared regime it should not be ignored. Then, a precise vector-fitting technique is utilized to decompose the analytical conductivity function into complex conjugate pairs. Every pair is connected to a complex surface current, while the frequency dispersion is evaluated via a proper recursive convolution scheme. Finally, a straightforward algebraic manipulation is conducted to eliminate the complex terms from the update equations of the electromagnetic components, concerning the finite-difference time-domain (FDTD) algorithm. The precision and efficiency of the proposed methodology are thoroughly validated by comparing the numerically extracted surface wave propagation characteristics to those obtained in terms of analytical expressions.

Extension of the Goldstone and the Englert-Brout-Higgs mechanisms to non-Hermitian theories

We discuss the extension of the Goldstone and Englert-Brout-Higgs mechanisms to non-Hermitian Hamiltonians that possess an antilinear PT symmetry. We study a model due to Alexandre, Ellis, Millington and Seynaeve and show that for the spontaneous breakdown of a continuous global symmetry we obtain a massless Goldstone boson in all three of the antilinear symmetry realizations: eigenvalues real, eigenvalues in complex conjugate pairs, and eigenvalues real but eigenvectors incomplete. In this last case we show that it is possible for the Goldstone boson mode to be a zero-norm state. For the breakdown of a continuous local symmetry the gauge boson acquires a non-zero mass by the Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the fact that the continuous local symmetry has been spontaneously broken, the gauge boson remains massless.

Degeneracies and symmetry breaking in pseudo-Hermitian matrices

Real eigenvalues of pseudo-Hermitian matrices, such as real matrices and $\mathcal{PT}\ensuremath{-}\mathrm{symmetric}$ matrices, frequently split into complex conjugate pairs. This is accompanied by the breaking of certain symmetries of the eigenvectors and, typically, also a drastic change in the behavior of the system. In this paper, we classify the eigenspace of pseudo-Hermitian matrices and show that such symmetry breaking occurs if and only if eigenvalues of opposite kinds collide on the real axis of the complex eigenvalue plane. This enables a classification of the disconnected regions in parameter space where all eigenvalues are real---which correspond, physically, to the stable phases of the system. These disconnected regions are surrounded by exceptional surfaces, which comprise all the real-valued exceptional points of pseudo-Hermitian matrices. The exceptional surfaces, together with the diabolic points created by their intersections, comprise all points of pseudo-Hermiticity breaking. In particular, this clarifies that the degeneracy involved in symmetry breaking is not necessarily an exceptional point. We also discuss how our study relates to conserved quantities and derive the conditions for when degeneracies caused by external symmetries are susceptible to thresholdless pseudo-Hermiticity breaking. We illustrate our results with examples from photonics, condensed matter physics, and mechanics.

On a generalization of time-accurate and highly-stable explicit operators for stiff problems

In this work we propose a generalization of the family of Time-Accurate and highly-Stable Explicit (TASE) operators recently introduced by Calvo, Montijano and Rández (2021). In this family the TASE operator of order p depends on p free real parameters. Here we consider operators that can also be defined by complex parameters occurring in conjugate pairs. Despite this choice the calculations continue to be done only in real arithmetic, thus not burdening the computational cost of the previous version of the family. Conversely, this generalization leads to improve both the accuracy and stability properties of explicit Runge-Kutta schemes supplemented with TASE operators. Numerical experiments showing the competitiveness of the methods proposed in this paper with respect to classical integrators for stiff problems are also presented.

Ultralinear 140-GHz FMCW signal generation with optical parametric wideband frequency modulation enabling 1-mm range resolution.

We demonstrate ultralinear and ultrawideband frequency-modulated continuous-wave (FMCW) signal generation using an optical parametric wideband frequency modulation (OPWBFM) method. The OPWBFM method optically expands the bandwidths of FMCW signals beyond the electrical bandwidths of optical modulators via a cascaded four-wave mixing (FWM) process. Compared to the conventional direct modulation approach, the OPWBFM method simultaneously achieves high linearity and a short measurement time of the frequency sweep. On the other hand, it is also known that the OPWBFM method expands the phase noise of idlers as well as their bandwidths if an input conjugate pair has different phase noise. To avoid this phase noise expansion, it is crucial to synchronize the phase of an input complex conjugate pair of an FMCW signal using an optical frequency comb. For demonstration, we successfully generated an ultralinear 140-GHz FMCW signal by using the OPWBFM method. Moreover, we employ a frequency comb in the conjugate pair generation process, leading to the mitigation of phase noise expansion. By using a 140-GHz FMCW signal, we achieve a range resolution of ∼1 mm through fiber-based distance measurement. The results show the feasibility of an ultralinear and ultrawideband FMCW system with a sufficiently short measurement time.

Research on the Sustainable Renewal of Architectural Heritage Sites from the Perspective of Extenics—Using the Example of Tulou Renovations in LantianVillage, Longyan City

Fujian Tulous in China are important international architectural heritage sites that reflect precious human cultural heritage. Currently, only a small number of Tulou buildings have been listed as world cultural heritage sites, resulting in a lack of attention and financial support for most Tulou buildings. Thus, it is difficult to effectively renovate and repair Tulou buildings to adapt to modern life, and therefore they are facing the severe challenge of abandonment and desolation. Due to the special conditions of Tulou buildings, there are significant limitations in renovation and repair work, with a number of problems such as the lack of innovative renovations. Therefore, through a problem model analysis of a design system for Tulou renovations, in this study, we adopt the methods of divergent tree, conjugate pair, correlative net, implied system, and split-merge chain analyses in extenics to carry out extension transformation and solve the problem and we verify its feasibility using the example of the Tulou renovation projects in Lantian Village, Longyan City. We explore an innovative methodology for scientific renovation of Tulou buildings, and we establish a design system for Tulou building renovations that enriches and supplements original renovation methods; thus, we provide a basis for the repair and reuse of Tulou buildings, to extend their service life and to realize the sustainable development of Tulou buildings. The research results show that extenics can be implemented in innovative renovations of Tulou buildings, and it is concluded that the essence of achieving sustainable renewal in Tulou building renovations is to solve contradictory problems, including contradictions in conditions, objectives, and design. This study verifies the possibility of applying extenics in the design of Tulou building renovations, makes corresponding contributions to the application of extension methods in the renovation and renewal of Tulou buildings, and also contributes to the renovation, renewal, and protection of other types of architectural heritage sites.

Monocular metasurface camera for passive single-shot 4D imaging

It is a grand challenge for an imaging system to simultaneously obtain multi-dimensional light field information, such as depth and polarization, of a scene for the accurate perception of the physical world. However, such a task would conventionally require bulky optical components, time-domain multiplexing, and active laser illumination. Here, we experimentally demonstrate a compact monocular camera equipped with a single-layer metalens that can capture a 4D image, including 2D all-in-focus intensity, depth, and polarization of a target scene in a single shot under ambient illumination conditions. The metalens is optimized to have a conjugate pair of polarization-decoupled rotating single-helix point-spread functions that are strongly dependent on the depth of the target object. Combined with a straightforward, physically interpretable image retrieval algorithm, the camera can simultaneously perform high-accuracy depth sensing and high-fidelity polarization imaging over an extended depth of field for both static and dynamic scenes in both indoor and outdoor environments. Such a compact multi-dimensional imaging system could enable new applications in diverse areas ranging from machine vision to microscopy.

Mapping classes are almost determined by their finite quotient actions

Given any connected compact orientable surface, a pair of mapping classes is said to be procongruently conjugate if it induces a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For example, this occurs if they induce conjugate outer automorphisms on every characteristic finite quotient of the fundamental group. In this paper, it is shown that every procongruent conjugacy class of mapping classes, as a subset of the surface mapping class group, is the disjoint union of at most finitely many conjugacy classes of mapping classes. For any pseudo-Anosov mapping class of a connected closed orientable surface, several topological features are confirmed to depend only on the procongurent conjugacy class of the mapping class, including the stretching factor, the topological type of the prong singularities, the transverse orientability of the invariant foliations, and the isomorphism type of the symplectic Floer homology.

Euler fluid in two dimensions: Statistical approach.

We use Kirchhoff's vortex formulation of 2D Euler fluid equationsto explore the equilibrium state to which a 2D incompressible fluid relaxes from an arbitrary initial flow. The vortex dynamics obeys Hamilton's equationsof motion with x and y coordinates of the vortex position forming a conjugate pair. A state of fluid can, therefore, be expressed in terms of an infinite number of infinitesimal vortices. If the vortex dynamics is mixing, the final equilibrium state of the fluid should correspond to the maximum of Boltzmann entropy, with the constraint that all the Casimir invariants of the fluid must be preserved. This is the fundamental assumption of Lynden-Bell's theory of collisionless relaxation. In this paper, we will present a Monte Carlo method which allows us to find the maximum entropy state of the fluid starting from an arbitrary initial condition. We will then compare this prediction with the results of molecular dynamics simulation and demonstrate that the final state to which the fluid evolves is, actually, very different from that corresponding to the maximum of entropy. This indicates that the mixing assumption is not correct. We will then present a different approach based on core-halo distribution which allows us to accurately predict the final state to which the fluid will relax, starting from an arbitrary initial condition.

The Conjusome—A Transient Organelle Linking Genome Rearrangements in the Parental and Developing Macronuclei

The conjusome plays an important role in the conjugation events that occur in Tetrahymena thermophila. The conjusome appears in the anterior of conjugant pairs during the early stages of new macronuclei (anlagen) development. It lacks a membrane, and is composed of a network of fibrous, electron dense material, containing background cytoplasm and ribosomes. Several proteins localize to this organelle, including Pdd1p, a chromodomain protein that participates in the formation of chromatin-containing structures in developing macronuclear anlagen, and is associated with the elimination of specific germ-line sequences from developing macronuclei. Conjugants lacking the PDD1 allele in the parental macronucleus do not show Pdd1p antibody staining in conjusomes. Investigations were performed using mutant cell lines, uniparental cytogamy and drug treatment, and show that the conjusome appears to be dependent on parental macronuclei condensation, and is a transitory organelle that traffics nuclear determinants from the parental macronucleus to the developing anlagen. These data, taken together with Pdd1p knockout experiments, suggest the conjusome is involved in the epigenetic phenomena that occur during conjugation and sexual reorganization. This is likely a conserved organelle. Conjusome-like structures were also observed in another Ciliate, Stylonichia. In general, conjusomes have features that resemble germ line P-granules.

Rational solitons for non-local Hirota equations: Robustness and cascading instability

The Hirota equation is a higher-order non-linear Schrödinger equation by incorporating third-order dispersion. Two pairs of non-local Hirota equations are studied. One is a parity transformed conjugate pair, and the other is a conjugate PT-symmetric pair. For the first pair, rational solitons are derived by the Darboux transformation, and are shown computationally to exhibit robust propagation properties. These rational solitons can exhibit both elastic and inelastic interactions. One particular case of an elastic collision between dark and “anti-dark” solitons is demonstrated. For the second pair, a “cascading mechanism” illustrating the growth of higher-order sidebands is elucidated explicitly for these non-local, conjugate PT-symmetric equations. These mechanisms provide a theoretical confirmation of the initial amplification phase of the growth-and-decay cycles of breathers. Such repeated patterns will serve as a manifestation of the classical Fermi-Pasta-Ulam-Tsingou recurrence.

Three-dimensional growth simulation of swellable soft materials based on CS-FEM

In this paper, a three-dimensional numerical framework for modeling growth of swellable soft materials at large deformation is established based on the cell-based smooth finite element method, and the multiplicative decomposition scheme of deformation gradient is given. The second P-K stress and Green's strain tensor are selected as work conjugate pairs, and the corresponding mathematical expressions of stiffness matrix and geometric stiffness matrix are derived. The numerical method is implemented based on Matlab platform, and the isotropic and anisotropic growth behaviors of swellable soft materials are simulated respectively. The results show that anisotropic growth will inhibit the deformation of expandable soft materials compared with isotropic growth. The simulation results are compared with the calculation results in the existing literature. The comparison results show that the characteristics and the morphological mode are in good agreement, which proves the effectiveness of the numerical framework in simulating the growth behavior of expandable soft materials at large deformation, and is able to reveal the mechanical mechanism of the plant growth phenomenon in nature.

Escape Mechanism with Shallow Ramp and Décollements in Southwest Taiwan

The escape structure in southwest Taiwan has long been discussed. The counterclockwise rotation in GPS ground motion is argued to be evidence of tectonic escape. However, tectonic escape events worldwide are always bounded by lithosphere-cutting strike-slip faults at the boundary of the lithospheric-scale rotating block, and these have not observed in Taiwan. In this study, we propose that the escape structure in southwest Taiwan is a thin-skinned deformation due to the open boundary in the continental slope, the ramp, and the shallow décollements. We tested this shallow escape hypothesis using a 3D numerical simulation with elastoplastic rheology. We found that a conjugate pair of forethrust and backthrust developed above the ramp. The strike and location of the forethrust mainly follow the ramp. However, the strike of the forethrust rotates perpendicular to the open boundary when in proximity. From north to south, the strike of the forethrust transfers from NNE to NE, and the deformation transfers from thrust to dextral thrust. This leads to a counterclockwise rotation in ground motion, which matches the GPS observation. This research provides a different explanation of the mechanism for the escape structure in southwest Taiwan.

The boundary between the Prime Gaps and the Interchangeable be-tween a Quantum Mechanics System & a Classical Chaos One

With reference to my last paper, this author found that there are always some discrete quantum constants like the Planck’s one laying between the real and the imaginary parts of the Riemann function (I.e when we fix or group the Riemann Zeros one by one and make the real part to be varying.) or the quantum mechan-ics system. Hence, the Riemann Zeta Zeros give us the discrete quantum con-stants together with the given discrete quantum constants may give you the wanted Riemann Zeta Zeros. This phenomenon may be explained by Quantum Riemann Zeta Epiphancy. In the present paper, this author tries to fix or group the real part of the Riemann Zeros and make the imaginary part of the Riemann Zeros to be varying from one to the tenth one or resulting a chaos like graph. Thus, it seems that we may interchange a quantum mechanics system with the classical chaos one by just varying one variable and fix another one (such as varying real part of the Riemann function’s zeors from 0.1 to 1 and fix the imaginary part of the Rie-mann zeros one group after one another). Certainly, the vice versa for the fix of the real part (one group after another) with different Riemann Zeros may also help. In other words, the quantum mechanics system is just the conjugate pair (varying x and fixed y variables) of the classical chaos one or the vice versa is al-so true (fixed x and varying y variables) & put back into the Zeta functions etc. Hence, it is a method (or transformation) for us so that we may easily to make a interchange between these two systems by only adjusting or selecting which vari-ables to be varying or fixed. Then we may turn the classical chaos system into a quantum mechanics one or the vice versa. This implies that it may be possible to apply controllable quantum systems for the investigation of our natural aspects. Finally, this author have also shown that there is a boundary laying be-tween the prime gap through the elementary calculation and mathematical deriva-tion without using the computer software etc. It is thus a contradiction to the fact that there is no bound for the prime gap. This tells us that there is a need to have a shift from zero to 0.5 for the real part of the Riemann Zeta Zeros as what this author has mentioned in my last paper etc.

Multitude of exceptional points in van der Waals magnets

Several works have recently addressed the emergence of exceptional points (EPs), i.e., degeneracies of non-Hermitian Hamiltonians, in the long-wavelength dynamics of coupled magnetic systems. Here, by focusing on the driven magnetization dynamics of a van der Waals ferromagnetic bilayer, we show that exceptional points can appear over extended portions of the first Brillouin zone as well. Furthermore, we demonstrate that the effective non-Hermitian magnon Hamiltonian, whose eigenvalues are purely real or come in complex conjugate pairs, respects an unusual wavevector-dependent pseudo-Hermiticity. Finally, for both armchair and zigzag nanoribbon geometries, we discuss both the complex and purely real spectra of the topological edge states and their experimental implications.

High Crosswind-Tolerance Airplane by Adjusting Dihedral Angle and Vertical Tail Volume

The present authors proposed the “quasi-neutral dihedral-effect and directional-stability” (QNDD) airplane, which is a modified neutral dihedral-effect and directional-stability (NDD) airplane for its realization. The QNDD airplane provides a superior tolerance to crosswinds compared to a conventional one by modifying geometric properties: dihedral angle of a main wing and vertical tail volume to achieve small values of and , which results in a small effect of crosswind gust on an airplane’s attitude. While the QNDD airplane yields small and , the NDD airplane yields and equal to zero. When the changes of and due to the wing deformations are considered, the NDD airplane is ideal, while the QNDD airplane is more practical. The QNDD airplane had four real eigenvalues rather than classically two reals and one pair of complex conjugates. The eigenvalues seen in a conventional airplane as a Dutch roll mode were seen to be reals in a QNDD airplane, and the mode was overdamped, which is different from a set of eigenvalues of a conventional one. From the Bode diagrams and unsteady simulations with a discrete gust applied as a crosswind, it was confirmed that the QNDD airplane had a higher tolerance to the crosswind.

Conjugator length in Thompson's groups

AbstractWe prove that Thompson's group has quadratic conjugator length function. That is, for any two conjugate elements of of length or less, there exists an element of of length that conjugates one to the other. Moreover, there exist conjugate pairs of elements of of length at most such that the shortest conjugator between them has length . This latter statement holds for and as well.