Quadratic Order Research Articles

Multifold Topological Point with Quadratic Order in Binary Skutterudite Rhodium Triarsenide

Multifold Topological Point with Quadratic Order in Binary Skutterudite Rhodium Triarsenide

Equivalent spring-like system for two nonlinear springs in series: application in metastructure units design

AbstractThe paper deals with the problem of design of unit in auxetic metastructure. The unit is modeled as a two-part spring-like system where each part is with individual stiffness. To overcome the problem of analyzing of each of parts separately, the equivalent spring is suggested to be introduced. In the paper, a method for obtaining the equivalent elastic force of the unit is developed. The method is the generalization of the procedure suggested for substitution of a hard and a soft spring in series with an equivalent one. The nonlinearity of original springs is of quadratic order. As a results, it is obtained that the equivalent elastic force for two equal springs remains of the same type as of the original springs (soft or hard). For two opposite type springs in series with equal coefficients, the equivalent force is soft. The method is applicable for any hard and soft nonlinear springs or spring-like systems. Thus the hexagonal auxetic unit which contains a soft and a hard part in series is analyzed. In the paper, a new analytic method for determination of the frequency of vibration for the unit under action of a constant compression force acting along the unit axis is introduced. The method is applied for units which contain two parts: hard–hard, soft–soft, hard–linear, soft–linear and opposite. The obtained approximate vibration results are compared with numerically obtained ones and show good agreement. The advantage of the method is its simplicity as it does not require the nonlinear equation of motion to be solved.

ANALYSIS OF PATH NONPARAMETRIC TRUNCATED SPLINE MAXIMUM CUBIC ORDER IN BANKING CREDIT OF RISK BEHAVIOR MODEL

Path analysis tests the relationship between variables through cause and effect. The assumption of linearity must be met before conducting further tests on path analysis. If the shape of the relationship is nonlinear and the shape of the curve is unknown, a nonparametric approach is used, one of which is a truncated spline. The purpose of this study is to estimate the function and obtain the best model on the nonparametric truncated spline path of linear, quadratic, and cubic orders with 1 and 2-knot points and determine the significance of the best function estimator in banking credit of risk behavior model through the jackknife resampling method. This study uses secondary data through questionnaires to KPR debtor consumers, as many as 100 respondents. Based on the results of the analysis, it is known that the best-truncated spline nonparametric path model is the quadratic order of 2 knots with a coefficient of determination of 85.50%; the significance of the best-truncated spline nonparametric path estimator shows that all exogenous variables have a significant effect on endogenous variables.

Computing Orientations from the Endomorphism Ring of Supersingular Curves and Applications

This work introduces several algorithms related to the computation of orientations in endomorphism rings of supersingular elliptic curves. This problem is at the heart of several results regarding the security of oriented-curves in isogeny-based cryptography. Under the Deuring correspondence, it can be expressed purely in terms of quaternion and boils down to representing integers by ternary quadratic forms. Our main contribution is to show that there exist efficient algorithms to solve this problem for quadratic orders of discriminant n up to O ( p 4 / 3 ) . Our approach improves upon previous results by increasing this bound from O ( p ) to O ( p 4 / 3 ) and removing some heuristics. We introduce several variants of our new algorithm and provide a careful analysis of their asymptotic running time (without heuristic when it is possible). The best proven asymptotic complexity of one of our variants is O ( n 3 / 4 / p ) in average. The best heuristic variant has a complexity of O ( p 1 / 3 ) for big enough n . We then introduce several results regarding the computation of ideals between oriented orders. The first application of this is a simplification of the known reduction from vectorization to computing the endomorphism ring, removing the assumption on the factorization of the discriminant. As a second application, we relate the problem of computing fixed-degree isogenies between supersingular curves to the problem of computing orientations in endomorphism rings, and we show that for a large range of degree d , our new algorithms improve on the state-of-the-art, and in important special cases, the range of degree d for which there exist a polynomial-time algorithm is increased. In the most special case we consider, when both curves are oriented by a small degree endomorphism, we show heuristically that our techniques allow the computation of isogenies of any degree, assuming they exist.

An Improved Divide-and-Conquer Approach to Estimating Mean Functional, with Application to Average Treatment Effect Estimation

Mean estimation is an important issue in statistical inference and machine learning. We are concerned with estimating mean functional that is a function of several nonparametric functions when there is a large amount of observations. Directly estimating such mean functional through nonparametric smoothing has the complexity of at least a quadratic order of the sample size, which is computationally prohibitive for massive data. The divide-and-conquer approach are thus readily used to alleviate the computational complexity issue, which however imposes a stringent condition on the sample size in each local machine if a locally optimal bandwidth is used. To address this issue, we suggest to use a globally optimal bandwidth in each local machine, which alleviates the restriction on the local sample sizes substantially. We show that the divide-and-conquer approach with a globally optimal bandwidth achieves the estimation efficiency bound as if all observations were pooled together. In terms of computational efficiency, our proposal outperforms the pooled algorithm dramatically. We demonstrate these properties through average treatment effect estimation from both the asymptotic and the non-asymptotic perspectives.

Analysis of models of measurement and correction of volumetric error of three-axis metal-cutting machine tool

The paper considers the improvement of the quality of products of machined production in connection with the quality of machining of various products and, accordingly, with the accuracy of the equipment used. It is noted that geometrical errors, which account for up to 40 % of the total error of a multi-axis, especially three-axis, metal-cutting machine tool, have the greatest influence on the accuracy of product machining. In order to improve the quality of product machining, it is necessary to correct its parameters according to the measurement information received during machine tool operation, for which the methods of mathematical modelling are used. Models of the volumetric error of a three-coordinate metal-cutting machine tool are developed, taking into account the summands of the first (linear) and second (quadratic) order of smallness. A comparative analysis of the developed models is carried out. The results of experiments on measurement of volumetric errors of the STAN S500 complex and their correction on the basis of the nonlinear theoretical model are given. It is found that in the range of angular errors 0–2.5′ the consideration of quadratic terms of the model in addition to linear ones does not lead to a significant reduction of the volumetric error. It is shown that when processing measurement information of multi-axis metal-cutting machines it is sufficient to limit the consideration of components of the volumetric error not higher than the first order of smallness. The research results are useful for the acceptance and periodic control of the volumetric error of metal-cutting machines, as well as for the programme reduction of the volumetric error.

Bosenovae with quadratically-coupled scalars in quantum sensing experiments

Ultralight dark matter (ULDM) particles of mass mϕ ≲ 1 eV can form boson stars in DM halos. Collapse of boson stars leads to explosive bosenova emission of copious relativistic ULDM particles. In this work, we analyze the sensitivity of terrestrial and space-based experiments to detect such relativistic scalar ULDM particles interacting through quadratic couplings with Standard Model constituents, including electrons, photons, and gluons. We highlight key differences with searches for linear ULDM couplings. Screening of ULDM with quadratic couplings near the surface of the Earth can significantly impact observations in terrestrial experiments, motivating future space-based experiments. We demonstrate excellent ULDM discovery prospects, especially for quantum sensors, which can probe quadratic couplings orders below existing constraints by detecting bosenova events in the ULDM mass range 10−23 eV ≲ mϕ ≲ 10−5 eV. We also report updated constraints on quadratic couplings of ULDM in case it comprises cold DM.

Stabilizing massless fields with fluxes in Landau-Ginzburg models

Recent work on flux compactifications suggests that the tadpole constraint generically allows only a limited number of complex structure moduli to become massive, i.e., be stabilized at quadratic order in the spacetime superpotential. We study the effects of higher-order terms systematically around the Fermat point in the 19 Landau-Ginzburg model. This model lives at strong coupling and features no Kähler moduli. We show that indeed massless fields can be stabilized in this fashion. We observe that, depending on the flux, this mechanism is more effective when the number of initially massless fields is large. These findings are compatible with both the tadpole conjecture and the massless Minkowski conjecture. Along the way, we complete the classification of integral flux vectors with small tadpole contribution. Thereby we are closing in on a future complete understanding of all possible flux configurations in the 19 Landau-Ginzburg model.

Observables from the spinning eikonal

We study the classical dynamics of spinning particles using scattering amplitudes and eikonal exponentiation. We show that observables are determined by a simple algorithm. A wealth of complexity arises in perturbation theory as positions, momenta and spins must be iteratively corrected at each order. Even though we restrict ourselves to one-loop computations at quadratic order in spin, nevertheless we encounter and resolve a number of subtle effects. Finally, we clarify the links between our work and various other eikonal approaches to spinning observables.

Cubic B-spline based numerical schemes for delayed time-fractional advection-diffusion equations involving mild singularities

This article presents two efficient layer-adaptive numerical schemes for a class of time-fractional advection-diffusion equations with a large time delay. The fractional derivative of order α with α ∈ (0, 1) is taken in the Caputo sense. The solution to this type of problem generally has a layer due to the mild singularity near the time t = 0. Consequently, the polynomial interpolation discretizing scheme degrades the convergence rate in the case of uniform meshes. In the presence of a singularity, the temporal fractional operator is discretized by employing the L1 technique on a layer-resolving mesh. In contrast, the cubic B-spline collocation method is used in the spatial direction. The convergence analysis and estimation of error are presented for the proposed scheme under reasonable regularity assumptions on the coefficients. The scheme achieves its optimal convergence rate (2 − α) for suitable choice of grading parameter (γ ≥ (2 − α)/α). Furthermore, we modified the proposed scheme by discretizing the fractional operator with the help of the L1-2 technique. The modified scheme gets a quadratic order convergence for γ ≥ 2/α. In addition, we extend the proposed schemes to solve the corresponding semilinear problem. Numerical examples demonstrate the efficiency and applicability of the proposed techniques.

Robust stability and H∞$$ {H}_{\infty } $$ performance evaluation for time‐varying‐delayed uncertain linear systems based on variable‐augmented‐based free‐weighting matrices

AbstractThe problem of the robust stability and performance evaluation of time‐varying‐delayed uncertain linear systems (TVDULSs) is studied in this article. At first, to obtain a modified robust delay‐dependent condition of TVDULSs, a novel augmented Lyapunov‐Krasovskii functional (LKF) is established, and a variable‐augmented‐based free‐weighting matrices (VAFWMs) technique is tailored to evaluate the functional derivative. Compared with the existing robust stability methods, the constructed functional takes fully time delay information of the system into consideration and provides more delay‐cross terms, where the corresponding matrices are used to relax the constraints of linear matrix inequalities in the stability condition. Also, the VAFWMs method fully considers the linear relation of some delayed integral terms in the functional derivative, which avoids the usage of a quadratic order or higher order terms of the time delay. Then, an improved performance analysis criterion of TVDULSs is obtained against the disturbance on the basis of the proposed robust stability condition. The admissible maximal delay upper bounds that the uncertain linear system can tolerate are provided by proposed criterion under a given performance level. At last, two numerical examples are presented to verify the effectiveness and the superiority of the methods obtained.

Finding orientations of supersingular elliptic curves and quaternion orders

AbstractAn oriented supersingular elliptic curve is a curve which is enhanced with the information of an endomorphism. Computing the full endomorphism ring of a supersingular elliptic curve is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is $$\mathfrak {O}$$ O -orientable for a fixed imaginary quadratic order $$\mathfrak {O}$$ O provides non-trivial information towards computing an endomorphism corresponding to the $$\mathfrak {O}$$ O -orientation. We provide explicit algorithms and in-depth complexity analysis. We also consider the question in terms of quaternion algebras. We provide algorithms which compute an embedding of a fixed imaginary quadratic order into a maximal order of the quaternion algebra ramified at p and $$\infty $$ ∞ . We provide code implementations in Sagemath (in Stein et al. Sage Mathematics Software (Version 10.0), The Sage Development Team, http://www.sagemath.org, 2023) which is efficient for finding embeddings of imaginary quadratic orders of discriminants up to O(p), even for cryptographically sized p.

A Replenishment Policy for an Inventory Model with Price-Sensitive Demand with Linear and Quadratic Back Order in a Finite Planning Horizon

A Replenishment Policy for an Inventory Model with Price-Sensitive Demand with Linear and Quadratic Back Order in a Finite Planning Horizon

ZH3: Quadratic Zonal Harmonics

Spherical Harmonics (SH) have been used widely to represent lighting in games and film. While the quadratic (SH3) and higher order spherical harmonics represent irradiance well, they are expensive to store and evaluate, requiring 27 coefficients per sample. Linear SH (SH2), requiring only 12 coefficients, are sometimes used, but they do not represent irradiance signals accurately and can have challenges with negative reconstruction. We introduce a new representation (ZH3) that augments linear SH with just the zonal coefficient of quadratic SH, yielding significant visual improvement with just 15 coefficients, and discuss how solving for a luminance zonal axis can significantly improve reconstruction accuracy and reduce color artifacts. We also discuss how, rather than storing the ZH3 coefficients explicitly, we can hallucinate them from the linear SH, improving reconstruction accuracy over linear SH at minimal extra cost.

Stability of saddles and choices of contour in the Euclidean path integral for linearized gravity: dependence on the DeWitt parameter

Due to the conformal factor problem, the definition of the Euclidean gravitational path integral requires a non-trivial choice of contour. The present work examines a generalization of a recently proposed rule-of-thumb [1] for selecting this contour at quadratic order about a saddle. The original proposal depended on the choice of an indefinite-signature metric on the space of perturbations, which was taken to be a DeWitt metric with parameter α = – 1. This choice was made to match previous results, but was otherwise admittedly ad hoc. To begin to investigate the physics associated with the choice of such a metric, we now explore contours defined using analogous prescriptions for α ≠ – 1. We study such contours for Euclidean gravity linearized about AdS-Schwarzschild black holes in reflecting cavities with thermal (canonical ensemble) boundary conditions, and we compare path-integral stability of the associated saddles with thermodynamic stability of the classical spacetimes. While the contour generally depends on the choice of DeWitt parameter α, the precise agreement between these two notions of stability found at α = – 1 continues to hold over the finite interval (– 2, – 2/d), where d is the dimension of the bulk spacetime. This agreement manifestly fails for α > – 2/d when the DeWitt metric becomes positive definite. However, we also find dramatic failures for α < – 2 that correlate with breakdowns of the de Donder-like gauge condition defined by α, and at which the relevant fluctuation operator fails to be diagonalizable. This provides criteria that may be useful in predicting metrics on the space of perturbations that give physically-useful contours in more general settings. Along the way, we also identify an interesting error in [1], though we show this error to be harmless.

Nonlinearity in the Medium for Electromagnetic Wave Propagation Induced by Quantum Mechanical Interaction of the Field With Finite-State Atoms

We consider the interaction between a classical time-varying electric field and the atomic electrical dipole moment operator of a finite state atom and calculate approximately the mixed state evolution of the atom with such an interaction Hamiltonian up to quadratic orders in the external classical electric field using Schrodinger’s equation for mixed state evolution. From this approximate expression for the state evolution, we evaluate the average atomic electric dipole moment as a linear-quadratic function of the electric field. Later on, we also take into account Brownian motion bath noise that couples to the atomic dynamics, so that the Schrodinger equation dynamics gets modified to the Lindblad dynamics of an open quantum system, i.e., the master equation. From the average atomic dipole moment, we obtain an expression for the average polarization field of atoms constituting the medium through which the classical electromagnetic wave propagates. This average polarization is a linear-quadratic causal functional of the electric field. The resulting Maxwell equations for wave propagation, taking into account this polarization current density, thus acquire a quadratic nonlinearity which can be used to explain higher harmonic frequencies at the output of an optical fiber when the input is excited by the electric field coming from a monochromatic laser. We are then also able to calculate the mean square fluctuations in the polarization field using the evolving atomic state, and hence the electric field propagating through the medium of atoms acquires quantum fluctuations which can be used to explain time-varying random shifts in the spectral lines. As mentioned above, we include in this article an extension of this problem to the case when there is white Gaussian noise corrupting the state evolution dynamics of the atom and also the case when there is a pairwise interaction between the spin dipoles of atoms at two different locations. In the former case, we derive an approximate formula for the polarization field as well as its temporal statistical correlations, and we show how this computation can be extended to the latter case too. When there is a pairwise interaction between the atomic spins, we use a version of the approximate quantum Boltzmann equation to calculate the first and second order marginals of the atomic states, which is then used to compute the average polarization as well as its spatio-temporal correlations. Temporal correlations using second marginals of the atomic states cannot be computed using classical probabilistic methods owing to Heisenberg uncertainty, so we propose a sequential measurement-based strategy involving computation of the joint probabilities of the atomic spin components at two different times by first measuring one component of the spin, allowing for state collapse, and then following it up with nonlinear time evolution from the collapsed two-particle state using the quantum Boltzmann equation, followed further by measurement of another component of the spin at the second time.

Implications of the weak gravity conjecture for de Sitter space decay by flux discharge

We examine the implications of the weak gravity conjecture for the mechanisms for discharging cosmological constant via membrane nucleations. Once screening fluxes and membranes which source them enter, and weak gravity bounds are enforced, a generic de Sitter space must be unstable. We show that when all the flux terms which screen and discharge the cosmological constant are dominated by quadratic and higher order terms, the bounds from weak gravity conjecture and naturalness lead toward anthropic outcomes. In contrast, when the flux sectors are dominated by linear flux terms, anthropics may be avoided, and the cosmological constant may naturally decay toward smallest possible values. Published by the American Physical Society 2024

Half-factorial real quadratic orders

Half-factorial real quadratic orders

The three-pion K-matrix at NLO in ChPT

The three-particle K-matrix, K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{K} $$\\end{document}df,3, is a scheme-dependent quantity that parametrizes short-range three-particle interactions in the relativistic-field-theory three-particle finite-volume formalism. In this work, we compute its value for systems of three pions in all isospin channels through next-to-leading order in Chiral Perturbation Theory, generalizing previous work done at maximum isospin. We obtain analytic expressions through quadratic order (or cubic order, in the case of zero isospin) in the expansion about the three-pion threshold.

On Computing General Root Algorithms Based on Binomial Expansion, Bernoulli’s Method of Continuous Compound Interest, Series Expansion and Other Modification Methods

Most of the Pth root algorithms exhibit high latency or small convergence rates when iteratively computing the roots. Here we present a slew of algorithms based on series expansions of binomial form, and exponential terms that show low latency or small computational cost for finding the Pth root. The latency decreases if the family of series taken are truncated at higher terms. We show that Babylonian method converges quadratically while the binomial series show an increase in convergence rate from quadratic, cubic and higher order O(t^N) convergence rate as the series is sequentially truncated at higher order terms.