Quadratic Forms Research Articles

Stability tests for neutral‐type delay systems: A delay Lyapunov matrix and piecewise linear approximation of the functional argument approach

AbstractNew necessary and sufficient stability tests for neutral‐type linear time delay systems are presented. They are based on functionals with prescribed derivative depending on the delay Lyapunov matrix and recent stability/instability theorems. A piecewise linear approximation of the initial function argument in the functional is employed. As a result, the functional is approximated via a quadratic form whose matrix provides stability criteria. This matrix involves the integrals based on the delay Lyapunov matrix which are handled via an efficient computational procedure. The main feature of these tests is that they are verified in a finite number of operations, which is achieved through rigorous bounding of the approximation error on a specific class of functions. Some examples which validate the obtained results are presented and discussed.

A Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸r

An integral quadratic form q : ℤn → ℤ is usually identified with a bilinear form 𝕓∨q:ℤn×ℤn→ℤ satisfying q(x)=𝕓∨q(x,x) for any vector x in ℤn, and such that its Gram matrix with respect to the canonical basis of ℤn is upper triangular. Two integral quadratic forms are called strongly (resp. weakly) Gram congruent if their corresponding upper triangular bilinear forms (resp. their symmetrizations) are equivalent. If q is unitary, such upper triangular bilinear form is unimodular, and one considers the associated Coxeter transformation and its characteristic polynomial, the so-called Coxeter polynomial of q with this identification. Two strongly Gram congruent quadratic unit forms are weakly Gram congruent and have the same Coxeter polynomial. Here we show that the converse of this statement holds for the connected non-negative case of Dynkin type 𝔸r (r ≥ 1) and arbitrary corank, and use this characterization to complete a combinatorial classification of such quadratic forms started in [Fundamenta Informaticae 184(1):49–82, 2021] and [Fundamenta Informaticae 185(3):221–246, 2022].

Two Extensions of the Sugeno Class and a Novel Constructed Method of Strong Fuzzy Negation for the Generation of Non-Symmetric Fuzzy Implications

In this paper, we present two new classes of fuzzy negations. They are an extension of a well-known class of fuzzy negations, the Sugeno Class. We use it as a base for our work for the first two construction methods. The first method generates rational fuzzy negations, where we use a second-degree polynomial with two parameters. We investigate which of these two conditions must be satisfied to be a fuzzy negation. In the second method, we use an increasing function instead of the parameter δ of the Sugeno class. In this method, using an arbitrary increasing function with specific conditions, fuzzy negations are produced, not just rational ones. Moreover, we compare the equilibrium points of the produced fuzzy negation of the first method and the Sugeno class. We use the equilibrium point to present a novel method which produces strong fuzzy negations by using two decreasing functions which satisfy specific conditions. We also investigate the convexity of the new fuzzy negation. We give some conditions that coefficients of fuzzy negation of the first method must satisfy in order to be convex. We present some examples of the new fuzzy negations, and we use them to generate new non-symmetric fuzzy implications by using well-known production methods of non-symmetric fuzzy implications. We use convex fuzzy negations as decreasing functions to construct an Archimedean copula. Finally, we investigate the quadratic form of the copula and the conditions that the coefficients of the first method and the increasing function of the second method must satisfy in order to generate new copulas of this form.

Scaling between elasticity and topological genus for random network nanomaterials

We explore the hypothesis that the variation of the effective, macroscopic Young’s modulus, Eeff, of a random network material with its scaled topological genus, g, and with the solid fraction, φ, can be decomposed into the product of g- and φ-dependent functions. Based on findings for nanoporous gold, supplemented by the Gibson–Ashby scaling law for Eeff, we argue that both functions are quadratic in bending-dominated structures. We present finite-element-modeling results for Eeff of coarsened microstructures, in which g and φ are decoupled. These results support the quadratic forms.

A truncated matrix variate gamma distribution

A truncated form of a matrix variate gamma distribution is introduced and a number of properties of this distribution such as cumulative distribution function, orthogonal invariance, moment generating function, marginal distribution of block matrices, and moments are derived. Some results on distribution of random quadratic forms are also derived.

Near universal property of a quadratic form of Hessian discriminant 4

A quadratic form is universal if it represents all totally positive integers in a number field. In classical number theory over the rational integers there are some results where a quadratic form takes on all positive integer values except for a certain restricted set. Here we show that one of the quaternary quadratic forms proven universal for ℚ(√5) takes on all totally positive values in ℚ(√2) except for a certain infinite set. The exceptional integers have even norm and the highest power of 2 dividing that norm is 2 to the first power.

New Coupled-Inductor High-Gain DC/DC Converter With Bipolar Outputs

This paper presents a new bipolar step-up DC/DC converter with the capability of decreasing the input current ripple for renewable energy applications. Two power switches with the simultaneous operation are used in the proposed converter. The suggested topology can provide positive and negative output voltages in a quadratic form with the same and different voltage levels and common ground. The bipolar outputs of the presented topology can be varied independently using the duty cycle and tuning the turn ratio of the coupled inductor (CI). Moreover, the maximum voltage stress across the switching components is much lower than the output voltage. This topology operates under soft-switching conditions for the second switch and all diodes. The steady-state and the main operating equations of the converter are also derived. Finally, a 210 W prototype with 110 V, -110 V, and 220 V output voltages is implemented to justify the theoretical analysis and performance operation of the proposed converter.

Bias in economic evaluation of variable rate application based on geographically weighted regression models with misspecified functional form

AbstractGeographically weighted regression (GWR) has been presented as a valuable tool for estimating site‐specific yield response functions to derive recommendations of variable rate input. This study employs Monte Carlo simulations to illustrate that if GWR assumes a quadratic yield response functional form while the actual yield‐input relationship is quadratic‐plateau, it can significantly overestimate the economic value of variable rate application compared to its true value. Practitioners in precision agriculture should exercise caution when utilizing GWR for site‐specific input recommendations. Statistical community is also encouraged to develop tools in software packages providing GWR that allow more flexibility in functional form assumptions.

Investments in Innovations and Market Structure: A Semi-parametric Approach

The economic literature suggests that there is an inverted-U type of relationship between innovation and product market competition. Existing empirical studies assume a quadratic functional form while estimating the relationship between the two variables. Using data on the Indian manufacturing industry, this article contributes by using a semi-parametric approach to test the ‘Inverted-U’ hypothesis without assuming a priori functional form. The estimation results suggest that the functional form of the relationship is contingent upon the choice of product market competition indicator in the model. When product market competition is defined as overall price competition, the empirical evidence confirms the inverted-U hypothesis. This implies that both escape competition and the Schumpeterian effect are observed as price competition increases. However, when product market competition is defined as domestic competition, only the escape competition effect is observed. JEL Codes: L10, L60, O30, O33, F10

Performance of an approximate Bayes factor based detector for environmentally informed active sonar

Developed here are a space of approximate Bayes Factor (BF) inference processors for high frequency broadband active sonar with short vertical arrays operating in shallow water ocean waveguides that exploit relatively depth invariant modes of propagation. The relevant information regarding the refractive media, rough surface and volume reverberation are incorporated to build the marginal likelihoods under each of the composite hypotheses of null and alternative. Acoustic scattering from a mobile object of interest under depth uncertainty characterizes the compound alternative hypothesis. Approximations are presented and inferences regarding the presence of the mobile body of interest are determined against a composite null hypothesis of reverberation and ambient acoustic noise. The approximate BF processor is shown to be a time-varying quadratic form of array observations over the beam-delay space. We demonstrate the sub-space processing of depth invariant modes at range and illustrate the BF inferential approach on a few representative waveguides. Performance in classic terms of probability of detection as a function of false alarm rate are presented. [This work supported by the Office of Naval Research.]

Permutation rational functions over quadratic extensions of finite fields

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over Fq2, whose numerators are so-called q-quadratic polynomials. To this end, we will first determine the exact number of zeros of a special q-quadratic polynomial in Fq2, by calculating character sums related to quadratic forms of Fq2/Fq. Then given some rational function, we can demonstrate whether it induces a permutation of Fq2.

A polynomial analogue of Berggren’s theorem on Pythagorean triples

Say that ( x , y , z ) (x, y, z) is a positive primitive integral Pythagorean triple if x , y , z x, y, z are positive integers without common factors satisfying x 2 + y 2 = z 2 x^2 + y^2 = z^2 . An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on ( 3 , 4 , 5 ) (3, 4, 5) and ( 4 , 3 , 5 ) (4, 3, 5) generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren’s theorem in the context of a one-variable polynomial ring over a field of characteristic ≠ 2 \neq 2 . As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring.

Model-Free Q-Learning for the Tracking Problem of Linear Discrete-Time Systems.

In this article, a model-free Q-learning algorithm is proposed to solve the tracking problem of linear discrete-time systems with completely unknown system dynamics. To eliminate tracking errors, a performance index of the Q-learning approach is formulated, which can transform the tracking problem into a regulation one. Compared with the existing adaptive dynamic programming (ADP) methods and Q-learning approaches, the proposed performance index adds a product term composed of a gain matrix and the reference tracking trajectory to the control input quadratic form. In addition, without requiring any prior knowledge of the dynamics of the original controlled system and command generator, the control policy obtained by the proposed approach can be deduced by an iterative technique relying on the online information of the system state, the control input, and the reference tracking trajectory. In each iteration of the proposed method, the desired control input can be updated by the iterative criteria derived from a precondition of the controlled system and the reference tracking trajectory, which ensures that the obtained control policy can eliminate tracking errors in theory. Moreover, to effectively use less data to obtain the optimal control policy, the off-policy approach is introduced into the proposed algorithm. Finally, the effectiveness of the proposed algorithm is verified by a numerical simulation.

An alternative multivariate exponential power distribution

In the existing multivariate Exponential Power Distribution (EPD), the shape parameter has the tendency to affect the variability in the data leading to distorted features of the distribution. This study resolves this problem by obtaining a generalized expression in place of the constant coefficient (0.5) of the quadratic form of the multivariate EPD as a function of the shape parameter, such that the variance-covariance matrix of the data is exactly equal to the scale parameter. The density function thus obtained is a more generalized multivariate EPD that provides flexibility in depicting the true underlying characteristics of the distribution for all sample sizes.

Exponentially harmonic maps between Kähler manifolds

In this paper, we show Liouville-type theorems for exponentially harmonic maps and p-harmonic maps between Kähler manifolds. We then study the quadratic form of an exponentially harmonic map of Kähler manifolds and its applications. We also prove that if [Formula: see text] is an exponentially harmonic map from a compact Kähler manifold [Formula: see text] into a Kähler manifold [Formula: see text] with constant energy density such that the curvature of [Formula: see text] is strongly semi-negative, then [Formula: see text] is holomorphic or antiholomorphic.

High-order linearly implicit exponential integrators conserving quadratic invariants with application to scalar auxiliary variable approach

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes.

Долгосрочное влияние силы репрессий в онлайн- и офлайн-среде на численность участников поствыборных протестов (Кросс-страновое эмпирическое исследование)

Although the relationship between state coercive measures and protests has long been studied in Political Science, there are a number of gaps in the modern research studies about the impact of repressions on street protest activity. A serious flaw in these studies is their focus on the influence of repressions on the existing protest campaigns, with little attention to their long-term implications. The impact of repressions on the scale of protest and number of protesters, as well as the role of the Internet as a factor mediating the impact of state sanctions on protests, is understudied. The article attempts to bridge these gaps. The author has first examined theoretical arguments in support of three possible forms of correlation between the severity of repressions and street protest activity — negative, positive, and parabolic (n-shaped), after which he tests relevant hypotheses on a sample of country cases that share the same trigger for protest — suspicions of electoral fraud. To test these hypotheses, the author utilized data on the maximum strength of repressions against civil society organizations, participants in protests and authors of anti-government messages on the Internet in the pre-electoral years, as well as data on protests in the first week following the elections. The study confirmed the influence of repressions on future protest activity. At the same time, the relationship between the severity of repressions and the number of protest participants in the long term has a quadratic n-shaped form: at low and high levels of repression, the number of protesters is minimal, at medium — it is maximal. As for the impact of the Internet, it was not detected on data on repression in the offline environment, but it was revealed on data on the strength of sanctions for political activity on the Web: if the share of Internet users in a country is low, repressions decrease the number of protest participants; if it is high, repressions of medium strength correspond to the maximum number of protesters.

Anisotropic media with singular slowness surfaces

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

An integer ambiguity resolution method based on baseline vector predictions in landslide monitoring

Accurate integer ambiguity resolution (IAR) is pivotal for high-precision and high-reliability Global Navigation Satellite Systems (GNSS) positioning. However, traditional IAR methodologies, encompassing integer estimation and validation, often face significant challenges in harsh landslide environments. This paper proposes a new IAR method based on baseline vector predictions. Initially, the real-time baseline vectors predictions are used to substitute the true values of baseline vectors in the mixed integer least-squares (MILS) estimation equation, and the MILS estimation criterion is constructed which contains the quadratic form of baseline vectors. Subsequently, a novel baseline vector validation procedure, built on real-time dynamic predictions, is established. Experiments were carried out at two GNSS landslide monitoring area using GPS + BDS dual-frequency measurement data. These stations underwent both stable deformation stages within complex measurement environments and accelerated deformation stages in open environments. Results indicate that the proposed method overcomes the limitations associated with conventional ambiguity validation methods based on hypothesis testing, addressing the errors of “abandoning trueness” and “accepting mistake” simultaneously. Notably, the ambiguity fixed rate improved by 80.7 % for single-frequency experiments and 39.8 % for dual-frequency experiments at one monitoring station located in a complex environment. This work provides valuable IAR approach for GNSS deformation monitoring applications.

On a Ramanujan-type series associated with the Heegner number 163

Using the Wolfram NumberTheory package and the Recognize command, together with numerical estimates involving the elliptic lambda and elliptic alpha functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type series related to the class number h(−d)=1 for a quadratic form with discriminant d=163. This conjectured series is of level one and has positive terms, and recalls the Chudnovsky brothers' alternating series of the same level, given the connection between the Chudnovsky–Chudnovsky formula and the Heegner number d=163 such that Q(−d) has class number one. We prove Bagis and Glasser's conjecture by proving evaluations for λ⁎(163) and α(163), which we derive using the Chudnovsky brothers' formula together with the analytic continuation of a formula due to the Borwein brothers for Ramanujan-type series of level one. As a byproduct of our method, we obtain an infinite family of Ramanujan-type series for 1π generalizing the Chudnovsky algorithm.