Difference Quotient Research Articles

Staircasing effect for minimizers of the one-dimensional discrete Perona-Malik functional

We consider the one-dimensional Perona-Malik functional, that is the energy associated to the celebrated forward-backward equation introduced by P. Perona and J. Malik in the context of image processing, with the addition of a forcing term. We discretize the functional by restricting its domain to a finite dimensional space of piecewise constant functions, and by replacing the derivative with a difference quotient. We investigate the asymptotic behavior of minima and minimizers as the discretization scale vanishes. In particular, if the forcing term has bounded variation, we show that any sequence of minimizers converges in the sense of varifolds to the graph of the forcing term, but with tangent component which is a combination of the horizontal and vertical directions. If the forcing term is more regular, we also prove that minimizers actually develop a microstructure that looks like a piecewise constant function at a suitable scale, which is intermediate between the macroscopic scale and the scale of the discretization.

Analysis on a High Accuracy Fully Implicit Solution for Strong Nonlinear Diffusion Problem - Convergence, Stability, and Uniqueness

Some fundamental properties are analyzed for a fully implicit finite difference (FIFD) solution of conservative strong nonlinear diffusion problem. The scheme is constructed by combining a second-order backward difference temporal discretization and a central finite difference spatial discretization, and therefore highly nonlinear. Theoretical analysis is carried out under a coercive condition according with the diffusion feature of the strong nonlinear diffusion model. Benefiting from the boundedness estimates of the FIFD solution itself and its first- and second-order spatial difference quotients, some novel argument techniques are developed to overcome the difficulties coming from the nonlinear approximation for the strong nonlinear conservative diffusion operator. Consequently, it is proved rigorously that the FIFD scheme is unconditionally stable, its solution is unique and convergent to the exact solution of the original problem with second-order space-time accuracy. Numerical examples are provided to confirm its advantages on precision and efficiency over its first-order time accurate counterpoint.

New estimates for a class of non-local approximations of the total variation

We consider a class of non-local functionals recently introduced by H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung, which offers a novel way to characterize functions with bounded variation.We give a positive answer to an open question related to these functionals in the case of functions with bounded variation. Specifically, we prove that in this case the liminf of these functionals can be estimated from below by a linear combination in which the three terms that sum up to the total variation (namely the total variation of the absolutely continuous part, of the jump part and of the Cantor part) appear with different coefficients. We prove also that this estimate is optimal in the case where the Cantor part vanishes, and we compute the precise value of the limit in this specific scenario.In the proof we start by showing the results in dimension one by relying on some measure theoretic arguments in order to identify sufficiently many disjoint rectangles in which the difference quotient can be estimated, and then we extend them to higher dimension by a classical sectioning argument.

From Halley to Secant: Redefining root finding with memory‐based methods including convergence and stability

Root‐finding methods solve equations and identify unknowns in physics, engineering, and computer science. Memory‐based root‐seeking algorithms may look back to expedite convergence and enhance computational efficiency. Real‐time systems, complicated simulations, and high‐performance computing demand frequent, large‐scale calculations. This article proposes two unique root‐finding methods that increase the convergence order of the classical Newton–Raphson (NR) approach without increasing evaluation time. Taylor's expansion uses the classical Halley method to create two memory‐based methods with an order of 2.4142 and an efficiency index of 1.5538. We designed a two‐step memory‐based method with the help of Secant and NR algorithms using a backward difference quotient. We demonstrate memory‐based approaches' robustness and stability using visual analysis via polynomiography. Local and semilocal convergence are thoroughly examined. Finally, proposed memory‐based approaches outperform several existing memory‐based methods when applied to models including a thermistor, path traversed by an electron, sheet‐pile wall, adiabatic flame temperature, and blood rheology nonlinear equation.

C1,α-regularity for solutions of degenerate/singular fully nonlinear parabolic equations

We establish the interior C1,α-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equationsut=|Du|γF(D2u)+fin Q1, where γ>−1 and f∈C(Q1)∩L∞(Q1). For this purpose, we prove the well-posedness of the regularized Cauchy-Dirichlet problem{ut=(1+|Du|2)γ/2F(D2u)in Q1u=φon ∂pQ1, where γ>−2. Our approach utilizes the Bernstein method with approximations in view of the difference quotient.

Second order error bounds for POD-ROM methods based on first order divided differences

This note proves for the heat equation that using BDF2 as time stepping scheme in POD-ROM methods with snapshots based on difference quotients gives both the optimal second order error bound in time and pointwise estimates.

The variational slope of quasar light curves is not a distance indicator

ABSTRACT When the time difference quotients, or variational slopes, of quasar light curves are plotted against their absolute magnitudes, there is a tight positive correlation of ∼0.16 dex in the variational slope direction or ∼0.5 dex in the absolute magnitude direction. This finding resulted in suggestions that a variational slope–luminosity relation could be used as a distance indicator. However, I show that this relation can be explained almost entirely from self-correlation with luminosity. After properly accounting for the self-correlation component, the relation has a true scatter of ∼1.5 dex in luminosity, consistent with established correlations for quasar variability amplitudes. Given this large scatter, correlation with variational slope or variability amplitude and luminosity is not by itself a suitable distance indicator for quasars.

Image-Evoked Emotion Recognition for Hearing-Impaired Subjects with EEG Signals

In recent years, there has been a growing interest in the study of emotion recognition through electroencephalogram (EEG) signals. One particular group of interest are individuals with hearing impairments, who may have a bias towards certain types of information when communicating with those in their environment. To address this, our study collected EEG signals from both hearing-impaired and non-hearing-impaired subjects while they viewed pictures of emotional faces for emotion recognition. Four kinds of feature matrices, symmetry difference, and symmetry quotient based on original signal and differential entropy (DE) were constructed, respectively, to extract the spatial domain information. The multi-axis self-attention classification model was proposed, which consists of local attention and global attention, combining the attention model with convolution through a novel architectural element for feature classification. Three-classification (positive, neutral, negative) and five-classification (happy, neutral, sad, angry, fearful) tasks of emotion recognition were carried out. The experimental results show that the proposed method is superior to the original feature method, and the multi-feature fusion achieved a good effect in both hearing-impaired and non-hearing-impaired subjects. The average classification accuracy for hearing-impaired subjects and non-hearing-impaired subjects was 70.2% (three-classification) and 50.15% (five-classification), and 72.05% (three-classification) and 51.53% (five-classification), respectively. In addition, by exploring the brain topography of different emotions, we found that the discriminative brain regions of the hearing-impaired subjects were also distributed in the parietal lobe, unlike those of the non-hearing-impaired subjects.

On the design of non-singular, energy-momentum consistent integrators for nonlinear dynamics using energy splitting and perturbation techniques

This work proposes a suite of numerical techniques to facilitate the design of structure-preserving integrators for nonlinear dynamics, with particular emphasis on many-body dynamics. The celebrated LaBudde-Greenspan integrator and various energy-momentum schemes adopt a difference quotient formula in their algorithmic force definitions, which suffers from numerical instability as the denominator gets close to zero. There is a need to develop structure-preserving integrators without invoking the quotient formula. In this work, the potential energy of a Hamiltonian system is split into two parts, and specially developed quadrature rules are applied separately to them. The resulting integrators can be regarded as classical ones perturbed with first- or second-order terms, and the energy split guarantees the dissipative nature in the numerical residual. In the meantime, the conservation of invariants is respected in the design. A complete analysis of the proposed integrators is given, with representative numerical examples provided to demonstrate their performance. They can be used either independently as energy-decaying and momentum-conserving schemes for nonlinear problems or as an alternate option with a conserving integrator, such as the LaBudde-Greenspan integrator, when the numerical instability in the difference quotient is detected.

Composition Analysis of Glass Artefacts Based on Spearman's Coefficient and Fuzzy Correlation

The identification and analysis of the composition of glass artefacts can support the related fields in exploring the history of glass and its processes, as well as the development and progress of glass composition. This paper identifies and analyses the composition of glass artefacts by collecting relevant data and information on glass artefacts through the Spearman correlation coefficient method and fuzzy correlation. In terms of data pre-processing, data with component ratios that did not add up to between 85% and 105% were removed, and then the degree of weathering on the surface of the glass artefacts was initially determined to be closely related to the glass type by applying a chi-square test, and then the highest correlation between the weathered surface of the glass and the glass type was determined by excluding possible interference caused by ornamentation and colour through partial correlation analysis. Fuzzy correlation analysis was then applied to remove abnormal data and calculate the mean values of each chemical composition for different types of glass at different degrees of weathering to discover statistical patterns. Finally, the mean values calculated by the fuzzy correlation analysis were summed using the difference to obtain the predicted results. Six chemical components were also screened out by the Spearman correlation coefficient method, and then the difference quotients of the same components were calculated by averaging these six components for different types of glass respectively. Finally, it was found that in addition to the three elements of potassium, lead and barium, calcium and strontium also have a greater influence on glass classification. The sample data were then clustered by the hierarchical clustering method based on the correlation coefficient method of calculating distances, and the optimal number of clusters was found by calculating the clustering contour coefficients. Then, the data of the clustered subclasses were analysed and subclasses such as soda-lime glass, sulphide glass and aluminosilicate glass were classified. Finally, the clustering clarity values were defined using the entropy weighting method to measure the clustering results, and the Held mean values at different p-values were adjusted for sensitivity analysis of the clustering clarity values.

Weak differentiability to nonuniform nonlinear degenerate elliptic systems under p,q-growth condition on the Heisenberg group

The paper concerns the weak differentiability of weak solutions to two kinds of nonuniform nonlinear degenerate elliptic systems under the p,q-growth condition on the Heisenberg Group. We use the iteration to fractional difference quotients on the Heisenberg Group to get the weak differentiability of weak solution u in the vertical direction (i.e., Lp(1<p<4) integrability of Tu) and then the second order weak differentiability of weak solution in the horizontal directions (i.e., L2 integrability of ∇H2u) and weak differentiability of gradient of weak solution in the vertical direction (i.e., L2 integrability of T∇Hu)).

Uniform Bounds with Difference Quotients for Proper Orthogonal Decomposition Reduced Order Models of the Burgers Equation

In this paper, we prove uniform error bounds for proper orthogonal decomposition (POD) reduced order modeling (ROM) of Burgers equation, considering difference quotients (DQs), introduced in Kunisch and Volkwein (Numer Math 90(1):117–148, 2001). In particular, we study the behavior of the DQ ROM error bounds by considering L^2(Omega ) and H^1_0(Omega ) POD spaces and l^{infty }(L^2) and natural-norm errors. We present some meaningful numerical tests checking the behavior of error bounds. Based on our numerical results, DQ ROM errors are several orders of magnitude smaller than noDQ ones (in which the POD is constructed in a standard way, i.e., without the DQ approach) in terms of the energy kept by the ROM basis. Further, noDQ ROM errors have an optimal behavior, while DQ ROM errors, where the DQ is added to the POD process, demonstrate an optimality/super-optimality behavior. It is conjectured that this possibly occurs because the DQ inner products allow the time dependency in the ROM spaces to make an impact.

A new approach to proper orthogonal decomposition with difference quotients

In a recent work (Koc et al., SIAM J. Numer. Anal. 59(4), 2163–2196, 2021), the authors showed that including difference quotients (DQs) is necessary in order to prove optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order models of the heat equation. In this work, we introduce a new approach to including DQs in the POD procedure. Instead of computing the POD modes using all of the snapshot data and DQs, we only use the first snapshot along with all of the DQs and special POD weights. We show that this approach retains all of the numerical analysis benefits of the standard POD DQ approach, while using a POD data set that has approximately half the number of snapshots as the standard POD DQ approach, i.e., the new approach requires less computational effort. We illustrate our theoretical results with numerical experiments.

A Weak-Type Expression of the Orlicz Modular

An equivalent expression of Orlicz modulars in terms of measure of level sets of difference quotients is established. The result in a sense complements the famous Maz’ya–Shaposhnikova formula for the fractional Gagliardo–Slobodeckij seminorm and its recent extension to the setting of Orlicz functions.

DIFFERENCE QUOTIENT ESTIMATES AND ACCURACY ENHANCEMENT OF DISCONTINUOUS GALERKIN METHODS FOR NONLINEAR CONVECTION-DIFFUSION EQUATIONS

In this paper, we apply the post-processing technique to the improvement of the superconvergence of the discontinuous Galerkin method for the nonlinear convection-diffusion equations. We firstly analyze the error estimate and convergence accuracy under <i>L</i>²-norm, and then demonstrate that the α-order difference quotient of DG error is of order <i>k</i>+3/2-<i>α</i>/2 when the upwind fluxes are used. By the duality argument, we construct an appropriate dual equation, and futher obtain superconvergence results of order in the negative-order norm, namely 2<i>k</i>+3/2-<i>α</i>/2 order superconvergence accuracy. Finally, we choose an appropriate kernel function and apply the SIAC filter to the nonlinear convection-diffusion equation to obtain at least 3<i>k</i>/2+1 order superconvergence for post-processed solutions. All theoretical results are proved by numerical experiments.

Quantitative and covariational reasoning opportunities provided by calculus textbooks: the case of the derivative

While research on the opportunity to learn about mathematics concepts provided by textbooks at the secondary level is well documented, there is still a paucity of similar research at the undergraduate level. Contributing towards addressing this knowledge gap, the present study examined opportunities to engage in quantitative and covariational reasoning, in the context of the derivative concept, provided by two textbooks commonly used in the teaching of calculus in the United States-a regular calculus textbook and an applied calculus textbook. Analysis of expository sections, examples, and exercises related to the derivative provided in these textbooks revealed three main results. First, the applied calculus textbook provides plenty of opportunities to engage in quantitative reasoning. Similar opportunities are limited in the regular calculus textbook. Second, opportunities to engage in covariational reasoning are not only minimal in both textbooks, but also limited to low levels of covariational reasoning, namely coordination, direction, and quantitative coordination. Third, the applied calculus textbook consistently defines ordinary and partial derivatives as rates of change while the regular calculus textbook consistently defines these concepts as limits of a difference quotient. Findings of this research have implications for several stakeholders, including calculus textbook authors and calculus instructors.

Sobolev spaces revisited

We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$ n, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis, and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies, and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo–Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the $L^p$ norm of functions in $L^p(\mathbb{R}^n)$.

On the characterization of constant functions through nonlocal functionals

We address a classical open question by H. Brezis and R. Ignat concerning the characterization of constant functions through double integrals that involve difference quotients. Our first result is a counterexample to the question in its full generality. This counterexample requires the construction of a function whose difference quotients avoid a sequence of intervals with endpoints that diverge to infinity. Our second result is a positive answer to the question when restricted either to functions that are bounded and approximately differentiable almost everywhere, or to functions with bounded variation. We also present some related open problems that are motivated by our positive and negative results.

Lipschitz type characterizations for Bergman-Orlicz spaces of eigenfunctions on hyperbolic space

We characterize Bergman-Orlicz spaces of eigenfunctions on hyperbolic space in terms of Lipschitz type conditions with respect to pseudo-hyperbolic, hyperbolic and Euclidean metrics. As an application, some properties of symmetric lifting operators defined by the difference quotients of eigenfunctions are obatined. Our results of this paper are extensions of the corresponding known ones in classical weighted Bergman spaces.

Aerodynamic system instability identification with sample entropy algorithm based on feature extraction

Based on the sample entropy algorithm in nonlinear dynamics, an improved sample entropy method is proposed in the aerodynamic system instability identification for the stall precursor detection based on the nonlinear feature extraction algorithm in an axial compressor. The sample entropy algorithm is an improved algorithm based on the approximate entropy algorithm, which quantifies the regularity and the predictability of data in time series. Combined with the spatial modes representing for the rotating stall in the circumferential direction, the recognition capacity of the sample entropy is displayed well on the detection of stall inception. The indications of rotating waves are extracted by the circumferential analysis from modal wave energy. The significant ascendant in the amplitude of the spatial mode is a pronounced feature well before the imminence of stall. Data processing with the spatial mode effectively avoids the problems of inaccurate identification of a single measuring point only depending on pressure. Due to the different selections of similarity tolerance, two kinds of sample entropy are obtained. The properties of the development process of the identification model show obvious mutation phenomena at the boundary of instability, which reveal the inherent characteristic in aerodynamic system. Then the dynamic difference quotient is computed according to the difference quotient criterion, after the smooth management by discrete wavelet. The rapid increase of difference quotient can be regarded as a significant feature of the system approaching the flow instability. It is proven that based on the principle of sample entropy algorithm, the nonlinear characteristic of rotating stall can be well described. The inception can be suggested by about 12–68 revolutions before the stall arrival. This prediction method presenting is accounted for the nonlinearity of the complex flow in stall, which is in a view of data fusion system of pressure for the spatial mode tracking.