Discrete Integration Research Articles

DC optimization for constructing discrete Sugeno integrals and learning nonadditive measures

Defined solely by means of order-theoretic operations meet (min) and join (max), weighted lattice polynomial functions are particularly useful for modelling data on an ordinal scale. A special case, the discrete Sugeno integral, defined with respect to a nonadditive measure (a capacity), enables accounting for the interdependencies between input variables. However, until recently the problem of identifying the fuzzy measure values with respect to various objectives and requirements has not received a great deal of attention. By expressing the learning problem as the difference of convex functions, we are able to apply DC (difference of convex) optimization methods. Here we formulate one of the global optimization steps as a local linear programming problem and investigate the improvement under different conditions. Abbreviation: DC: difference of convex; LP: linear programming

Some discrete soliton solutions and interactions for the coupled Ablowitz–Ladik equations with branched dispersion

The coupled Ablowitz-Ladik lattice equations are the integrable discretizations of the Schrödinger equation, which can be used to model the propagation of an optical field in a tight binding waveguide array. In this paper, the discrete N-fold Darboux transformation(DT) is used to derive the discrete breather and bright soliton solutions of coupled Ablowitz–Ladik equations. Soliton interaction structures of obtained solutions are shown graphically. Based on 4 × 4 discrete Lax pairs, the transformation matrix T of DT is constructed. Then, we derive novel discrete one-soliton and two-soliton with the zero and nonzero seed solutions. And the dynamic features of breather and bright solutions are displayed, some soliton interaction phenomena are shown in the coupled Ablowitz-Ladik lattice equations. These results may be useful to explain some nonlinear wave phenomena in certain electrical and optical systems.

Normal approximation for sums of weighted $U$-statistics – application to Kolmogorov bounds in random subgraph counting

We derive normal approximation bounds in the Kolmogorov distance for sums of discrete multiple integrals and weighted $U$-statistics made of independent Bernoulli random variables. Such bounds are applied to normal approximation for the renormalized subgraph counts in the Erdős–Rényi random graph. This approach completely solves a long-standing conjecture in the general setting of arbitrary graph counting, while recovering recent results obtained for triangles and improving other bounds in the Wasserstein distance.

Discrete differentiators based on sliding modes

Sliding-mode-based differentiation of the input f(t) yields exact estimations of the derivatives ḟ,…,f(n), provided an upper bound L(t) of |f(n+1)(t)| is available in real-time. In practice it involves discrete sampling and numerical integration of the internal variables between the measurements. Accuracy asymptotics of different discretization schemes are calculated for discrete noisy sampling, whereas sampling and integration steps are independently variable or constant. Proposed discrete differentiators restore the optimal accuracy asymptotics of their continuous-time counterparts. Event-triggered sampling is considered. Extensive numeric experiments are presented and analyzed.

A posteriori error estimation in maximum norm for a two-point boundary value problem with a Riemann–Liouville fractional derivative

A Riemann–Liouville two-point boundary value problem is considered. An integral discretization scheme is developed to approximate the Volterra integral equation transformed from the Riemann–Liouville boundary value problem. The stability results and a posteriori error analysis are given. A solution-adaptive algorithm based on a posteriori error analysis is designed by equidistributing arc-length monitor function. Numerical experiments are presented to support the theoretical result.

Accelerating equilibrium isotope effect calculations. II. Stochastic implementation of direct estimators.

Path integral calculations of equilibrium isotope effects and isotopic fractionation are expensive due to the presence of path integral discretization errors, statistical errors, and thermodynamic integration errors. Whereas the discretization errors can be reduced by high-order factorization of the path integral and statistical errors by using centroid virial estimators, two recent papers proposed alternative ways to completely remove the thermodynamic integration errors: Cheng and Ceriotti [J. Chem. Phys. 141, 244112 (2015)] employed a variant of free-energy perturbation called "direct estimators," while Karandashev and Vaníček [J. Chem. Phys. 143, 194104 (2017)] combined the thermodynamic integration with a stochastic change of mass and piecewise-linear umbrella biasing potential. Here, we combine the former approach with the stochastic change in mass in order to decrease its statistical errors when applied to larger isotope effects and perform a thorough comparison of different methods by computing isotope effects first on a harmonic model and then on methane and methanium, where we evaluate all isotope effects of the form CH4-xDx/CH4 and CH5-xDx +/CH5 +, respectively. We discuss the reasons for a surprising behavior of the original method of direct estimators, which performed well for a much larger range of isotope effects than what had been expected previously, as well as some implications of our work for the more general problem of free energy difference calculations.

A modified integral discretization scheme for a two-point boundary value problem with a Caputo fractional derivative

In this paper, we consider a two-point boundary value problem with a Caputo fractional derivative, where the exact solution may have weak singularity. First, the boundary value problem is turned into an initial value problem by using a shooting method based on the secant iterative method. Second, the initial value problem is transformed into an equivalent integral–differential equation with a weakly singular kernel. Next, a modified integral discretization scheme is proposed to approximate the integral–differential equation. It is proved that the modified integral discretization scheme converges with order O(N−min{2,2δ−1}) in the discrete maximum norm, which improves previous results. Numerical experiments are provided to demonstrate the theoretical results.

Construction and Numerical Realization of a Magnetization Model for a Magnetostrictive Actuator Based on a Free Energy Hysteresis Model

Giant magnetostrictive actuators (GMA) driven by giant magnetostrictive material (GMM) has some advantages such as a large strain, high precision, large driving force, fast response, high reliability, and so on, and it has become the research hotspot in the field of microdrives. Research shows there is a nonlinear, intrinsic relationship between the output signal and the input signal of giant magnetostrictive actuators because of the strong coupling characteristics between the machine, electromagnetic field, and heat. It is very complicated to construct its nonlinear eigenmodel, and it is the basis of the practical process of giant magnetostrictive material to construct its nonlinear eigenmodel. Aiming at the design of giant magnetostrictive actuators, the magnetization model based on a free-energy hysteresis model has been deeply researched, constructed, and put forward by Smith, which combines Helmholtz–Gibbs free energy and statistical distribution theory, to simulate the hysteresis model at medium or high driving strengths. Its main input and output parameters include magnetic field strength, magnetization, and mechanical strain. Then, numerical realization and verification of the magnetization model are done by the Gauss–Legendre integral discretization method. The results show that the magnetization model and its numerical method are correct, and the research results provide a theoretical basis for the engineering application of giant magnetostrictive material and optimized structure of giant magnetostrictive material actuators, which have an important practical application value.

Integrable discretizations and numerical simulation for a modified coupled integrable dispersionless equation

In this paper, we propose an integrable semi-discrete and full-discrete analogues of the modified coupled integrable dispersionless (mCID) equation that was derived recently as a two-component generalization of CID equation. The key of the integrable discretization is to find the relation between the mCID equation and the sine-Gordon equation. We complete the integrable discrete analogues based on the discrete sine-Gordon equation. Numerical simulation of one-soliton, two-soliton and breather solution are conducted based on the integrable semi-discrete scheme. We also present the Casorati determinant solution in parametric form of N-soliton solutions for the semi-discrete analogue of the mCID equation. We show that the continuum limit of the full-discrete mCID equation leads to the semi-discrete mCID equation.

Fitting Sugeno integral for learning fuzzy measures using PAVA isotone regression

The discrete Sugeno integral is an aggregation function particularly suited to aggregation of ordinal inputs. It accounts for inputs interactions, such as redundancy and complementarity, and its learning from empirical data is a challenging optimisation problem. The methods of ordinal regression involve an expensive objective function, whose complexity is quadratic in the number of data. We formulate ordinal regression using a much less expensive objective computed in linear time by the pool-adjacent-violators algorithm. We investigate the learning problem numerically and show the superiority of the new algorithm.

Discrete exponential type systems on a quad graph, corresponding to the affine Lie algebras

The article deals with the problem of the integrable discretization of the well-known Drinfeld–Sokolov hierarchies related to the Kac–Moody algebras. A class of discrete exponential systems connected with the Cartan matrices has been suggested earlier in Garifullin et al (2012 SIGMA 8 33) which coincide with the corresponding Drinfeld–Sokolov systems in the continuum limit. It was conjectured that the systems in this class are all integrable and the conjecture has been proven by numerous examples. In the present article we study those systems from this class which are related to the algebras . We found the Lax pair for arbitrary N, briefly discussed the possibility of using the method of formal diagonalization of Lax operators for describing a series of local conservation laws and illustrated the technique using the example of N = 3. Higher symmetries of the system are presented in both characteristic directions. The recursion operator for the case N = 3 is found. It is interesting to note that this operator is not weakly nonlocal.

Integral norm discretization and related problems

The problem is discussed of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure. This problem is investigated for elements of finite-dimensional spaces. Also, discretization of the uniform norm of functions in a given finite-dimensional subspace of continuous functions is studied. Special attention is given to the case of multivariate trigonometric polynomials with frequencies (harmonics) in a finite set with fixed cardinality. Both new results and a survey of known results are presented. Bibliography: 47 titles.

Spontaneous symmetry breaking in 2D supersphere sigma models and applications to intersecting loop soups

Two-dimensional sigma models on superspheres are known to flow to weak coupling in the IR when r − 2s < 2. Their long-distance properties are described by a free ‘Goldstone’ conformal field theory with r − 1 bosonic and 2s fermionic degrees of freedom, where the symmetry is spontaneously broken. This behavior is made possible by the lack of unitarity.The purpose of this paper is to study logarithmic corrections to the free theory at small but non-zero coupling . We do this in two ways. On the one hand, we perform perturbative calculations with the sigma model action, which are of special technical interest since the perturbed theory is logarithmic. On the other hand, we study an integrable lattice discretization of the sigma models provided by vertex models and spin chains with symmetry. Detailed analysis of the Bethe equations then confirms and completes the field theoretic calculations. Finally, we apply our results to physical properties of dense loop soups with crossings.

Integrable discretizations of AB system and multi-soliton solutions

Different integrable discretizations of AB system are proposed in this paper. Discrete AB (dAB) system and two different versions of semi-discrete AB (sdAB) systems are obtained by discretization of associated Lax pair of continuous integrable AB system. Discrete multiple soliton solutions are constructed for dAB system and sdAB systems by applying Darboux transformation. In order to explain our construction we also computed discrete single and double soliton solutions for dAB and sdAB systems. Dynamics of one discrete bright and dark soliton solution and interaction of two discrete bright-bright and dark-dark soliton solutions for dAB system have been presented.

Дискретизация интегральной нормы и близкие задачи

В статье обсуждается задача о замене интегральной нормы по заданной вероятностной мере соответствующей интегральной нормой по дискретной мере. Указанная задача изучается для элементов конечномерных пространств. Также рассматривается дискретизация равномерной нормы для функций из заданного конечномерного подпространства непрерывных функций. Особое внимание уделено случаю многомерных тригонометрических полиномов со спектрами из конечных множеств заданной мощности. Мы приводим как новые результаты, так и обзор известных результатов. Библиография: 47 названий.

On the weak consistency of finite volumes schemes for conservation laws on general meshes

The aim of this paper is to develop some tools in order to obtain the weak consistency of (in other words, an analogue of the Lax–Wendroff theorem for) finite volume schemes for balance laws in the multi-dimensional case and under minimal regularity assumptions for the mesh. As in the seminal Lax–Wendroff paper, our approach relies on a discrete integration by parts of the weak formulation of the scheme. Doing so, a discrete gradient of the test function appears; the central argument for the scheme consistency is to remark that this discrete gradient is convergent in $$L^\infty $$ weak $$\star $$ .

SOWAT: High-resolution imaging with only partial AO correction.

Observations of dense stellar systems such as globular clusters (GCs) are limited in resolution by the optical aberrations induced by atmospheric turbulence (atmospheric seeing). At the example of holographic speckle imaging, we now study, to which degree image reconstruction algorithms are able to remove residual aberrations from a partial adaptive optics (AO) correction, such as delivered from ground-layer AO (GLAO) systems. Simultaneously, we study, how such algorithms benefit from being applied to pre-corrected instead of natural point-spread functions (PSFs). We find that using partial AO corrections already lowers the demands on the holography reference star by ∼ 3 mag, what makes more fields accessible for this technique, and also that the discrete integration times may be chosen about 2 - 3× longer, since the effective wavefront evolution is slowed down by removing the perturbation power.

Three Approaches to Detecting Discrete Integrability

A class of discrete equations is considered from three perspectives corresponding to three measures of complexity of the solutions: the (hyper-) order of meromorphic solutions in the sense of Nevanlinna, the degree growth of iterates over a function field and the height growth of iterates over the rational numbers. In each case, low complexity implies a form of singularity confinement which results in a known discrete Painlevé equation.

A Second-Order Bandpass &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$\Delta\Sigma$ &lt;/tex-math&gt; &lt;/inline-formula&gt; Time-to-Digital Converter With Negative Time-Mode Feedback

This paper presents an all-digital bandpass $\Delta\Sigma $ time-to-digital converter (BP) $\Delta \Sigma $ TDC) for IF data conversion. The proposed TDC is based on a second-order resonator implemented by a cascade of two time-mode lossless discrete integrators (LDI), and a digital-to-time converter (DTC) in a feedback loop. The DTC is based on a new topology of double-edge voltage-controlled delay unit. The proposed TDC achieves bandpass quantization noise shaping and it does not require any complex calibration circuit to compensate for timing errors. To realize the time-mode LDI-based resonator, time-latches with some digital gates are employed. Moreover, a direct feed-forward compensation is utilized in the TDC to attain high signal-to-noise and distortion ratio (SNDR) by reducing the internal time-mode swing, thus relaxing the speed and area requirements on digital circuits leading to low-voltage operation. The prototype BP $\Delta \Sigma $ TDC achieves a 39.5-dB peak SNDR over a 0.2 MHz signal bandwidth at 42.8 MS/s while consuming less than 5 mW in a 1.2 V IBM 130-nm CMOS process. The proposed TDC is highly digital and occupies a core area of only 0.048 mm2.

Performance Evaluation Using the Discrete Choquet Integral: Higher Education Sector

Performance evaluation functions as an essential tool for decision makers in the field of measuring and assessing the performance under the multiple evaluation criteria aspect of the systems such as management, economy, and education system. Besides, academic performance evaluation is one of the critical issues in higher institution of learning. Even though the academic evaluation criteria are inherently dependent, most of the traditional evaluation methods take no account of the dependency. Currently, the discrete Choquet integral can be proposed as a useful and effective aggregation operator due to being capable of considering the interactions among the evaluation criteria. In this paper, it is aimed to solve an academic performance evaluation problem of students in a university in Turkey using the discrete Choquet integral with the complexity-based method and the entropy-based method. Moreover, the k-means method, which has been widely used for evaluating students’ performance over 50 years, is used to compare the effectiveness and the success of two different frameworks based on discrete Choquet integral in the robustness check. Our results indicate that the entropy-based Choquet integral outperforms the complexity-based Choquet and k-means method in most of the cases.