Integral Weight Research Articles

On non-superperfection of edge intersection graphs of paths

The routing and spectrum assignment problem in modern flexgrid elastic optical networks asks for assigning to given demands a route in an optical network and a channel within an optical frequency spectrum so that the channels of two demands are disjoint whenever their routes share a link in the optical network. This problem can be modeled in two phases: firstly, a selection of paths in the network and, secondly, an interval coloring problem in the edge intersection graph of these paths. The interval chromatic number equals the smallest size of a spectrum such that a proper interval coloring is possible, the weighted clique number is a natural lower bound. Graphs where both parameters coincide for all possible non-negative integral weights are called superperfect. Therefore, the occurrence of non-superperfect edge intersection graphs of routing paths can provoke the need of larger spectral resources. In this work, we examine the question which minimal non-superperfect graphs can occur in the edge intersection graphs of routing paths in different underlying networks: when the network is a path, a tree, a cycle, or a sparse planar graph with small maximum degree. We show that for any possible network (even if it is restricted to a path) the resulting edge intersection graphs are not necessarily superperfect. We close with a discussion of possible consequences and of some lines of future research.

An adaptive weighted integration method for multipath ultrasonic flowmeter

Abstract The error introduced by the integration method is an important factor affecting the measurement accuracy of the multipath ultrasonic flowmeter. An adaptive weighted integration method (adaptive weighted integration method for velocity profile of circular section) is proposed to reduce the integration error, taking a DN400 double-side eight-path ultrasonic flowmeter as an example. This method is based on the velocity distribution information in the full flow range and the integration weights are determined by the principle of minimum error. The applicability of this method is verified by numerical simulation and actual fluid flow experiments. The results show that the integration error of the proposed method is superior to the Gauss–Jacobi and optimized weighted integration method for circular section integration methods, and the maximum integration error is reduced from 0.0877% and −0.0355% to 0.0220% in the flow range of 125–2500 t h−1.

Lie semisimple algebras of derivations and varieties of PI-algebras with almost polynomial growth

We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras U T 2 ( W λ ) UT_2(W_\lambda ) or E n d ( W μ ) End(W_\mu ) for some integral dominant weight λ , μ \lambda ,\mu with μ ≠ 0 \mu \neq 0 . In the special case L = s l 2 L=\mathfrak {sl}_2 we prove that this is a sufficient condition too.

On the Shintani lifting of integral weight modular forms to half-integral weight modular forms

On the Shintani lifting of integral weight modular forms to half-integral weight modular forms

A Fock space approach to socle series of Grothendieck group over Lie superalgebra o s p ( 2 | 2 )

We explore two modules F and K over a subalgebra U of the quantum group U q ( g l ( ∞ ) ) . The first U -module is the Fock space related to the natural module over the quantum group. If we let O + to be the category of finite-dimensional o s p ( 2 | 2 ) -modules of integral weights, the second U -module K is the vector space obtained by tensoring the Grothendieck group K ( O + ) by the field Q . Using thick tensor ideals and a U -isomorphism, we compute the socles of F and K .

On the optimality and decay of p-Hardy weights on graphs

We construct optimal Hardy weights to subcritical energy functionals h associated with quasilinear Schrödinger operators on infinite graphs. Here, optimality means that the weight w is the largest possible with respect to a partial ordering, and that the corresponding shifted energy functional h-w\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h-w$$\\end{document} is null-critical. Moreover, we show a decay condition of Hardy weights in terms of their integrability with respect to certain integral weights. As an application of the decay condition, we show that null-criticality implies optimality near infinity. We also briefly discuss an uncertainty-type principle, a Rellich-type inequality and examples.

Representations of the cyclotomic oriented Brauer-Clifford supercategory

Let k be an algebraically closed field with characteristic p different from 2. We generalize the notion of a weakly triangular decomposition in [7] to the super case called a super weakly triangular decomposition. We show that the underlying even category of locally finite-dimensional left A-supermodules is an upper finite fully stratified category in the sense of [6, Definition 3.34] if the superalgebra A admits an upper finite super weakly triangular decomposition. In particular, when A is the locally unital superalgebra associated with the cyclotomic oriented Brauer-Clifford supercategory in [1], the Grothendieck group of the category of left A-supermodules admitting finite standard flags has a g-module structure that is isomorphic to the tensor product of an integrable lowest weight and an integrable highest weight g-module, where g is the complex Kac-Moody Lie algebra of type A2ℓ(2) (resp., B∞) if p=2ℓ+1 (resp., p=0).

Graph Aggregating-Repelling Network: Do Not Trust All Neighbors in Heterophilic Graphs

Graph neural networks (GNNs) have demonstrated exceptional performance in processing various types of graph data, such as citation networks and social networks, etc. Although many of these GNNs prove their superiority in handling homophilic graphs, they often overlook the other kind of widespread heterophilic graphs, in which adjacent nodes tend to have different classes or dissimilar features. Recent methods attempt to address heterophilic graphs from the graph spatial domain, which try to aggregate more similar nodes or prevent dissimilar nodes with negative weights. However, they may neglect valuable heterophilic information or extract heterophilic information ineffectively, which could cause poor performance of downstream tasks on heterophilic graphs, including node classification and graph classification, etc. Hence, a novel framework named GARN is proposed to effectively extract both homophilic and heterophilic information. First, we analyze the shortcomings of most GNNs in tackling heterophilic graphs from the perspective of graph spectral and spatial theory. Then, motivated by these analyses, a Graph Aggregating-Repelling Convolution (GARC) mechanism is designed with the objective of fusing both low-pass and high-pass graph filters. Technically, it learns positive attention weights as a low-pass filter to aggregate similar adjacent nodes, and learns negative attention weights as a high-pass filter to repel dissimilar adjacent nodes. A learnable integration weight is used to adaptively fuse these two filters and balance the proportion of the learned positive and negative weights, which could control our GARC to evolve into different types of graph filters and prevent it from over-relying on high intra-class similarity. Finally, a framework named GARN is established by simply stacking several layers of GARC to evaluate its graph representation learning ability on both the node classification and image-converted graph classification tasks. Extensive experiments conducted on multiple homophilic and heterophilic graphs and complex real-world image-converted graphs indicate the effectiveness of our proposed framework and mechanism over several representative GNN baselines.

Multinomial Logistic Regression with Adaptive Regularization for Cancer Subtype Classification via Multi-omics Data

Background: Integrating multi-omics data for cancer classification brings complementary biological insights while also facing challenges such as data integration, gene grouping, and adaptive weight construction. Objective: This paper aims to address the challenges faced by the cancer subtype classification and gene screening based on multi-omics data. Methods: Multinomial logistic regression with adaptive regularization (MLRAR) was proposed by integrating DNA methylation, gene mutation, and RNA-seq information. A data preprocessing strategy that effectively utilizes multi-omics information was presented, and the local maximum quasiclique merging (lmQCM) algorithm was implemented to group genes. Biological pathway information was utilized to evaluate the significance of gene groups, while the significance of each gene within a group was evaluated by integrating mutation information, information theory, and methylation information. Results: Compared to MRlasso, MRGL, MSGL, MROGL, AMRSOGL, and AGLRMR, the proposed method yielded improvements in subtype classification accuracy of breast cancer by 2.6%, 2.9%, 3.5%, 2.3%, 2.0%, and 1.8%, respectively. In addition, MLRAR also achieved significant improvements in ovarian cancer by 8.2%, 5.0%, 6.8%, 5.2%, 12.7%, and 6.3%, respectively. Conclusion: The proposed method can effectively deal with data integration, gene grouping, and adaptive weight construction.

On the Fourier Coefficients for General Product 𝐿-Functions

Let 𝑓 be a normalized primitive cusp form of even integral weight for the full modular group Γ = 𝑆𝐿(2, ℤ). In this paper, we investigate upper bounds for the error terms related to the average behavior of Fourier coefficients 𝜆𝑓 ⊗𝑓 ⊗⋯⊗𝑙𝑓 (𝑛) of 𝑙-fold product 𝐿-functions, where 𝑓 ⊗ 𝑓 ⊗ ⋯ ⊗𝑙 𝑓 denotes the 𝑙-fold product of 𝑓. These results improves and generalizes the recent developments of Venkatasubbareddy and Sankaranarayanan [41]. We also provide some other similar results related to the error terms of general product 𝐿-functions.

Period-like polynomials for L-series associated with half-integral weight cusp forms

Given the L-series of a half-integral weight cusp form, we construct polynomials behaving similarly to the classical period polynomial of an integral weight cusp form. We also define a lift of half-integral weight cusp forms to integral weight cusp forms that are compatible with the L-series of the respective forms.

On the sign changes of the coefficients attached to triple product L-functions

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights [Formula: see text] and [Formula: see text] for the full modular group SL[Formula: see text], respectively. In this paper, we investigate the sign changes of the sequence [Formula: see text] in short intervals, where [Formula: see text] denote the normalized coefficients of triple product L-functions [Formula: see text] attached to f and [Formula: see text], respectively. We provide quantitative results for the number of sign changes of the sequences mentioned above in short intervals.

Preliminary analysis of global column-averaged CO2 concentration data from the spaceborne aerosol and carbon dioxide detection lidar onboard AEMS

In contrast to the passive remote sensing of global CO2 column concentrations (XCO2), active remote sensing with a lidar enables continuous XCO2 measurements throughout the entire atmosphere in daytime and nighttime. The lidar could penetrate most cirrus and is almost unaffected by aerosols. Atmospheric environment monitoring satellite (AEMS, also named DQ-1) aerosol and carbon dioxide detection Lidar (ACDL) is a novel spaceborne lidar that implements a 1572 nm integrated path differential absorption (IPDA) method to measure the global XCO2 for the first time. In this study, special methods have been developed for ACDL data processing and XCO2 retrieval. The CO2 measurement data products of ACDL, including the differential absorption optical depth between the online and offline wavelengths, the integral weighting function, and XCO2, are presented. The results of XCO2 measurements over the period from 1st June 2022 to 30th June 2022 (first month data of ACDL) are analyzed to demonstrate the measurement capabilities of the spaceborne ACDL system.

On the density hypothesis for $L$-functions associated with holomorphic cusp forms

We study the range of validity of the density hypothesis for the zeros of L -functions associated with cusp Hecke eigenforms f of even integral weight, and prove that N_{f}(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon} holds for \sigma \geq 1407/1601 . This improves upon a result of Ivić, who had previously shown the zero-density estimate in the narrower range \sigma\geq 53/60 . Our result relies on an improvement of the large value estimates for Dirichlet polynomials based on mixed moment estimates for the Riemann zeta function. The main ingredients in our proof are the Halász–Montgomery inequality, Ivić’s mixed moment bounds for the zeta function, Huxley’s subdivision argument, Bourgain’s dichotomy approach, and Heath-Brown’s bound for double zeta sums.

A boundary point interpolation method for acoustic problems with particular emphasis on the calculation of Cauchy principal value integrals

A boundary point interpolation method (BPIM) is presented in this paper for numerical solving acoustic problems. The BPIM is a boundary-type meshless method that combines the point interpolation technique for constructing approximate functions with boundary integral equations for reformulating boundary value problems. Since effective numerical integration techniques are crucial in the BPIM and boundary integral equation methods for calculating regular and singular integrals, two Clenshaw-Curtis integration techniques (CCITs) for regular and weakly singular integrals are derived in a unified manner with the same integration points on quadratic integration cells. In addition, a unilateral CCIT, a power series expansion technique, and an improved CCIT are proposed to effectively calculate strongly singular integrals or Cauchy principal value integrals in the BPIM. These integration techniques provide direct and efficient formulas for calculating boundary integrals on quadratic integration cells, and can be directly applied to the calculation of regular and singular integrals in the boundary element method and other meshless boundary integral equation methods. Explicit expressions of integration weights in all CCITs are derived, and the values of integration points and weights are listed. Numerical results are finally provided to validate the effectiveness of the present meshless method and integration techniques.

Is Single Enough? A Joint Spatiotemporal Feature Learning Framework for Multivariate Time Series Prediction.

A fuzzy cognitive map (FCM) is a simple but effective tool for modeling and predicting time series. This article focuses on the problem of multivariate time series prediction (TSP), which is essential and challenging in data mining. Although several FCM-based approaches have been designed to solve this problem, their feature extraction module designed for single mode falls short in capturing the nonlinear spatiotemporal dependencies among variates, thereby resulting in low prediction accuracy in forecasting multivariate time series, which shows that the single mode learning is not enough. Therefore, in this article, we propose a joint spatiotemporal feature learning framework for multivariate TSP, where a mix-resolution spatial module consisting of multiple sparse autoencoders (SAEs) is designed to extract the feature series with different spatial resolutions, and a mix-order spatiotemporal module concluding multiple high-order FCMs (HFCMs) is designed to model the spatiotemporal dynamics of these feature series. Finally, the outputs of the two modules are concatenated to predict future values. We refer to this framework as the spatiotemporal FCM (STFCM). Especially, an efficient learning algorithm is designed to update the integral weights of STFCM based on the batch gradient descent algorithm when it deems necessary. We validate the performance of the STFCM on four real-world datasets. Compared with the existing state-of-the-art (SOTA) methods, the experimental results not only show the advantages of the two designed modules in the STFCM but also show the excellent performance of the STFCM.

The R-matrix presentation for the rational form of a quantized enveloping algebra

Let Uq(g) denote the rational form of the quantized enveloping algebra associated to a complex simple Lie algebra g. Let λ be a nonzero dominant integral weight of g, and let V be the corresponding type 1 finite-dimensional irreducible representation of Uq(g). Starting from this data, the R-matrix formalism for quantum groups outputs a Hopf algebra URλ(g) defined in terms of a pair of generating matrices satisfying well-known quadratic matrix relations. In this paper, we prove that this Hopf algebra admits a Chevalley–Serre type presentation which can be recovered from that of Uq(g) by adding a single invertible quantum Cartan element. We simultaneously establish that URλ(g) can be realized as a Hopf subalgebra of the tensor product of the space of Laurent polynomials in a single variable with the quantized enveloping algebra associated to the lattice generated by the weights of V. The proofs of these results are based on a detailed analysis of the homogeneous components of the matrix equations and generating matrices defining URλ(g), with respect to a natural grading by the root lattice of g compatible with the weight space decomposition of End(V).

On certain partial sums involving squares of Hecke eigenvalues

Let $a(n)$ be the $n$th Fourier coefficient of a cuspidal Hecke eigenform of even integral weight $k\ge 2$ and trivial character that is a normalized new form for some level $N$. We show that the partial sums$$H_n=\sum_{m=1}^n a(m)^2/m^k$$are not integral for $n\ge n_0$.

Optimizing Controls to Track Moving Targets in an Intelligent Electro-Optical Detection System

Electro-optical detection systems face numerous challenges due to the complexity and difficulty of targeting controls for “low, slow and tiny” moving targets. In this paper, we present an optimal model of an advanced n-step adaptive Kalman filter and gyroscope short-term integration weighting fusion (nKF-Gyro) method with targeting control. A method is put forward to improve the model by adding a spherical coordinate system to design an adaptive Kalman filter to estimate target movements. The targeting error formation is analyzed in detail to reveal the relationship between tracking controller feedback and line-of-sight position correction. Based on the establishment of a targeting control coordinate system for tracking moving targets, a dual closed-loop composite optimization control model is proposed. The outer loop is used for estimating the motion parameters and predicting the future encounter point, while the inner loop is used for compensating the targeting error of various elements in the firing trajectory. Finally, the modeling method is substituted into the disturbance simulation verification, which can monitor and compensate for the targeting error of moving targets in real time. The results show that in the optimal model incorporating the nKF-Gyro method with targeting control, the error suppression was increased by up to 36.8% compared to that of traditional KF method and was 25% better than that of the traditional nKF method.

Optimal Hermite-Fejér interpolation of algebraic polynomials and the best one-sided approximation on the interval [−1,1

In this paper, we give the relations of the optimal Hermite-Fejér interpolation and the best one-sided approximation to the smooth function classes BC2r,+, r∈N in weighted space L1,ω([−1,1]), with a positive, continuous and integrable weight function ω on (−1,1). We proved that the Hermite-Fejér interpolation based on the set of the zeros of some orthogonal polynomials is optimal respectively in the space L1,ω([−1,1]) and the space L∞([−1,1]) and gave the exact constants of the approximation errors of these Hermite-Fejér interpolation. We also obtained the exact constant of the approximation errors of the optimal Hermite-Fejér interpolation when the endpoints of the interval [−1,1] are included in the considered sets of the nodes.