Integral Weight Research Articles

Novel Generalized Predictive Control for Photoelectric Tracking System Based on Improved Objective Function and Predictive Value Correction

Novel generalized predictive control (GPC) is proposed to deal with the problems of cumulative deviation of the predicted value, long control process and large noise that often encountered in conventional GPC. First, the objective function of GPC is improved based on integral time square error (ITSE) criterion, and the output expression under soft constraint is deduced. Then, the predicted value is corrected by adopting the integral weighting method, the convergence of which is demonstrated. Finally, both the simulations and hardware experiments are conducted in a practical photoelectric tracking system. It turns out that the superiority of the improved GPC in control precision, dynamic response and robustness is verified.

Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules

We consider the Cauchy problem of the nonlinear Landau equation of Maxwellian molecules, under the perturbation frame work to global equilibrium. We show that if $H^r_x(L^2_v), r >3/2$ norm of the initial perturbation is small enough, then the Cauchy problem of the nonlinear Landau equation admits a unique global solution which becomes analytic with respect to both position $x$ and velocity $v$ variables for any time $t>0$. This is the first result of analytic smoothing effect for the spatially inhomogeneous nonlinear kinetic equation. The method used here is microlocal analysis and energy estimates. The key point is adopting a time integral weight associated with the kinetic transport operator.

Converse theorems for automorphic distributions and Maass forms of level N

We investigate the relations for $L$-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and half-integral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of $SL(2)$. Our goal is converse theorems for automorphic distributions and Maass forms of level $N$ characterizing them by analytic properties of the associated $L$-functions. As an application of our converse theorems, we construct Maass forms from the two-variable zeta functions related to quadratic forms studied by Peter and the fourth author.

Approximating connected safe sets in weighted trees

For a graph G and a non-negative integral weight function w on the vertex set of G, a set S of vertices of G is w-safe if w(C)≥w(D) for every component C of the subgraph of G induced by S and every component D of the subgraph of G induced by the complement of S such that some vertex in C is adjacent to some vertex of D. The minimum weight w(S) of a w-safe set S is the safe number s(G,w) of the weighted graph (G,w), and the minimum weight of a w-safe set that induces a connected subgraph of G is its connected safe number cs(G,w). Bapat et al. showed that computing cs(G,w) is NP-hard even when G is a star. For a given weighted tree (T,w), they described a polynomial time 2-approximation algorithm for cs(T,w) as well as a polynomial time 4-approximation algorithm for s(T,w). Addressing a problem they posed, we present a PTAS for the connected safe number of a weighted tree. Our PTAS partly relies on an exact pseudopolynomial time algorithm, which also allows to derive an asymptotic FPTAS for restricted instances. Finally, we extend a bound due to Fujita et al. from trees to block graphs.

Shimura and Shintani liftings of certain cusp forms of half-integral and integral weights

In [4], W. Kohnen constructed explicit Shintani lifts from the space of cusp forms of weight $2k$ on $Γ_0(N)$, which are mapped into the plus space $S^+_{k+1/2}(Γ_0(4N))$, where $N$ is an odd integer. This lifting is adjoint to the (modified) Shimura map defined by him with respect to the Petersson scalar product. Using this construction along with the theory of newforms on $S^+_{k+1/2}(Γ_0(4N))$ (where $N$ is an odd square-free natural number) developed in [3], he derived explicit Waldspurger theorem for the newforms belonging to the space $S^+_{k+1/2}(Γ_0(4N))$. Further in this work, Kohnen considered the $\pm 1$ eigen subspaces corresponding to the Atkin-Lehner involution and showed that the intersection of this $\pm$ spaces with the corresponding newform spaces are isomorphic under the Shimura correspondence. A natural question is whether a similar result can be obtained for the intersection of this $(\pm1)$ subspaces with the space of oldforms. In this direction, S. Choi and C. H. Kim [2] considered the case where $N$ is an odd prime $p$ and constructed similar Shimura and Shintani maps between subspaces of forms of half-integral and integral weights. The subspaces considered by Choi and Kim are nothing but $+1$ eigenspace under the Fricke involution. In this paper, we generalise the work of Choi and Kim to the case where $N$ is an odd square-free natural number.

A Fock space model for decomposition numbers for quantum groups at roots of unity

In this paper we construct an "abstract Fock space" for general Lie types that serves as a generalisation of the infinite wedge $q$-Fock space familiar in type $A$. Specifically, for each positive integer $\ell$, we define a $\mathbb{Z}[q,q^{-1}]$-module $\mathcal{F}_{\ell}$ with bar involution by specifying generators and "straightening relations" adapted from those appearing in the Kashiwara-Miwa-Stern formulation of the $q$-Fock space. By relating $\mathcal{F}_{\ell}$ to the corresponding affine Hecke algebra we show that the abstract Fock space has standard and canonical bases for which the transition matrix produces parabolic affine Kazhdan-Lusztig polynomials. This property and the convenient combinatorial labeling of bases of $\mathcal{F}_{\ell}$ by dominant integral weights makes $\mathcal{F}_{\ell}$ a useful combinatorial tool for determining decomposition numbers of Weyl modules for quantum groups at roots of unity.

The Integral Weight and Superposition Operators Between Bloch-Type Spaces

Using the notion of the integral weight, we characterize all entire functions that transform a Bloch-type space $${\mathcal {B}}^{\mu _1}$$ into another space of the same kind $${\mathcal {B}}^{\mu _2}$$ by superposition for very general weights $$\mu _1$$ and $$\mu _2$$, satisfying a growth condition.

Sensitivity analysis and correction algorithms for atmospheric CO2 measurements with 1.57-µm airborne double-pulse IPDA LIDAR.

In this study, a 1.57-µm airborne double-pulse integrated-path differential absorption (IPDA) light detection and ranging (LIDAR) system was developed for CO2 measurements. This airborne IPDA LIDAR is integrated with a real-time frequency monitoring system, an integrated sensor for temperature, pressure, and humidity, an inertial navigation system, and a global positioning system. The random errors of the LIDAR system, which are caused by the signal noise, background noise, and detector noise, among other factors, are analyzed for different target reflectivities at a flight altitude of 8 km. After parametric optimization, the signal is unsaturated at high target reflectivity. Further, it can be detected at low target reflectivity by adjusting the detector gain. After the averaging of 148 shots, the relative random error (RRE) was 0.057% for a typical target reflectivity of 0.1 sr-1. Moreover, the systematic errors caused by the laser pulse energy, linewidth, spectral purity, and frequency drift, as well as the atmospheric parameters related to the flight experiments are also investigated. The relative system error (RSE) was 0.214% as determined based on an analysis of the systematic errors, which are primarily caused by the frequency drift. Two methods are proposed to reduce the RSE caused by the frequency drift. The first is the averaging of 148 shots, which can reduce the RSE to 0.096%. The other method involves calculating the integral weight function (IWF) using real-time frequency. However, this is a time-consuming and computationally intensive process. Hence, look-up tables for the absorption cross-section were created to overcome this issue, resulting in a decrease in the RSE to 0.096%. Using actual aircraft attitude angles, velocity, and position data from flight experiments, the relative errors (REs) in the IWF caused by the uncorrected integral path and Doppler shift were determined to be 0.273% and 0.479%, respectively. However, it was found that corrections to the integral path and Doppler shift based on accurate calculations of the IWF cause the airborne platform to turn in such a way that the REs are eliminated. Hence, this study confirms the validity of system parameters and provides a reference for other researchers who study similar IPDA LIDAR systems. Further, the sensitivity analysis of the airborne IPDA LIDAR system can provide a reference to future data inversions. Moreover, the proposed correction algorithms for the integral path and Doppler shift contribute to more accurate inversion results.

Ranking tournaments with no errors II: Minimax relation

Abstract A tournament T = ( V , A ) is called cycle Mengerian (CM) if it satisfies the minimax relation on packing and covering cycles, for every nonnegative integral weight function defined on A. The purpose of this series of two papers is to show that a tournament is CM iff it contains none of four Mobius ladders as a subgraph; such a tournament is referred to as Mobius-free. In the first paper we have given a structural description of all Mobius-free tournaments, and have proved that every CM tournament is Mobius-free. In this second paper we establish the converse by using our structural theorems and linear programming approach.

Shifted convolution sums of Fourier coefficients with squarefull kernel functions

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth normalized Fourier coefficient (Hecke eigenvalue) by λf (n). Let a(n) be the function with squarefull kernel. In this paper, we establish that $$ {\sum}_{n\leqslant x}a(n){\lambda}_f^2\left(n+1\right)= Cx+O\left({x}^{13/14+\upepsilon}\right) $$, where C is a constant that can be explicitly evaluated.

On comparing Hecke eigenvalues of cusp forms

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth Hecke eigenvalue or normalized Fourier coefficient by \(\lambda_{f}(n)\). Let g(z) be another distinct primitive holomorphic cusp form of even integral weight \(\ell\) for the full modular group. In this paper, we establish that the set $$\{p| \lambda_f(p^j)<\lambda_g(p^j)\} \quad \mbox{with } 1\leq j \leq 8$$ has analytic density at least \(\frac{1}{16[\frac{j+1}{2}]^{2}}\), and the set $$\{p| \lambda_f(p^j)^2<\lambda_g(p^j)^2\} \quad \mbox{with } 1\leq j \leq 4$$ has analytic density at least \(\frac{1}{4j(j+1)^{2}}\).

A new absolute nodal coordinate formulation beam element with multilayer circular cross section

A systematic numerical integration method is applied to the absolute nodal coordinate formulation (ANCF) fully parameterized beam element with smooth varying and continuous cross section. Moreover, the formulation for the integration points and weight coefficients are given in the method which is used to model the multilayer beam with a circular cross section. To negate the effect of the bending stiffness for the element used to model the high-voltage electrical wire, the general continuum mechanical approach is adjusted. Additionally, the insulation cover for some particular types of the wire is described by the nearly incompressible Mooney–Rivlin material model. Finally, a static problem is presented to prove the accuracy and convergence properties of the element, and a dynamic problem of a flexible pendulum is simulated whereby the balance of the energy can be ensured. An experiment is carried out in which a wire is released as a pendulum and falls on a steel rod. The configurations of the wire are captured by a high-speed camera and compared with the simulation results. The feasibility of the wire model can therefore be demonstrated.

A transport point method for complex flow problems with free surface

The material point method (MPM) has been widely used in a broad area of engineering. However, it still suffers from the accuracy problem. One of the most important sources of the accuracy problem is the error of particle quadrature. The discontinuity of the gradient of the shape function causes severe crossing-mesh error. This paper proposes a new transport point method (TPM) which employs a mixed quadrature scheme combining Gaussian quadrature in the internal cells and particle quadrature in the boundary cells. The main distinction between the TPM and MPM is the quantities carried by particles. The transport points in TPM do not have volume and only carry intensive quantities and have dual properties of both Euler and Lagrange. The MLS method is used to reconstruct quantities at cell centers and nodes, and the integral weight is the cell volume. A point rearrangement algorithm is proposed so that the transport points can be added, moved or deleted arbitrarily as long as the accuracy of the reconstructed flow field is maintained, which can be used to eliminate the numerical fracture and impose inlet condition.

Polyhedral realizations of crystal bases B(λ) for quantum algebras of nonexceptional affine types

We give explicit forms of the crystal bases B(λ) for the integrable highest weight modules of the quantum affine algebras for A2n(2), A2n−1(2), Bn(1), Cn(1), and Dn+1(2).

Fusion hierarchies, T-systems and Y-systems for the dilute loop models

The fusion hierarchy, T-system and Y-system of functional equations are the key to exact solvability for 2d lattice models. We derive these equations for the generic dilute loop models. The fused transfer matrices are associated with nodes of the infinite dominant integral weight lattice of . For generic values of the crossing parameter , the T- and Y-systems do not truncate. For the case rational so that is a root of unity, we find explicit closure relations and derive closed finite T- and Y-systems. The thermodynamics Bethe ansatz (TBA) integral equations and the associated TBA diagrams of the Y-systems are not of simple Dynkin type. They involve nodes if p is even and nodes if p is odd and are related to the TBA diagrams of models at roots of unity by a folding which originates from the addition of crossing symmetry. In an appropriate regime, the known central charges are . Prototypical examples of the loop models, at roots of unity, include critical dense polymers with central charge c = −2, and loop fugacity and critical site percolation on the triangular lattice with c = 0, and . Solving the TBA equations for the conformal data will determine whether these models lie in the same universality classes as their counterparts. More specifically, it will confirm the extent to which bond and site percolation lie in the same universality class as logarithmic conformal field theories.

A novel comprehensive risk assessment method for sudden water accidents in the Middle Route of the South–North Water Transfer Project (China)

The Middle Route of the South–North Water Transfer Project in China consists of a long open canal and complex hydraulic structures. It provides drinking water for Beijing, Shijiazhuang, and other cities under extremely strict water quality requirements. In the recent decades, water pollution accidents have frequently occurred in water transfer projects. Scientific and effective risk assessment is needed to assess the impact on the overall emergency management, which should be considered to incorporate social, economic, and environmental issues in the timely response to and management of emergencies. In this study, we combine the Drivers–Pressures–State–Impact–Response model, fuzzy comprehensive evaluation method, and coordinated development degree model into a comprehensive risk assessment tool. This new approach was tested on an emergency drilling simulation related to a sudden MRP water pollution accident in 2016. Based on the combined integration weight ranking, “water delivery status,” “pollution accident characteristics,” “town size,” and “public satisfaction” play prominent roles in the risk assessment. Especially, “town size” is identified as the most important influent factor. The Drivers–Pressures–State–Impact–Response model index system and comprehensive risk assessment method can be used to evaluate accidents more scientifically and versatile, which helps managers or experts to make faster and more efficient decisions.

Ranking tournaments with no errors I: Structural description

In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments.

A local model for the trianguline variety and applications

We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck's simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial description of the completed local rings of the fiber over the weight map, etc. Combined with the patched Hecke eigenvariety (under the usual Taylor-Wiles assumptions), these results in turn have several global consequences: classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, existence of singularities on global Hecke eigenvarieties.

Neutrino masses and mixing from double covering of finite modular groups

We extend the even weight modular forms of modular invariant approach to general integral weight modular forms. We find that the modular forms of integral weights and level N can be arranged into irreducible representations of the homogeneous finite modular group {Gamma}_N^{prime } which is the double covering of ΓN. The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of {Gamma}_3^{prime } ≅ T′. The modular forms of weights 2, 3, 4, 5 and 6 are presented. We build a model of lepton masses and mixing based on T′ modular symmetry.

Weyl–Kac type character formula for admissible representations of Borcherds–Kac–Moody Lie superalgebras

We prove that Borcherds–Kac–Moody Lie superalgebras yield Weyl–Kac type character formulas for some specific types of admissible representations. Our results generalize the results of Kac and Wakimoto (Proc Natl Acad Sci USA 85:4956–4960, 1988) by extending the Weyl–Kac character formula for integrable irreducible highest weight modules over Kac–Moody algebras to the case of admissible irreducible highest weight modules over Borcherds–Kac–Moody Lie superalgebras.