Integral Weight Research Articles

A Study on the Integration and Practice of Integrated Transfer Learning Strategies in Japanese Language Teaching and Learning

Abstract Addressing the prevalent issue of inefficiency in Japanese language learning, this study focuses on innovating classroom teaching methodologies for the Japanese language. Initially, the research employs a stochastic gradient descent algorithm to enhance the performance of a transfer learning algorithm, aiming to accurately identify student behaviors within the Japanese language classroom. Subsequently, these behaviors are quantified into integration weights using a predefined integration strategy. Behaviors demonstrating significant similarities are consolidated and analyzed to yield the results of the student behavior identification process. Furthermore, this behavior recognition approach is seamlessly incorporated into Japanese language pedagogy to develop a refined model for teaching, thereby potentially increasing the efficacy of language acquisition. Using this paper’s algorithm to identify students’ classroom behaviors and correlate them with their grades, it is found that when the number of positive behaviors in the classroom is higher, the student’s academic performance is also higher (r=0.9986, p=0.001<0.05). The average percentage of students’ attentive listening behaviors could reach 52.24% after the intervention, as determined by behavioral recognition results in the Japanese language teaching classroom. The behavior recognition algorithm in this paper provides an effective method for analyzing and intervening in students’ behavior in the classroom, and the teaching model constructed based on the algorithm offers a reference way to improve the effectiveness of Japanese language teaching.

The Path of “Dual-Teacher” Teacher Cultivation in Applied Local Higher Teacher Colleges under the Background of Big Data

Abstract The cultivation path of the applied “dual-teacher” teacher team in this paper mainly relies on the qualification standard and evaluation system. Big data technology is used to collect data, deeply mine fragmented data, and further extract index factors that affect the quality of teachers. The multi-level fuzzy hierarchical evaluation model realizes the quantification of the decision-making process with less quantitative information, and determines the weights through the multi-level comparison of importance, to test and evaluate the teaching ability of “dual-teacher” teachers. The results show that, among the 166 survey respondents, 77 respondents belong to the “ Dual-teacher” teachers, accounting for about 45.03%. The primary needs and ways for most of the teachers to participate in the training of “dual-teacher” teachers are to improve the practical ability of industry and enterprise positions (score 0.86) and the school-based training of dual-teacher teachers (0.93), which is centralized in the college. In evaluating “dual-teacher” teaching, the weight of industry-teaching integration is 0.3378, so the ability of industry-teaching integration is a more meaningful indicator. College students are satisfied with “dual-teacher teaching”, and the score shows that the essential satisfaction is more than half of the total number of samples, which is 343.

Brezis-Kamin type results involving locally integrable weights

Brezis-Kamin type results involving locally integrable weights

Pengaruh berat telur, luas permukaan, dan berat jenis, terhadap shape index telur puyuh (Coturnix coturnix japonica)

The research was held from 6th April 2023 to 6th Mei 2023 in Laboratory of Poultry Poduction, Padjadjaran University. The purpose of this research was to figure out the corellations between weight, surface area, and specific gravity on the shape index of quail eggs (Coturnix Coturnix Japonica). The method used in this research is corellational, and path analysis, with exogenous variables they are weight (X1), surface area (X2), specific gravity (X3) and one endogenous variable is shape index(Y). Sampling collects by purposive sampling with the criteria of eggshell integrity and uniform egg weight. The data obtained from the measurement results were then carried out the normality test, linearity test, Pearson correlation and path analysis. The results showed that the weight and surface area of the quail egg shape index has a correlation value of -0.194. Specific gravity with shape index quail eggs showed a correlation value of 0.013. Weight gives the greatest effect on the shape index of quail eggs (Coturnix Coturnix Japonica) by 7.51%, surface area gives the effect on the shape index of quail eggs (Coturnix Coturnix Japonica) by 0.62% and specific gravity gives a low effect on the shape index of quail eggs (Coturnix Coturnix Japonica) 0.03%.

A comparison of Rosenbrock–Wanner and Crank–Nicolson time integrators for atmospheric modelling

AbstractNonhydrostatic atmospheric models often use semi‐implicit temporal discretisations in order to negate the time‐step limitation of explicitly resolving the fast acoustic and gravity waves. Solving the resulting system to machine precision using Newton's method is considered prohibitively expensive, and so the nonlinear solver is typically truncated to a fixed number of iterations, often using an approximate Jacobian matrix that is reassembled only once per time step. The present article studies the impact of using various third‐order, four stage Rosenbrock–Wanner schemes, where integration weights are chosen to meet specific stability and order conditions, in comparison to a Crank–Nicolson time discretisation, as is done in the UK Met Office's LFRic model. Rosenbrock–Wanner schemes present a promising alternative on account of their ability to preserve their temporal order with only an approximate Jacobian, and may be constructed to be stiffly‐stable, so as to ensure the decay of fast unresolved modes. These schemes are compared for the 2D rotating shallow‐water equations and the 3D compressible Euler equations at both planetary and nonhydrostatic scales and are shown to exhibit improved results in terms of their energetic profiles and stability. Results in terms of computational performance are mixed, with the Crank–Nicolson method allowing for longer time steps and faster time to solution for the baroclinic instability test case at planetary scales, and the Rosenbrock–Wanner methods allowing for longer time steps and faster time to solution for a rising bubble test case at nonhydrostatic scales.

The continuous stochastic gradient method: part II–application and numerics

In this contribution, we present a numerical analysis of the continuous stochastic gradient (CSG) method, including applications from topology optimization and convergence rates. In contrast to standard stochastic gradient optimization schemes, CSG does not discard old gradient samples from previous iterations. Instead, design dependent integration weights are calculated to form a convex combination as an approximation to the true gradient at the current design. As the approximation error vanishes in the course of the iterations, CSG represents a hybrid approach, starting off like a purely stochastic method and behaving like a full gradient scheme in the limit. In this work, the efficiency of CSG is demonstrated for practically relevant applications from topology optimization. These settings are characterized by both, a large number of optimization variables and an objective function, whose evaluation requires the numerical computation of multiple integrals concatenated in a nonlinear fashion. Such problems could not be solved by any existing optimization method before. Lastly, with regards to convergence rates, first estimates are provided and confirmed with the help of numerical experiments.

The continuous stochastic gradient method: part I–convergence theory

In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function requires some form of integration, e.g., expected values. Since approximating the integration by a fixed quadrature rule can introduce artificial local solutions into the problem while simultaneously raising the computational effort, stochastic optimization schemes have become increasingly popular in such contexts. However, known stochastic gradient type methods are typically limited to expected risk functions and inherently require many iterations. The latter is particularly problematic, if the evaluation of the cost function involves solving multiple state equations, given, e.g., in form of partial differential equations. To overcome these drawbacks, a recent article introduced the CSG method, which reuses old gradient sample information via the calculation of design dependent integration weights to obtain a better approximation to the full gradient. While in the original CSG paper convergence of a subsequence was established for a diminishing step size, here, we provide a complete convergence analysis of CSG for constant step sizes and an Armijo-type line search. Moreover, new methods to obtain the integration weights are presented, extending the application range of CSG to problems involving higher dimensional integrals and distributed data.

Multiplicities of maximal weights of the sℓ ̂(n)-module V (kΛ0)

Consider the affine Lie algebra [Formula: see text] with null root [Formula: see text], weight lattice [Formula: see text] and set of dominant weights [Formula: see text]. Let [Formula: see text] [Formula: see text] denote the integrable highest weight [Formula: see text]-module with level [Formula: see text] highest weight [Formula: see text]. Let [Formula: see text] denote the set of weights of [Formula: see text]. A weight [Formula: see text] is a maximal weight if [Formula: see text]. Let [Formula: see text] denote the set of maximal dominant weights which is known to be a finite set. The explicit description of the weights in the set [Formula: see text] is known [R. L. Jayne and K. C. Misra, On multiplicities of maximal dominant weights of [Formula: see text]-modules, Algebr. Represent. Theory 17 (2014) 1303–1321]. In papers [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb. 22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the [Formula: see text]-modules [Formula: see text], Algebr. Represent. Theory 25 (2022) 477–490], the multiplicities of certain subsets of [Formula: see text] were given in terms of some pattern-avoiding permutations using the associated crystal base theory. In this paper the multiplicity of all the maximal dominant weights of the [Formula: see text]-module [Formula: see text] are given generalizing the results in [R. L. Jayne and K. C. Misra, Lattice paths, Young tableaux, and weight multiplicities, Ann. Comb. 22 (2018) 147–156; R. L. Jayne and K. C. Misra, Multiplicities of some maximal dominant weights of the [Formula: see text]-modules [Formula: see text], Algebr. Represent. Theory 25 (2022) 477–490].

Prime representations in the Hernandez–Leclerc category: classical decompositions

Abstract We use the dual functional realization of loop algebras to study the prime irreducible objects in the Hernandez–Leclerc (HL) category for the quantum affine algebra associated with $\mathfrak {sl}_{n+1}$ . When the HL category is realized as a monoidal categorification of a cluster algebra (Hernandez and Leclerc (2010, Duke Mathematical Journal 154, 265–341); Hernandez and Leclerc (2013, Symmetries, integrable systems and representations, 175–193)), these representations correspond precisely to the cluster variables and the frozen variables are minimal affinizations. For any height function, we determine the classical decomposition of these representations with respect to the Hopf subalgebra $\mathbf {U}_q(\mathfrak {sl}_{n+1})$ and describe the graded multiplicities of their graded limits in terms of lattice points of convex polytopes. Combined with Brito, Chari, and Moura (2018, Journal of the Institute of Mathematics of Jussieu 17, 75–105), we obtain the graded decomposition of stable prime Demazure modules in level two integrable highest weight representations of the corresponding affine Lie algebra.

A note on odd zeta values over any number field and extended Eisenstein series

In this article, we study transformation formulas of zeta function at odd integers over an arbitrary number field which in turn generalizes Ramanujan's identity for the Riemann zeta function. Our transformation formula for the Dedekind zeta function at odd integers leads to a new number field extension of Eisenstein series, which satisfies the transformation z↦−1/z like an integral weight modular form over SL2(Z). The results provide a number of important applications, which are important in studying the behavior of odd zeta values as well as Lambert series in an arbitrary number field.

Representation type of cyclotomic quiver Hecke algebras of type [formula omitted

We first investigate a connected quiver consisting of all dominant maximal weights for an integrable highest weight module in affine type A. This quiver provides an efficient method to obtain all dominant maximal weights. Then, we completely determine the representation type of cyclotomic Khovanov-Lauda-Rouquier algebras of arbitrary level in affine type A, by using the quiver we construct. This result gives a complete classification for the representation type of blocks of cyclotomic Hecke algebras since cyclotomic KLR algebras of type Aℓ(1) form a one-parameter family and cyclotomic Hecke algebras occur at a special parameter, i.e., t=−2 if ℓ=1 and t=(−1)ℓ+1 if ℓ≥2.

Scarcity of congruences for the partition function

abstract: The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0$ $({\rm mod}\;\ell)$ for the primes $\ell=5,7,11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+\beta)\equiv 0$ $({\rm mod}\;\ell)$ where $Q\geq 5$ is prime and $m\geq 3$. This leaves open the question of the existence of such congruences when $m=1$ or $m=2$ (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve $X(\ell Q)$, Galois representations and the arithmetic large sieve.

Scalar‐valued depth two Eichler–Shimura integrals of cusp forms

AbstractGiven cusp forms and of integral weight , the depth two holomorphic iterated Eichler–Shimura integral is defined by , where is the Eichler integral of and are formal variables. We provide an explicit vector‐valued modular form whose top components are given by . We show that this vector‐valued modular form gives rise to a scalar‐valued iterated Eichler integral of depth two, denoted by , that can be seen as a higher depth generalization of the scalar‐valued Eichler integral of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol–Popa. We show that can be expressed in terms of sums of products of components of vector‐valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form . This allows for effective computation of .

The double Gelfand–Cetlin system, invariance of polarization, and the Peter–Weyl theorem

The bundle map T⁎U(n)⟶U(n) provides a real polarization of the cotangent bundle T⁎U(n), and yields the geometric quantization Q1(T⁎U(n))=L2(U(n)). We use the Gelfand–Cetlin systems of Guillemin and Sternberg to show that T⁎U(n) has a different real polarization with geometric quantization Q2(T⁎U(n))=⨁αVα⊗Vα⁎, where the sum is over all dominant integral weights α of U(n). The Peter–Weyl theorem, which states that these two quantizations are isomorphic, may therefore be interpreted as an instance of “invariance of polarization” in geometric quantization.

Components of $$V(\rho ) \otimes V(\rho )$$ and Dominant Weight Polyhedra for Affine Kac–Moody Lie Algebras

AbstractKostant asked the following question: Let $$\mathfrak {g}$$ g be a simple Lie algebra over the complex numbers. Let $$\lambda $$ λ be a dominant integral weight. Then, $$V(\lambda )$$ V ( λ ) is a component of $$V(\rho )\otimes V(\rho )$$ V ( ρ ) ⊗ V ( ρ ) if and only if $$\lambda \le 2 \rho $$ λ ≤ 2 ρ under the usual Bruhat–Chevalley order on the set of weights. In an earlier work with R. Chirivi and A. Maffei, the second author gave an affirmative answer to this question up to a saturation factor. The aim of the current work is to extend this result to untwisted affine Kac–Moody Lie algebra $$\mathfrak {g}$$ g associated with any simple Lie algebra $$\mathring{\mathfrak {g}}$$ g ˚ (up to a saturation factor). In fact, we prove the result for affine $$sl_n$$ s l n without any saturation factor. Our proof requires some additional techniques including the Goddard–Kent–Olive construction and study of the characteristic cone of non-compact polyhedra.

Toric Periods and 𝑝-adic Families of Modular Forms of Half-Integral Weight

The primary goal of this work is to construct p p -adic families of modular forms of half-integral weight, by using Waldspurger’s automorphic framework to make the results as comprehensive and precise as possible. A secondary goal is to clarify the role of test vectors as defined by Gross-Prasad in the elucidation of general formulae for the Fourier coefficients of modular forms of half-integral weight in terms of toric periods of the corresponding modular forms of integral weight. As a consequence of our work, we develop a generalization of a classical formula due to Shintani, and make precise the conditions under which Shintani’s lift vanishes. We also give a number of results on test vectors for ramified representations which are of independent interest.

Continuous Dependence of Szegő Kernel on a Weight of Integration

The weighted Szegő kernel was investigated in a few papers (see Nehari in J d’Analyse Mathématique 2:126–149, 1952; Alenitsin in Zapiski Nauchnykh Seminarov LOMI 24:16–28, 1972; Uehara and Saitoh in Mathematica Japonica 29:887–891, 1984; Uehara in Mathematica Japonica 42:459–469, 1995). In all of these, however, only continuous weights were considered. The aim of this paper is to show that the Szegő kernel depends in a continuous way on a weight of integration in the case when the weights are not necessarily continuous. A topology on the set of admissible weights will be constructed and Pasternak’s theorem (see Pasternak-Winiarski in Studia Mathematica 128:1, 1998) on the dependence of the orthogonal projector on a deformation of an inner product will be used in the proof of the main theorem.

Forecast combination using grey prediction with fuzzy integral and time-varying weighting in tourism

PurposeTourism demand forecasting is vital for the airline industry and tourism sector. Combination forecasting has the advantage of fusing several forecasts to reduce the risk of inappropriate model selection for analyzing decisions. This paper investigated the effects of a time-varying weighting strategy on the performance of linear and nonlinear forecast combinations in the context of tourism.Design/methodology/approachThis study used grey prediction models, which did not require that the available data satisfy statistical assumptions, to generate forecasts. A quality-control technique was applied to determine when to change the combination weights to generate combined forecasts by using linear and nonlinear methods.FindingsThe empirical results showed that except for when the Choquet fuzzy integral was used, forecast combination with time-varying weights did not significantly outperform that with fixed weights. The Choquet integral with time-varying weights significantly outperformed that with fixed weights for all model combinations, and had a superior forecasting accuracy to those of other combination methods.Practical implicationsThe tourism sector can benefit from the use of the Choquet integral with time-varying weights, by using it to formulate suitable strategies for tourist destinations.Originality/valueCombining forecasts with time-varying weights may improve the accuracy of the predictions. This study investigated incorporating a time-varying weighting strategy into combination forecasting by using CUSUM. The results verified the effectiveness of the time-varying Choquet integral for tourism forecast combination.

Ramanujan systems of Rankin–Cohen type and hyperbolic triangles

Abstract In the first part of the paper, we characterize certain systems of first-order nonlinear differential equations whose space of solutions is an s ⁢ l 2 ⁢ ( C ) \mathfrak{sl}_{2}(\mathbb{C}) -module. We prove that such systems, called Ramanujan systems of Rankin–Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin–Cohen structure. In the second part of the paper, we consider triangle groups Δ ⁢ ( n , m , ∞ ) \Delta(n,m,\infty) . By means of modular embeddings, we associate to every such group a number of systems of nonlinear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on Δ ⁢ ( n , m , ∞ ) \Delta(n,m,\infty) are generated by the solutions of one such system (which is of Rankin–Cohen type). As a corollary, we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non-classical setting, we construct the space of integral weight twisted modular form on Δ ⁢ ( 2 , 5 , ∞ ) \Delta(2,5,\infty) from solutions of systems of nonlinear ODEs.

Effect of an efficient numerical integration technique on the element-free Galerkin method

The reproducing kernel gradient smoothing integration (RKGSI) is an efficient technique to tackle the integration problem and optimal convergence in meshless methods. In this paper, the effect of the RKGSI on the element-free Galerkin (EFG) method is studied for elliptic boundary value problems with mixed boundary conditions. Theoretical results of smoothed gradients in the RKGSI are provided. Fundamental criteria on how to determine integration points and weights of quadrature rules are established according to necessary algebraic precision. By using the Nitsche's technique to impose Dirichlet boundary condition, the existence, uniqueness and error estimations of the solution of the EFG method with numerical integration are analyzed. Numerical results validate the theoretical analysis and the optimal convergence of the method.