Integral Weight Research Articles

Eighth-order explicit two-step hybrid methods with symmetric nodes and weights for solving orbital and oscillatory IVPs

The construction of new two-step hybrid (TSH) methods of explicit type with symmetric nodes and weights for the numerical integration of orbital and oscillatory second-order initial value problems (IVPs) is analyzed. These methods attain algebraic order eight with a computational cost of six or eight function evaluations per step (it is one of the lowest costs that we know in the literature) and they are optimal among the TSH methods in the sense that they reach a certain order of accuracy with minimal cost per step. The new TSH schemes also have high dispersion and dissipation orders (greater than 8) in order to be adapted to the solution of IVPs with oscillatory solutions. The numerical experiments carried out with several orbital and oscillatory problems show that the new eighth-order explicit TSH methods are more efficient than other standard TSH or Numerov-type methods proposed in the scientific literature.

Extending the WEDBA to the fuzzy multi criteria decision making environment

This paper extends the Weighted Euclidean Distance Based Approach (WEDBA) to the fuzzy environment in MCDM. Fuzzy concepts are introduced, in which alternative ratings and criteria weights are described by linguistic terms. WEDBA features the usage of integrated criteria weights, objective and subjective, which enhances its performance and overcomes the drawbacks of methods that only use one type of weight such as AHP and TOPSIS. Two examples are provided to highlight the procedure of the proposed method named Fuzzy WEDBA. The first example is a fuzzy MCDM problem, which aims to evaluate five suppliers of an automotive production company. The second is a real-life case study to select a contractor for a construction project. A comparative analysis is conducted based on several factors. The proposed method has greater time complexity, due to using integrated criteria weights, but performs similarly to F-TOPSIS in most of the evaluation factors, and both perform better than F-AHP with regard to the remaining factors.

Cloud manufacturing service evaluation based on modular neural network

In order to solve the evaluation problem of cloud manufacturing service and then minimise the time of choosing the best service, a comprehensive evaluation index system of cloud manufacturing services is constructed in the light of the characteristics of cloud manufacturing service. The service attributes, manufacturing attributes and network attributes are taken into account from two views of the online characteristics and the offline characteristics of cloud manufacturing service in the evaluation index system. And on this basis, a quantitative evaluation model based on modular neural network (MNN) is proposed. Firstly, the clustering algorithm subtractive-k-means is introduced and used to divide the task into several subtasks based on the idea of modularisation. Then, the subtasks are processed through the trained BP network. Finally, the self-adaptive genetic algorithm is used to optimise the integration weights of each subtask processing module. Combining with the cloud manufacturing evaluation index system, this modular neural network model can be used to effectively evaluate the cloud manufacturing service and help customers quickly select the best cloud manufacturing service.

Shimura lifts of weakly holomorphic modular forms

We show how to realize the Shimura lift of arbitrary level and character using the vector-valued theta lifts of Borcherds. Using the regularization of Borcherds' lift we extend the Shimura lift to take weakly holomorphic modular forms of half-integral weight to meromorphic modular forms of even integral weight having poles at CM points.

Representations of superconformal algebras and mock theta functions

It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $\hat{\frak{g}}$ span an $SL_2(\mathbf{Z})$-invariant space. This result extends to admissible $\hat{\frak{g}}$-modules, where $\frak{g}$ is a simple Lie algebra or $osp_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $\hat{\frak{g}}$-modules when $\frak{g} =sl_2$ (resp. $=osp_{1|2}$) one obtains minimal series modules over the Virasoro (resp. $N=1$ superconformal algebras), which form modular invariant families. Another instance of modular invariance occurs for boundary level admissible modules, including when $\frak{g}$ is a basic Lie superalgebra. For example, if $\frak{g}=sl_{2|1}$ (resp. $=osp_{3|2}$), we thus obtain modular invariant families of $\hat{\frak{g}}$-modules, whose QHR produces the minimal series modules for the $N=2$ superconformal algebras (resp. a modular invariant family of $N=3$ superconformal algebra modules). However, in the case when $\frak{g}$ is a basic Lie superalgebra different from a simple Lie algebra or $osp_{1|n}$, modular invariance of normalized supercharacters of admissible $\hat{\frak{g}}$-modules holds outside of boundary levels only after their modification in the spirit of Zwegers' modification of mock theta functions. Applying the QHR, we obtain families of representations of $N=2,3,4$ and big $N=4$ superconformal algebras, whose modified (super)characters span an $SL_2(\mathbf{Z})$-invariant space.

Use of Legendre multiwavelets to solve Carleman type singular integral equations

• Multiscale representation of Cauchy singular integral operator and weight ω ( x ) ( = x α ( 1 − x ) β , α , β > − 1 ) is given in LMW bases. • Estimation formula of local Holder exponent helps to recast the equation with bounded solution throughout the domain. • A posteriori L 2 error of approximate solution u J M S ( x ) is given in terms of wavelet coefficients of resolutions J 2, J 1. • Numerical examples from the theory of elasticity having real as well as complex exponents of the solution are presented. • The stress intensity factor at the tip of the crack can be efficiently estimated with the aid of the approximate solution. This paper is concerned with obtaining approximate numerical solutions of Carleman type singular integral equations by using Legendre multiwavelet basis. The properties of Legendre multiwavelets are first given and the low- and high-pass filters for two-scale relations involving Legendre multiwavelets have been derived. The scheme proposed here consists of two basic steps. In the first step the multiscale representation of the given integral operator is obtained and explicit expressions for the elements of the matrix associated with the multiscale representation are derived. In the second step the given integral equation is converted into a system of linear algebraic equations. The convergence of the method is proved in L 2 space. Two illustrative examples from elasticity are given to show the efficiency of the method developed here.

REPRESENTATION-THEORETIC INTERPRETATION OF CHEREDNIK-ORR’S RECURSION FORMULA FOR THE SPECIALIZATION OF NONSYMMETRIC MACDONALD POLYNOMIALS AT T = ∞

We give a representation-theoretic (or rather, crystal-theoretic) proof of Cherednik-Orr's recursion formula of Demazure type for the specialization at t = ∞ of the nonsymmetric Macdonald polynomials Ewλ(q, t), w ∈ W, where λ is a dominant integral weight and W is a finite Weyl group.

Specialization of nonsymmetric Macdonald polynomials at 𝑡=∞ and Demazure submodules of level-zero extremal weight modules

In this paper, we give a representation-theoretic interpretation of the specialization E w ∘ λ ( q , ∞ ) E_{w_\circ \lambda } (q,\infty ) of the nonsymmetric Macdonald polynomial E w ∘ λ ( q , t ) E_{w_\circ \lambda }(q,t) at t = ∞ t=\infty in terms of the Demazure submodule V w ∘ − ( λ ) V_{w_\circ }^- (\lambda ) of the level-zero extremal weight module V ( λ ) V(\lambda ) over a quantum affine algebra of an arbitrary untwisted type. Here, λ \lambda is a dominant integral weight, and w ∘ w_\circ denotes the longest element in the finite Weyl group W W . Also, for each x ∈ W x \in W , we obtain a combinatorial formula for the specialization E x λ ( q , ∞ ) E_{x \lambda } (q, \infty ) at t = ∞ t=\infty of the nonsymmetric Macdonald polynomial E x λ ( q , t ) E_{x \lambda } (q,t) and also a combinatorical formula for the graded character gch ⁡ V x − ( λ ) \operatorname {gch} V_{x}^- (\lambda ) of the Demazure submodule V x − ( λ ) V_{x}^- (\lambda ) of V ( λ ) V(\lambda ) . Both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape λ \lambda .

Gelfand–Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $\boldsymbol{{SU}(p,q)}$

Abstract Let $(G,K)$ be an irreducible Hermitian symmetric pair of non-compact type with $G={SU}(p,q)$, and let $\lambda$ be an integral weight such that the simple highest weight module $L(\lambda)$ is a Harish-Chandra $({\mathfrak{g}},K)$-module. We give a combinatorial algorithm for the Gelfand–Kirillov (GK) dimension of $L(\lambda)$. This enables us to prove that the GK dimension of $L(\lambda)$ decreases as the integer $\langle{\lambda+\rho},{\beta}^{\vee} \rangle$ increases, where $\rho$ is the half sum of positive roots and $\beta$ is the maximal non-compact root. Finally by the combinatorial algorithm, we obtain a description of the associated variety of $L(\lambda)$.

Irreducibility of induced supermodules for general linear supergroups

In this note we determine when an induced supermodule HG0(λ), corresponding to a dominant integral highest weight λ of the general linear supergroup G=GL(m|n), is irreducible. Using the contravariant duality given by the supertrace we obtain a characterization of irreducibility of Weyl supermodules V(λ). This extends the result of Kac ([12], [13]) who proved that, for ground fields of characteristic zero, V(λ) is irreducible if and only if λ is typical.

Sign changes of Fourier coefficients of Siegel cusp forms of degree two on Hecke congruence subgroups

In this paper, we give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. We also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus was dealt in an earlier work of Choie, the first author and Kohnen.

Newforms in the Kohnen plus space

In this paper we want to define the Kohnen plus space for Hilbert modular forms with a odd square-free level and a quadratic character by a representation-theoretic way. We will show that in the classical case the one we defined is the same with the one given by Kohnen. Also, we will interpret the actions of the Hecke operators on the new forms in the plus space and give a characterization for the new forms using Hecke operators. All the results with respect to Hecke operators can are comparable with the integral weight case.

MACKAY FUNCTIONS AND EXACT CUTTING IN SPACES OF MODULAR FORMS

In the article we consider structure problems in the theory of modular forms. The phenomenon of the exact cutting for the spaces Sk(0(N), ), where is a quadratic character with the condition (1) = = (1)k . We prove that for the levels N ̸= 3, 17, 19 the cutting function is a multiplicative etaproduct of an integral weight. In the article we give the table of the cutting functions. We prove that the space of an cutting function is onedimensional. Dimensions of the spaces are calculated by the Cohen-Oesterle formula, the orders in cusps are calculated by the Biagioli formula.

Reconstruction from cone integral transforms

The paper contains new reconstruction formulas for a function on 3D space from data of its cone integrals with fixed opening and integrable weight. In the case of cone integrals with the (non integrable) weight modelling photometric law, a reconstruction is obtained for the non redundant data of cones with the apex running on a curve.

Synthesis and primary evaluation of the hepatoprotective properties of novel pyrimidine derivatives

Based on the active ingredient of the drug Ximedon (1,2-dihydro-4,6-dimethyl-1-N-(2-hydroxyethyl)pyrimidone-2, referred below to as pyrimidine (I), novel derivatives containing biogenic acids: succinic, L-ascorbic, para-aminobenzoic, nicotinic, and L-2-amino-4-(methylthio)butanoic (L-methionine) acids have been synthesized. The parameters of acute toxicity (LD50) have been studied. The antitoxic effect of the compounds upon the injury by the hepatotropic poison carbon tetrachloride has been examined as the primary evaluation of their hepatoprotective properties. It has been found that, according to toxicological safety, the compounds synthesized belong to classes III and IV (moderately and little toxic compounds). The conjugates of pyrimidine (I) with ascorbic acid and methionine (LD50 more than 5400 mg/kg) are least toxic. Pyrimidine (I) and its derivatives possess the antitoxic activity upon acute poisoning with carbon tetrachloride; the combined injection of carbon tetrachloride with pyrimidine (I) or its derivatives leads to an increase in the survival of animals and the normalization of the integral functional parameters, weight and body temperature, which decrease upon toxic injury. In addition, pyrimidine (I) and some of its derivatives (conjugates with L-ascorbic, succinic, para-aminobenzoic, and nicotinic acids) decrease the weight coefficients of the liver and kidneys (the organ-to-body-weight ratio) and the activity of transaminases, the markers of hepatic cytolysis, which increase upon toxic injury with carbon tetrachloride. The area of the pathological injury of the liver by steatosis and necrosis decreases by the action of pyrimidine (I) and its novel derivatives (conjugates with L-ascorbic, succinic, and nicotinic acids) two to three times. Advantages of pyrimidine (I) and its novel derivatives over the hepatoprotective drug Thiotriazolin have been revealed.

Higher level vertex operators for $$U_q \left( \widehat{\mathfrak {sl}}_2\right) $$ U q sl ^ 2

We study graded nonlocal $\underline{\mathsf{q}}$-vertex algebras and we prove that they can be generated by certain sets of vertex operators. As an application, we consider the family of graded nonlocal $\underline{\mathsf{q}}$-vertex algebras $V_{c,1}$, $c\geq 1$, associated with the principal subspaces $W(c\Lambda_0)$ of the integrable highest weight $U_q (\hat{\mathfrak{sl}}_2)$-modules $L(c\Lambda_0)$. Using quantum integrability, we derive combinatorial bases for $V_{c,1}$ and compute the corresponding character formulae.

Solving Solar-Wind Power Station Location Problem Using an Extended Weighted Aggregated Sum Product Assessment (WASPAS) Technique with Interval Neutrosophic Sets

As one of the promising renewable energy resources, solar-wind energy has increasingly become a regional engine in leading the economy and raising competitiveness. Selecting a solar-wind power station location can contribute to efficient utilization of resource and instruct long-term development of socio-economy. Since the selection procedure consists of several location alternatives and many influential criteria factors, the selection can be recognized as a multiple criteria decision-making (MCDM) problem. To better express multiple uncertainty information during the selection procedure, fuzzy set theory is introduced to manage that issue. Interval neutrosophic sets (INSs), which are characterized by truth-membership, indeterminacy-membership and falsity-membership functions in the interval numbers (INs) form, are feasible in modeling more uncertainty of reality. In this paper, a newly extended weighted aggregated sum product assessment (WASPAS) technique, which involves novel three procedures, is utilized to handle MCDM issues under INSs environment. Some modifications are conducted in the extended method comparing with the classical WASPAS method. The most obvious improvement of the extended method relies on that it can generate more realistic criteria weight information by an objective and subjective integrated criteria weight determination method. A case study concerning solar-wind power station location selection is implemented to demonstrate the applicability and rationality of the proposed method in practice. Its validity and feasibility are further verified by a sensitivity analysis and a comparative analysis. These analyses effectively reveal that the extended WASPAS technique can well match the reality and appropriately handle the solar-wind power station location selection problem.

Decision Supplier Package System Using Fuzzy Subjective and Objective Integrated Weights Method (case Study: PT Garudafood Putra Putri Jaya)

The purchasing department is the part that plays an important and influential role in a company even can be said mostof the business process comes from this section. Because of its nature as a procurement of goods and services then oneof its duties and responsibilities is to choose suppliers of procurement of goods and services for production operationalprocesses within the company. In this case PT Garudafood Putra - Putri Jaya often have difficulty in determining thebest product packaging supplier because of the performance instability of each supplier. For that the company needsa system whose purpose is to help decide the best supplier determination. Decision support system is a computer-basedinformation system that generates various decision alternatives to assist management in handling various structuredor unstructured problems using data or models. There are many methods used in a decision support system such asFuzzy Subjective And Objective Integrated Weights. Fuzzy can solve the problem of uncertainty in determining theweight of each supplier's criteria. The result of the implementation of this decision support system resulted in thesupplier of Mandhara Adhitama Utama as the best supplier. Where the results of calculations with Fuzzy SubjectiveAnd Objective Integrated Weights this supplier obtained the highest value with the number 1.742.

Generalization and reverses of the left Fejer inequality for convex functions

In this paper we establish a generalization of the left Fejer inequality for general Lebesgue integral on measurable spaces as well as various upper bounds for the di¤erence 1 R b a g (x) dx Z b a h (x) g (x) dx h a+ b 2 where h : [a; b] ! R is a convex function and g : [a; b] ! [0;1) is an integrable weight. Applications for discrete means and Hermite-Hadamard type inequalities are also provided.

A new framework for probabilistic analysis of the performance of a supported excavation in clay considering spatial variability

This technical note presents a new framework for probabilistic analysis of the performance of a geotechnical system such as a diaphragm wall-supported excavation. The new framework consists of two components: a sampling algorithm and an integration method. With the new sampling algorithm, samples of uncertain variables are ‘uniformly’ generated in the possible domain of uncertain variables; the new integration method is then used to construct the cumulative distribution function of the system performance, based upon the integration weights of the generated samples of uncertain variables. To demonstrate this new framework, an illustrative example focusing on a diaphragm wall-supported excavation in clay is analysed; the system performance of the diaphragm wall-supported excavation is studied with Flac (a finite-difference computer code) and the spatial variability of soil properties, in the context of numerical modelling, is modelled with random field theory. Finally, a parametric analysis is conducted to examine the influence of the statistical characterisation of soil properties on the results of the probabilistic analysis. The innovation of the new framework is demonstrated with the results presented.