Multidimensional Case Research Articles

Stability of planar diffusion wave for the three dimensional full bipolar Euler–Poisson system

In the paper, we consider a three-dimensional full bipolar classical hydrodynamic model. This model takes the form of non-isentropic bipolar Euler–Poisson with the electric field and the relaxation term added to the momentum equations. Based on the diffusive wave phenomena of the one dimensional full non-isentropic bipolar Euler–Poisson equations, we show the nonlinear stability of the planar diffusive wave for the initial value problem of the three dimensional non-isentropic bipolar Euler–Poisson system. Moreover, the convergence rates in L2-norm and L∞-norm are also obtained. The proofs are finished by some elaborate energy estimates. The study generalizes the result of [Y.-P. Li, J. Differential Equations, 250(2011), 1285–1309] to multi-dimensional case.

Family of central bicompact schemes with spectral resolution property for hyperbolic equations

For the numerical solution of nonstationary quasilinear hyperbolic equations, a family of central semidiscrete bicompact schemes based on collocation polynomials is constructed in the one- and multidimensional cases. A dispersion analysis of semidiscrete bicompact schemes of fourth to eighth orders of accuracy in space is performed. Numerical examples are presented that demonstrate the ability of the bicompact schemes to adequately simulate wave propagation, including short waves, on highly nonuniform grids at long times. The properties of solutions of bicompact schemes in the problem of transfer of a stepwise initial profile are also considered.

The arithmetic of multidimensional Z-number

The Z-number has become a new representation of evaluation information because of its superiority in describing reliability measure. Current studies focuses mainly on the unidimensional Z-number like Z = (about 25 min, usually). Studies on multidimensional cases have not been reported. However, bec ause people often describe things from various aspects, using only one aspect of information to describe uncertain events fully is difficult. In this paper, we propose the concept of the multidimensional Z-number, such as ((about 20 miles, about 25 min), usually) to handle complex information. For this purpose, we first define the basic operations of the multidimensional Z-number. We then propose a feasible comparison method. The effectiveness of the proposed method is demonstrated by a series of numerical examples.

Multidimensional Test Assembly Using Mixed-Integer Linear Programming: An Application of Kullback-Leibler Information.

Many educational testing programs require different test forms with minimal or no item overlap. At the same time, the test forms should be parallel in terms of their statistical and content-related properties. A well-established method to assemble parallel test forms is to apply combinatorial optimization using mixed-integer linear programming (MILP). Using this approach, in the unidimensional case, Fisher information (FI) is commonly used as the statistical target to obtain parallelism. In the multidimensional case, however, FI is a multidimensional matrix, which complicates its use as a statistical target. Previous research addressing this problem focused on item selection criteria for multidimensional computerized adaptive testing (MCAT). Yet these selection criteria are not directly transferable to the assembly of linear parallel test forms. To bridge this gap the authors derive different statistical targets, based on either FI or the Kullback-Leibler (KL) divergence, that can be applied in MILP models to assemble multidimensional parallel test forms. Using simulated item pools and an item pool based on empirical items, the proposed statistical targets are compared and evaluated. Promising results with respect to the KL-based statistical targets are presented and discussed.

A note on the existence and stability of an inverse problem for a SIS model

In this note we discuss the existence and stability of an inverse problem arising from the determination of the reaction coefficients for an SIS model. The study is motivated by a remark regarding the final discussion of the recent paper by Xiang and Liu (2015). The weak assumption of the work of H. Xiang and B. Liu is that the proofs of existence and stability results are valid only for the one-dimensional case. Here, we introduce an appropriate framework which is also valid in the multidimensional case and that generalizes the previous results.

Multi-dimensional periodic problems for higher-order linear hyperbolic equations

Abstract For higher-order linear hyperbolic equations the problem on periodic solutions is investigated. The concepts of associated problems, and α-well-posedness are introduced. Necessary and sufficient conditions of well-posedness in the two-dimensional case, as well as unimprovable sufficient conditions of well-posedness and α-well-posedness in the multi-dimensional case are established.

On Spectral Approximations with Nonstandard Weight Functions and Their Implementations to Generalized Chaos Expansions

In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained, e.g, by Gram–Schmidt orthogonalization. However, in most cases, they are not eigenfunctions of some singular Sturm–Liouville problem, as is the case for known polynomials, such as the Jacobi polynomials. Therefore, the standard convergence theorems do not apply. Furthermore, since in general multidimensional cases the weight functions are not a tensor product of one-dimensional functions, the orthogonal polynomials are not a product of one-dimensional orthogonal polynomials, as well. This work provides a way of estimating the convergence rate using a comparison lemma. We also present a spectrally convergent, multidimensional, integration method. Numerical examples demonstrate the efficacy of the proposed method. We also show that the use of non-standard weight functions can allow for efficient integration of singular functions. We demonstrate the use of this method to uncertainty quantification problem using Generalized Polynomial Chaos Expansions in the case of dependent random variables, as well.

Girsanov’s theorem in vector lattices

In this paper we formulate and proof Girsanov’s theorem in vector lattices. To reach this goal, we develop the theory of cross-variation processes, derive the cross-variation formula and the Kunita–Watanabe inequality. Also needed and derived are properties of exponential processes, Ito’s rule for multi-dimensional processes and the integration by parts formula for martingales. After proving Girsanov’s theorem for the one-dimensional case, we also discuss the multi-dimensional case.

Invariant Manifolds and Rate Constants in Driven Chemical Reactions.

Reaction rates of chemical reactions under nonequilibrium conditions can be determined through the construction of the normally hyperbolic invariant manifold (NHIM) [and moving dividing surface (DS)] associated with the transition state trajectory. Here, we extend our recent methods by constructing points on the NHIM accurately even for multidimensional cases. We also advance the implementation of machine learning approaches to construct smooth versions of the NHIM from a known high-accuracy set of its points. That is, we expand on our earlier use of neural nets and introduce the use of Gaussian process regression for the determination of the NHIM. Finally, we compare and contrast all of these methods for a challenging two-dimensional model barrier case so as to illustrate their accuracy and general applicability.

Test Eşitlemede Çok Boyutluluğun Eş Zamanlı ve Ayrı Kalibrasyona Etkisi

The aim of this study is to investigate the effects of multidimensionality on equating results which obtained from separate and concurrent calibration methods. The study was conducted with using simulated data. In the scope of research, totally 40 simulation conditions [5 (degree of multidimensionality: 0.90, 0.75, 0.50, 0.25, and 0.00) x 2 calibration methods (separate and concurrent) x 2 (scale transformation methods: Stocking-Lord and Haebara) x 2 (test equating methods: IRT true score equating and observed score equating)] were examined. Multidimensionality was constructed as assuming the two test forms measuring Ɵ1 and Ɵ2 abilities. While the simulation condition which has correlation between abilities 0.90 represents weak multidimensional case, the correlation between abilities 0.00 represents the severe multidimensional case. Tests were equated under common-item non-equivalent groups design. Equating results were evaluated by using bias, standard deviation and RMSE evaluation criteria. The results showed that, generally under all conditions equating results provided from concurrent calibration more biased and had higher RMSE values than equating results provided by separate calibration. Based on standard deviation criteria, when the degree of multidimensionality was low, equating results which got from concurrent calibration and separate calibration with Stocking-Lord or Haebara scale transformation methods showed similar performance but when the degree of multidimensionality was severe equating results which had lowest random error were provided by concurrent calibration.

On the study of a rational expectation model with lagged endogenous variables

ABSTRACTA rational expectation model with lagged endogenous variables is used to describe how the current price level is influenced by the expectation and historic price level. The time domain of the rational expectation model is extended to a complex discrete time domain which is a collection of points along the real number line. The rational expectation model with lagged endogenous variables is solved in multi-dimensional cases where the agents possess multiple assets, and the current price of each asset is related to the expected price and historical prices. An example about price determination process of storable commodities is given to illustrate the advantages of the rational expectation model on isolate time domain.

Multi-dimensional particle filter-based estimation of inter-system phase biases for multi-GNSS real-time integer ambiguity resolution

In multi-GNSS integration, fixing inter-system double-difference ambiguities to integers is still a challenge due to the existence of inter-system biases (ISB) when mixed types of GNSS receivers are used. It has been shown that when ISB is known, the inter-system ambiguities can be fixed and the reliability of ambiguity fixing can be improved significantly, especially under poor conditions when the number of observed satellites is small. In traditional methods, the intra-system ambiguity is fixed first; then, the ISB is estimated to ultimately fix the inter-system ambiguity. In our work, we use the particle filter-based method to estimate the ISB parameter and fix the inter-system ambiguities to integers at the same time. This method shows higher reliability and higher ambiguity fixing rate. Nevertheless, the existing particle filter approach for ISB parameter estimation is a one-dimensional algorithm. When satellites from three or more systems are observed, there are two or more ISB parameters. We extend the current one-dimensional particle filter approach to multi-dimensional case and estimate multi-ISB parameters in this study. We first present a multi-dimensional particle filter approach that can estimate multi-ISB parameters simultaneously. We also show that the RATIO values can be employed to judge the quality of multi-dimensional ISB values. Afterward, a two-dimensional particle filter approach is taken as an example to validate this approach. For example, in the experiment of GPS L5, Galileo E5a and QZSS L5 integration with 6 satellites using the IGS baseline SIN0-SIN1, only three ambiguities are resolved to integer when the ISBs are unknown. The integer ambiguity fixing rate is 41.0% with 53% of the ambiguity-fixed solutions having positioning errors larger than 3 cm. However, when our approach is adopted, the number of integer ambiguity parameters increases to five. The integer ambiguity fixing rate increases to 99.7% with 100% of ambiguity-fixed solutions having positioning errors smaller than 3 cm.

On Some Families of New Constructed Polynomials

The purpose of this paper is to introduce some new polynomials obtained from second- and third-order algebraic equations by using a simple iterative method. One-variable polynomials obtained in this study deal with special form of Poschl–Teller potential with constant energy, and the two-variable polynomials are related to time-dependent wave equations. We present some recurrence relations, Binet formula and get various families of linear, multilinear and multilateral generating functions for these polynomials. In addition, we derive some special cases. At the end of the paper we also give an extension to the multidimensional case of our results.

Poincaré and Log-Sobolev Inequalities for Mixtures.

This work studies mixtures of probability measures on and gives bounds on the Poincaré and the log–Sobolev constants of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the -distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincaré constant stays bounded in the mixture parameter, whereas the log–Sobolev may blow up as the mixture ratio goes to 0 or 1. This observation generalizes the one by Chafaï and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.

Theoretical Foundations of Multi-Product Prices Optimization Model (Part1)

The paper investigates and analyses the technologies and the methods for price optimization in a multi-product environment, where the firms face more difficulties to make an optimal decisions. Multi-Product Prices Optimization (MPPO) problem emerges specific problems, because of both prices and demands, being random variables, may exhibit different covariate relationships, determined by means of substitutional and complimentarity effects. The system of covariate relationships, determined by means of substitution and complimentarity processes, is represented as the system of multivariance regression equations. MPPO problem have been reduced to the maximization problem of the objective function of Total Revenue, which is a quadratic form. In multidimensional cases one faces the problem of absence of local maximum of Total Revenue function. The simple criterion of existing the optimal solution ofthe MPPO problem is suggested. In the case of absence of the strict maximum of Total Revenue function, parameters should be calculated on the base of OPPO model, that is on the base of independent paired problems. The represented theory can be used as a basis for development of a relevant software.Keywords: complementarity, multi-product pricing, price optimization, single-product pricing, substitution, total revenueJEL: M21

IHP: improving the utility in differential private histogram publication

Differential privacy (DP) is a promising tool for preserving privacy during data publication, as it provides strong theoretical privacy guarantees in face of adversaries with arbitrary background knowledge. Histogram, as the result of a set of count queries, serves as a core statistical tool to report data distributions and is in fact viewed as the fundamental method for many other statistical analysis such as range queries. It is an important form for data publishing. In this paper, we consider the scenario of publishing sensitive histogram data with differential privacy scheme. Existing work in this field has justified that, comparing to directly applying DP techniques (i.e., injecting noise) over the counts in histogram bins, grouping bins before noise injection is more effective (i.e., with higher utility) as it introduces much less error over the sanitized histogram given the same privacy budget. However, state-of-the-art works have not unveiled how the overall utility of a sanitized histogram can be affected by the balance between the privacy budget distributed between grouping and noise injection phases. In this work, we conduct a theoretical study towards how the probability of getting better groups can be improved such that the overall error introduced in sanitized histogram can be further reduced, which directly leads to a higher utility for the sanitized histograms. In particular, we show that the probability of achieving better grouping can be affected by two factors, namely privacy budget assigned in grouping and the normalized utility function used for selecting groups. Motivated by that, we propose a new DP histogram publishing scheme, namely Iterative Histogram Partition, in which we carefully assign privacy budget between grouping and injection phases based on our theoretical study. We also theoretically prove that \(\epsilon \)-differential privacy can be achieved according to our new scheme. Moreover, we also show that, under the same privacy budget, our scheme exhibits less errors in the sanitized histograms comparing with state-of-the-art methods. We also extends the model to multi-dimensional histogram publication cases. Finally, empirical study over four real-world datasets also justifies that our scheme achieves the least error among series of state-of-the-art baseline methods.

Applications of the Newton-Raphson Method in Decision Sciences and Education

The Newton-Raphson (NR)methodis one ofthemostimportant and popularmethods to determine an optimal solution in many applications in the decision sciences and education. The NR method can be used for an optimal solution to obtain estimates in regression models, the maxima or minima of many functionsin both the one-dimensional and multi-dimensional cases, or to solve systems of equations with many unknowns in both the one-dimensional and multi-dimensional cases. Applications of the NR method cover a wide variety of interesting cases, including the decisionsciences and the teaching of mathematics-related subjects. The primary purpose of the paper is to provide a universal approach to the theory and practice of the NR method. In addition, we focus the discussion on applications in decision sciences, statistics, portfolio optimization, and education, among others Interesting potential research directions of the NR method are also discussed.

Formal Groups over Sub-Rings of the Ring of Integers of a Multidimensional Local Field

We constructed so-called convergence rings for the ring of integers of a multidimensional local field. The convergence ring is a sub-ring of the ring of integers with the property that any power series with coefficients from the sub-ring converges when replacing a variable by an arbitrary element of the maximal ideal. The properties of convergence rings and an explicit formula for their construction are derived. Note that the multidimensional case is fundamentally different from the case of the classical (one-dimensional) local field, where the whole ring of integers is the convergence ring. Next, we consider a multidimensional local field with zero characteristics of the penultimate residue field. For each convergence ring of such a field, we introduce a homomorphism that allows us to construct a formal group over the same ring with a logarithm having coefficients from the field for a power series with coefficients from the ring, and we give an explicit formula for the coefficients. In addition, by isogeny with coefficients from this ring, we construct a generalization of the formal Lubin—Tate group over this ring, study the endomorphisms of these formal groups, and derive a criterion for their isomorphism. We prove a one-to-one correspondence between formal groups created by ring homomorphism and by isogeny. Also, for any finite extension of a multidimensional local field with zero characteristic of the penultimate residue field, we consider the point group generated by the corresponding Lubin—Tate formal group.

Polarization Tensors for Massive Arbitrary-Spin Particles and the Behrends–Fronsdal Projection Operator

Based on theWigner unitary representations for the covering Poincare group ISL(2,ℂ), we construct spin–tensor wave functions of free massive arbitrary-spin particles satisfying the Dirac–Pauli–Fierz equations. We obtain polarization spin–tensors and indicate conditions that fix the density matrices (Behrends–Fronsdal projection operators), which determine the numerators in the propagators of the fields of such particles. Using such conditions extended to the multidimensional case, we construct a generalization of Behrends–Fronsdal projection operators (for any number D >2 of space–time dimensions) corresponding to a symmetric representation of the D-dimensional Poincare group.

Estimations and Optimal Tests in Some Parametric Models

In the present paper, we introduce an efficient method for the estimation in the multidimensional case. The key idea is based on a good assessment of the error without using confidence intervals. The consistency of the proposed estimate is established. Consequently, we discuss the estimation in statistical tests corresponding to parametric context, and prove that this kind of estimators ensures the optimality of statistical tests. We partially extend the scope of our study to some processes. In order to examine the performance of our methodology, finite sample results are performed. This work completes and extends in nontrivial way the results obtained by Lounis (2017).