Double Sylvester sums for subresultants and multi-Schur functions

Sylvester has announced formulas expressing the subresultants (or the successive polynomial remainders for the Euclidean division) of two polynomials, in terms of some double sums over the roots of the two polynomials. We prove Sylvester formulas using the techniques of multivariate polynomials involving multi-Schur functions and divided differences.     

An elementary proof of Sylvester’s double sums for subresultants

In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester’s formula was also recently proved by Lascoux and Pragacz using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants.     

Sylvester’s double sums: The general case

In 1853 Sylvester introduced a family of double-sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic polynomials associated with these sets. A question naturally arises: What are the other members of the family? This paper provides a complete answer to this question. The technique that we developed to answer the question turns out to be general enough to characterize all members of the family, providing a uniform method.     

A note on double splittings of different monotone matrices

Comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners. For single splittings of different monotone matrices, Elsner et al. (Linear Algebra Appl. 363:65–80, 2003) gave out comparison theorems for spectral radii. For double splittings, some convergence and comparison theorems of a monotone matrix are presented by Shen et al. (Comput. Math. Appl. 51:1751–1760, 2006). In this note we give the comparison theorem for the spectral radii of matrices arising from double splittings of different monotone matrices.     

Code OK1—Simulation of multi-beam irradiation on a spherical target in heavy ion fusion

Code OK1 is a fast and precise three-dimensional computer program designed for simulations of heavy ion beam (HIB) irradiation on a direct-driven spherical fuel pellet in heavy ion fusion (HIF). OK1 provides computational capabilities of a three-dimensional energy deposition profile on a spherical fuel pellet and the HIB irradiation non-uniformity evaluation, which are valuables for optimizations of the beam parameters and the fuel pellet structure, as well for further HIF experiment design. The code is open and complete, and can be easily modified or adapted for users' purposes in this field.

Program summary

Title of program: OK1Catalogue identifier: ADSTProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADSTProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandComputer: PC (Pentium 4, ∼1 GHz or more recommended)Operating system: Windows or UNIXProgram language used: C++Memory required to execute with typical data: 911 MBNo. of bits in a word: 32No. of processors used: 1 CPUHas the code been vectorized or parallelized: NoNo. of bytes in distributed program, including test data: 16 557Distribution format: tar gzip fileKeywords: Heavy ion beam, inertial confinement fusion, energy deposition, fuel pelletNature of physical problem: Nuclear fusion energy may have attractive features as one of our human energy resources. In this paper we focus on heavy ion inertial confinement fusion (HIF). Due to a favorable energy deposition behavior of heavy ions in matter [J.J. Barnard et al., UCRL-LR-108095, 1991; C. Deutsch et al., J. Plasma Fusion Res. 77 (2001) 33; T. Someya et al., Fusion Sci. Tech. (2003), submitted] it is expected that heavy ion beam (HIB) would be one of energy driver candidates to operate a future inertial confinement fusion power plant. For a successful fuel ignition and fusion energy release, a stringent requirement is imposed on the HIB irradiation non-uniformity, which should be less than a few percent [T. Someya et al., Fusion Sci. Tech. (2003), submitted; M.H. Emery et al., Phys. Rev. Lett. 48 (1982) 253; S. Kawata et al., J. Phys. Soc. Jpn. 53 (1984) 3416]. In order to meet this requirement we need to evaluate the non-uniformity of a realistic HIB irradiation and energy deposition pattern. The HIB irradiation and non-uniformity evaluations are sophisticated and difficult to calculate analytically. Based on our code one can numerically obtain a three-dimensional profile of energy deposition and evaluate the HIB irradiation non-uniformity onto a spherical target for a specific HIB parameter value set in HIF.Method of solution: OK1 code is based on the stopping power of ions in matter [J.J. Barnard et al., UCRL-LR-108095, 1991; C. Deutsch et al., J. Plasma Fusion Res. 77 (2001) 33; T. Someya et al., Fusion Sci. Tech. (2003), submitted; M.H. Emery et al., Phys. Rev. Lett. 48 (1982) 253; S. Kawata et al., J. Phys. Soc. Jpn. 53 (1984) 3416; T. Mehlhorn, SAND80-0038, 1980; H.H. Andersen, J.F. Ziegler, Pergamon Press, 1977, p. 3]. The code simulates a multi-beam irradiation, obtains the 3D energy deposition profile of the fuel pellet and evaluates the deposition non-uniformity.Restrictions on the complexity of the problem: NoTypical running time: The execution time depends on the number of beams in the simulated irradiation and its characteristics (beam radius on the pellet surface, beam subdivision, projectile particle energy and so on). In almost of the practical running tests performed, the typical running time for one beam deposition is less than 2 s on a PC with a CPU of Pentium 4, 2.2 GHz (e.g., in Test 2 when the number of beams is 600, the running time is about 18 minutes).Unusual features of the program: No     

Numerical study of fourth-order linearized compact schemes for generalized NLS equations

The fourth-order compact approximation for the spatial second-derivative and several linearized approaches, including the time-lagging method of Zhang et al. (1995), the local-extrapolation technique of Chang et al. (1999) and the recent scheme of Dahlby et al. (2009), are considered in constructing fourth-order linearized compact difference (FLCD) schemes for generalized NLS equations. By applying a new time-lagging linearized approach, we propose a symmetric fourth-order linearized compact difference (SFLCD) scheme, which is shown to be more robust in long-time simulations of plane wave, breather, periodic traveling-wave and solitary wave solutions. Numerical experiments suggest that the SFLCD scheme is a little more accurate than some other FLCD schemes and the split-step compact difference scheme of Dehghan and Taleei (2010). Compared with the time-splitting pseudospectral method of Bao et al. (2003), our SFLCD method is more suitable for oscillating solutions or the problems with a rapidly varying potential.     

Regression analysis of clustered interval-censored failure time data with informative cluster size

Correlated or clustered failure time data often occur in medical studies, among other fields ( [Cai and Prentice, 1995] and [Kalbfleisch and Prentice, 2002]), and sometimes such data arise together with interval censoring (Wang et al., 2006). Furthermore, the failure time of interest may be related to the cluster size. For example, Williamson et al. (2008) discussed such an example arising from a lymphatic filariasis study. A simple and common approach to the analysis of these data is to simplify or convert interval-censored data to right-censored data due to the lack of proper inference procedures for direct analysis of these data. In this paper, two procedures are presented for regression analysis of clustered failure time data that allow both interval censoring and informative cluster size. Simulation studies are conducted to evaluate the presented approaches and they are applied to a motivating example.     

Eigenvectors of tridiagonal matrices of Sylvester type

Eigenvectors of the tridiagonal matrices of Sylvester type are explicitly determined. These are closely related to orthogonal polynomials named after Krawtchouk, (dual) Hahn and Racah as well as to q-Racah polynomials.      

Comments on “A Gaussian approximation recursive filter for nonlinear systems with correlated noises” [Automatica 48 (2012) 2290–2297]

This note points out that the framework proposed in (Wang et al., 2012) is equivalent to the conventional de-coupling framework introduced in some textbooks; see e.g. (Bar-Shalom et al., 2001).     

A mixed effects log-linear model based on the Birnbaum-Saunders distribution

In lifetime data analysis and particularly in engineering reliability contexts, the Birnbaum-Saunders (BISA) density is often suggested as a suitable model; see Birnbaum and Saunders (1969), Mann et al. (1974), and Desmond (1985). A linear regression model, obtained from a logarithmic transformation of the response variable, is useful in studying the effect of covariates on the response variable; see Rieck and Nedelman (1991), Tsionas (2001) and Galea et al. (2004). In this paper, an extension of the log-linear regression model of Rieck and Nedelman (1991), which considers random effects, is introduced. From a Monte Carlo simulation study, the performance of various estimation and prediction methods are studied. The usefulness of the mixed log-linear model is stressed and compared to the pure fixed effects log-linear regression BISA model. The new model is used to analyze a real data set, for which a fixed effects model is inappropriate.     

Soft sets and soft rings

Molodtsov (1999) introduced the concept of soft sets in [1]. Then, Maji et al. (2003) defined some operations on soft sets in [2]. Akta? and Ça?man (2007) defined the notion of soft groups in [3]. Finally, soft semirings are defined by Feng et al. (2008) in [5]. In this paper, we have introduced initial concepts of soft rings.     

Intuitionistic fuzzy soft modules

Molodtsov (1999) initiated the concept of soft sets in [1]. Maji et al. (2003) defined some operations on soft sets in [22]. Akta? and Ça?man (2007) generalized soft sets by defining the concept of soft groups in [16]. After them, Sun et al. (2008) gave soft modules in [21]. In this paper, the concept of an intuitionistic fuzzy soft module is introduced and some operations on intuitionistic fuzzy soft sets are given. Finally, some of its basic properties are studied.     

Convergence Results of the Biconjugate Residual Algorithm for Solving Generalized Sylvester Matrix Equation

In this article, we investigate a variant of the biconjugate residual (BCR) algorithm to solve the generalized Sylvester matrix equation which includes the well‐known Lyapunov, Stein and Sylvester matrix equations. We show that the BCR algorithm with any (special) initial matrix pair can smoothly compute the (least Frobenius norm) solution pair of the generalized Sylvester matrix equation within a finite number of iterations in the absence of round‐off errors. Finally the accuracy and effectiveness of the BCR algorithm in comparison to some existing algorithms are demonstrated by two numerical examples.     

Sensitivity of the Indian Ocean circulation to phytoplankton forcing using an ocean model

Most ocean general circulation models (OGCMs) do not take into account the effect of space- and time-varying phytoplankton on solar radiation penetration, or do it in a simplistic way using a constant attenuation depth, even though one-dimensional experiments have shown potential significant effect of phytoplankton on mixed-layer dynamics. Since some ocean basins are biologically active, it is necessary for an OGCM to take water turbidity into account, even if it is not coupled with a biological model. Sensitivity experiments carried out with the Massachusetts Institute of Technology (MIT) OGCM with spatially and temporally-varying pigment concentration from Sea-viewing Wide Field-of-view Sensor (SeaWiFS) data during 1998-2003 reveal the effect of ocean turbidity on tropical Indian Ocean circulation. Variations of light-absorbing phytoplankton pigments change the vertical distribution of solar heating in the mixed layer, thereby affecting upper-ocean circulation. A simulation was performed from 1948 to 2003 with a constant minimum pigment concentration of 0.02 mg m^− 3 while another simulation was performed from September 1997 to December 2003 with variable pigment concentration, and the differences between these two simulations allow us to quantify the effects of phytoplankton on solar radiation penetration in the ocean model. Model results from a period of 6 years (1998-2003) show large seasonal variability in the strength of the meridional overturning circulation (MOC), meridional heat transports (MHT), and equatorial under current (EUC). The MOC mass transport changes by 2 to 5 Sv (1 Sv = 10⁶ m³ s^− 1) between boreal winter (January) and boreal summer (July), with a corresponding change in the MHT of ∼ 0.05 PW (1 PW = 10¹⁵ W) in boreal winter, which is close to the expected change associated with a significant climate change [Shell, K., Frouin, R., Nakamoto, S., & Somerville, R.C.J. (2003): Atmospheric response to solar radiation absorbed by phytoplankton. Journal of Geophysical Research, 108(D15), 4445. doi:10.1029/2003JD003440.]. In addition, changes in phytoplankton pigments concentration are associated with a reduction in the EUC by ∼ 3 cm s^− 1. We discuss the possible physical mechanisms behind this variability, and the necessity of including phytoplankton forcing in the OGCM.     

Some series identities involving the generalized Apostol type and related polynomials

A unification (and generalization) of various Apostol type polynomials was introduced and investigated recently by Luo and Srivastava [Q.-M. Luo, H.M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217 (2011) 5702-5728]. In this paper, we prove several symmetry identities for these generalized Apostol type polynomials by using their generating functions. As special cases and consequences of our results, we obtain the corresponding symmetry identities for the Apostol-Euler polynomials of higher order, the Apostol-Bernoulli polynomials of higher order and the Apostol-Genocchi polynomials of higher order, and also for another family of generalized Apostol type polynomials which were investigated systematically by Ozden et al. [H. Ozden, Y. Simsek, H.M. Srivastava, A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials, Comput. Math. Appl. 60 (2010) 2779-2787]. We also derive several relations between the Apostol type polynomials, the generalized sum of integer powers and the generalized alternating sum. It is shown how each of these results would extend the corresponding known identities.     

On the spectrum of projective norm-graphs

We show that the projective norm-graphs of Alon et al. [J. Combin. Theory Ser. B 76 (1999) 280-290] are pseudorandom in the sense that their second eigenvalue is as small as the square root of the degree. Our approach is simple, it only uses the evaluation of Gaussian sums and determines the complete spectrum exactly.     

Partial order reduction for state/event LTL with application to component-interaction automata

Software systems assembled from a large number of autonomous components become an interesting target for formal verification due to the issue of correct interplay in component interaction. State/event LTL (Chaki et al. (2004, 2005) [1] and [2]) incorporates both states and events to express important properties of component-based software systems.The main contribution of this paper is a partial order reduction technique for verification of state/event LTL properties. The core of the partial order reduction is a novel notion of stuttering equivalence which we call state/event stuttering equivalence. The positive attribute of the equivalence is that it can be resolved with existing methods for partial order reduction. State/event LTL properties are, in general, not preserved under state/event stuttering equivalence. To this end we define a new logic, called weak state/event LTL, which is invariant under the new equivalence.To bring some evidence of the method’s efficiency, we present some of the results obtained by employing the partial order reduction technique within our tool for verification of component-based systems modelled using the formalism of component-interaction automata (Brim et al. (2005) [3]).