Supplement : Efficient weak second-order stochastic Runge-Kutta methods for non-commutative Stratonvich stochastic differential equations

This paper gives a modification of a class of stochastic Runge–Kutta methods proposed in a paper by Komori (2007). The slight modification can reduce the computational costs of the methods significantly.

High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the Linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative Linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order I Runge-Kutta methods with d + 1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches. (C) 1999 Elsevier Science B.V. All rights reserved.

Three primary school students’ cognition about 3D rotation in a virtual reality learning environment

This paper reports on three primary school students’ explorations of 3D rotation in a virtual reality learning environment (VRLE) named VRMath. When asked to investigate if you would face the same direction when you turn right 45 degrees first then roll up 45 degrees, or when you roll up 45 degrees first then turn right 45 degrees, the students found that the different order of the two turns ended up with different directions in the VRLE. This was contrary to the students’ prior predictions based on using pen, paper and body movements. The findings of this study showed the difficulty young children have in perceiving and understanding the non-commutative nature of 3D rotation and the power of the computational VRLE in giving students experiences that they rarely have in real life with 3D manipulations and 3D mental movements.

A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems

In order to simulate stiff biochemical reaction systems, an explicit exponential Euler scheme is derived for multidimensional, non-commutative stochastic differential equations with a semilinear drift term. The scheme is of strong order one half and A-stable in mean square. The combination with this and the projection method shows good performance in numerical experiments dealing with an alternative formulation of the chemical Langevin equation for a human ether a-go-go related gene ion channel mode

Proton-transfer and non-transfer compounds of the multi-purpose drug dapsone [4-(4-Aminophenylsulfonyl)aniline] with 3,5-dinitrosalicylic acid and 5-nitroisophthalic acid

The structures of the compounds from the reaction of the drug dapsone [4-(4-aminophenylsulfonyl)aniline] with 3,5-dinitrosalicylic acid, the salt hydrate [4-(4-aminohenylsulfonyl)anilinium 2-carboxy-4,6-dinitrophenolate monohydrate] (1) and the 1:1 adduct with 5-nitroisophthalic acid [4-(4-aminophenylsulfonyl)aniline 5-nitrobenzene-1,3-dicarboxylic acid] (2) have been determined. Crystals of 1 are triclinic, space group P-1, with unit cell dimensions a = 8.2043(3), b = 11.4000(6), c = 11.8261(6)Å, α = 110.891(5), β = 91.927(3), γ = 98.590(4)deg. and Z = 4. Compound 2 is orthorhombic, space group Pbcn, with unit cell dimensions a = 20.2662(6), b = 12.7161(4), c = 15.9423(5)Å and Z = 8. In 1, intermolecular analinium N-H…O and water O-H…O and O-H…N hydrogen-bonding interactions with sulfone, carboxyl, phenolate and nitro O-atom and aniline N-atom acceptors give a two-dimensional layered structure. With 2, the intermolecular interactions involve both aniline N-H…O and carboxylic acid O-H…O and O-H…N hydrogen bonds to sulfone, carboxyl, nitro and aniline acceptors, giving a three-dimensional network structure. In both structures π--π aromatic ring associations are present.

Post-quantum key exchange for the TLS protocol from the ring learning with errors problem

Lattice-based cryptographic primitives are believed to offer resilience against attacks by quantum computers. We demonstrate the practicality of post-quantum key exchange by constructing cipher suites for the Transport Layer Security (TLS) protocol that provide key exchange based on the ring learning with errors (R-LWE) problem, we accompany these cipher suites with a rigorous proof of security. Our approach ties lattice-based key exchange together with traditional authentication using RSA or elliptic curve digital signatures: the post-quantum key exchange provides forward secrecy against future quantum attackers, while authentication can be provided using RSA keys that are issued by today's commercial certificate authorities, smoothing the path to adoption. Our cryptographically secure implementation, aimed at the 128-bit security level, reveals that the performance price when switching from non-quantum-safe key exchange is not too high. With our R-LWE cipher suites integrated into the Open SSL library and using the Apache web server on a 2-core desktop computer, we could serve 506 RLWE-ECDSA-AES128-GCM-SHA256 HTTPS connections per second for a 10 KiB payload. Compared to elliptic curve Diffie-Hellman, this means an 8 KiB increased handshake size and a reduction in throughput of only 21%. This demonstrates that provably secure post-quantum key-exchange can already be considered practical.

Genome-wide association study of ankylosing spondylitis identifies non-MHC susceptibility loci

To identify susceptibility loci for ankylosing spondylitis, we undertook a genome-wide association study in 2,053 unrelated ankylosing spondylitis cases among people of European descent and 5,140 ethnically matched controls, with replication in an independent cohort of 898 ankylosing spondylitis cases and 1,518 controls. Cases were genotyped with Illumina HumHap370 genotyping chips. In addition to strong association with the major histocompatibility complex (MHC; P 10 800), we found association with SNPs in two gene deserts at 2p15 (rs10865331; combined P = 1.9 × 10 19) and 21q22 (rs2242944; P = 8.3 × 10 20), as well as in the genes ANTXR2 (rs4333130; P = 9.3 × 10 8) and IL1R2 (rs2310173; P = 4.8 × 10 7). We also replicated previously reported associations at IL23R (rs11209026; P = 9.1 × 10 14) and ERAP1 (rs27434; P = 5.3 × 10 12). This study reports four genetic loci associated with ankylosing spondylitis risk and identifies a major role for the interleukin (IL)-23 and IL-1 cytokine pathways in disease susceptibility. © 2010 Nature America, Inc. All rights reserved.

Correcting for bias when estimating the cost of hospital-acquired infection : an analysis of lower respiratory tract infections in non-surgical patients

Hospital acquired infections (HAI) are costly but many are avoidable. Evaluating prevention programmes requires data on their costs and benefits. Estimating the actual costs of HAI (a measure of the cost savings due to prevention) is difficult as HAI changes cost by extending patient length of stay, yet, length of stay is a major risk factor for HAI. This endogeneity bias can confound attempts to measure accurately the cost of HAI. We propose a two-stage instrumental variables estimation strategy that explicitly controls for the endogeneity between risk of HAI and length of stay. We find that a 10% reduction in ex ante risk of HAI results in an expected savings of £693 ($US 984).

A Petrov-Galerkin method for a singularly perturbed ordinary differential equation with non-smooth data

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.