A New class of rational multistep methods for solving initial value problem

There exists initial value problem whose solution possesses singularity.Studies show that conventional numerical method such as multistep method fail woefully near the singular point when solving problem whose solution possesses singularity.This is because a multistep method is based on the local representation of polynomial of the theoretical solution of an initial value problem.Therefore, a natural step would appear to be the replacement of the polynomial function for a multistep method, by a rational function due to its smooth behaviour in the neighbourhood of singularity.In this paper, we have developed a new class of two step numerical methods that are based on rational functions in solving general initial value problem and problem whose solution possesses singularity.These new methods are called rational multistep methods.The developments of these rational multistep methods, as well as the local truncation error and stability analysis for each rational multistep method are presented.We have found out that only the second order, third order and fourth order rational multistep methods are A-stable.Numerical experiments have showed that all newly developed rational multistep methods presented in this paper are suitable to solve general initial value problem, stiff problem and problem whose solution possesses singularity

Numerical solution of first order initial value problem using 7-stage tenth order Gauss-Kronrod-Lobatto IIIA method

In this paper, a new implicit Runge-Kutta method which based on a 7-point Gauss-Kronrod-Lobatto quadrature formula is developed.The resulting implicit method is a 7-stage tenth order Gauss-Kronrod-Lobatto IIIA method, or in brief as GKLM(7,10)-IIIA. GKLM(7,10)-IIIA requires seven function of evaluations at each integration step and it gives accuracy of order ten.In addition, GKLM(7,10)-IIIA has stage order seven and being A-stable. Numerical experiments compare the accuracy between GKLM(7,10)-IIIA and the classical 5-stage tenth order Gauss-Legendre method in solving some test problems. Numerical results reveal that GKLM(7,10)-IIIA is more accurate than the 5-stage tenth order Gauss-Legendre method because GKLM(7,10)-IIIA has higher stage orde

Pharmacological modulation of oncogenic Ras by natural products and their derivatives: renewed hope in the discovery of novel anti-Ras drugs

Oncogenic rat sarcoma (Ras) is linked to the most fatal cancers such as those of the pancreas, colon, and lung. Decades of research to discover an efficacious drug that can block oncogenic Ras signaling have yielded disappointing results; thus, Ras was considered “undruggable” until recently. Inhibitors that directly target Ras by binding to previously undiscovered pockets have been recently identified. Some of these molecules are either isolated from natural products or derived from natural compounds. In this review, we described the potential of these compounds and other inhibitors of Ras signaling in drugging Ras. We highlighted the modes of action of these compounds in suppressing signaling pathways activated by oncogenic Ras, such as mitogen-activated protein kinase (MAPK) signaling and the phosphoinositide-3-kinase (PI3K) pathways. The anti-Ras strategy of these compounds can be categorized into four main types: inhibition of Ras–effector interaction, interference of Ras membrane association, prevention of Ras–guanosine triphosphate (GTP) formation, and downregulation of Ras proteins. Another promising strategy that must be validated experimentally is enhancement of the intrinsic Ras–guanosine triphosphatase (GTPase) activity by small chemical entities. Among the inhibitors of Ras signaling that were reported thus far, salirasib and TLN-4601 have been tested for their clinical efficacy. Although both compounds passed phase I trials, they failed in their respective phase II trials. Therefore, new compounds of natural origin with relevant clinical activity against Ras-driven malignancies are urgently needed. Apart from salirasib and TLN-4601, some other compounds with a proven inhibitory effect on Ras signaling include derivatives of salirasib, sulindac, polyamine, andrographolide, lipstatin, levoglucosenone, rasfonin, and quercetin

New rational methods for the numerical solution of first order initial value problem

Exponentially-fitted numerical methods are appealing because L-stability is guaranteed when solving initial value problems of the form Y¹= ʎy,y(a)=ɳ, ʎ є C , Rc(λ)<0. Such numerical methods also yield the exact solution when solving the above-mentioned problem. Whilst rational methods have been well established in the past decades, most of them are not ‘completely’ exponentially fitted.Recently, a class of one-step exponential-rational methods (ERMs) were discovered.Analyses showed that all ERMs are exponentially-fitted, hence implying L-stability.Several numerical experiments showed that ERMs is more accurate than existing rational methods in solving general initial value problem. However, ERMs have several weaknesses: i) every ERM is non-uniquely defined; ii) may return complex values; and iii) less accurate numerical solution when solving problem whose solution possesses singularity.Therefore, the first purpose of this study is to modify the original ERMs so that the first two weaknesses will be overcomed. Theoretical analyses such as consistency, stability and convergence of the modified ERMs are presented.Numerical experiments showed that the modified ERMs and the original ERMs are found to have comparable accuracy; hence modified ERMs are preferable to original ERMs.The second purpose of this study is to overcome the third weakness of the original ERMs where a variable step-size strategy is proposed to improve the accuracy ERMs.The procedures of the strategy are detailed out in this report.Numerical experiments have revealed that the affects from the implementation of the strategy is less obvious

Block multistep methods based on rational approximants

In this study, the concept of block multistep methods based on rational approximants is introduced for the numerical solution of first order initial value problems. These numerical methods are also called rational block multistep methods.The main reason to consider block multistep methods in rational setting, is to improve the numerical accuracy and absolute stability property of existing block multistep methods that are based on polynomial approximants.For this pilot study, a 2-point explicit rational block multistep method is developed.Local truncation error and stability analysis for this new method are included as well.Numerical experimentations and results using some test problems are presented.Numerical results are satisfying in terms of numerical accuracy. Finally, future issues on the developments of rational block multistep methods are discussed

An A-stable explicit rational block method for the numerical solution of initial value problem

In this paper,a 2-point explicit rational block method for the numerical solution of first order initial value problem is proposed.The main reason to consider rational block method is to improve the numerical accuracy and absolute stability property of esisting block multistep methods that are based 011 polynomial approximants. The proposed method is found to possess A-stability.Local truncation error is included as well.Numerical experimentations and results using some test problems are presented.Numerical results are satisfying in terms of numerical accuracy.Finally,a conclusion is included

Intratumoral Decorin Gene Delivery by AAV Vector Inhibits Brain Glioblastomas and Prolongs Survival of Animals by Inducing Cell Differentiation

Glioblastoma multiforme (GBM) is the most malignant cancer in the central nervous system with poor clinical prognosis. In this study, we investigated the therapeutic effect of an anti-cancer protein, decorin, by delivering it into a xenograft U87MG glioma tumor in the brain of nude mice through an adeno-associated viral (AAV2) gene delivery system. Decorin expression from the AAV vector in vitro inhibited cultured U87MG cell growth by induction of cell differentiation. Intracranial injection of AAV-decorin vector to the glioma-bearing nude mice in vivo significantly suppressed brain tumor growth and prolonged survival when compared to control non-treated mice bearing the same U87MG tumors. Proteomics analysis on protein expression profiles in the U87MG glioma cells after AAV-mediated decorin gene transfer revealed up- and down-regulation of important proteins. Differentially expressed proteins between control and AAV-decorin-transduced cells were identified through MALDI-TOF MS and database mining. We found that a number of important proteins that are involved in apoptosis, transcription, chemotherapy resistance, mitosis, and fatty acid metabolism have been altered as a result of decorin overexpression. These findings offer valuable insight into the mechanisms of the anti-glioblastoma effects of decorin. In addition, AAV-mediated decorin gene delivery warrants further investigation as a potential therapeutic approach for brain tumors

SARS in Hospital Emergency Room

Thirty-one cases of severe acute respiratory syndrome (SARS) occurred after exposure in the emergency room at the National Taiwan University Hospital. The index patient was linked to an outbreak at a nearby municipal hospital. Three clusters were identified over a 3-week period. The first cluster (5 patients) and the second cluster (14 patients) occurred among patients, family members, and nursing aids. The third cluster (12 patients) occurred exclusively among healthcare workers. Six healthcare workers had close contact with SARS patients. Six others, with different working patterns, indicated that they did not have contact with a SARS patient. Environmental surveys found 9 of 119 samples of inanimate objects to be positive for SARS coronavirus RNA. These observations indicate that although transmission by direct contact with known SARS patients was responsible for most cases, environmental contamination with the SARS coronavirus may have lead to infection among healthcare workers without documented contact with known hospitalized SARS patients