Random extremal solutions of differential inclusions

Given a Lipschitz continuous multifunction $F$ on ${\mathbb{R}}^{n}$, we construct a probability measure on the set of all solutions to the Cauchy problem $\dot x\in F(x)$ with $x(0)=0$. With probability one, the derivatives of these random solutions take values within the set $ext F(x)$ of extreme points for a.e.~time $t$. This provides an alternative approach in the analysis of solutions to differential inclusions with non-convex right hand side.

Effect of nitric oxide donors on blood pressure and pulse pressure in acute and subacute stroke

High blood pressure (BP), pulse pressure (PP), and rate pressure product (RPP) areeach associated independently with a poor outcome in acute ischemic stroke. Whereas nitric oxide (NO) donors, such as glyceryl trinitrate (GTN), lower blood pressure in acute ischemic stroke, their effect on other hemodynamic measures is not known. We performed a systematic review of the effects of NO donors on systemic hemodynamic measures in patients with acute/subacute stroke. Randomized controlled trials were identified from searches of the Cochrane Library, Pubmed, and Embase. Information on hemodynamic measures, including systolic BP (SBP), diastolic BP (DBP), and heart rate, were assessed, and hemodynamic derivatives of these were calculated: PP (PP SBP DBP), mean arterial pressure (MAP DBP PP/3), mid blood pressure (MBP (SBP DBP)/2), pulse pressure index (PPI PP/MAP), and RPP (RPP SBP HR). The effect of treatment on hemodynamic measures was calculated as the weighted mean difference (WMD) between treated and control groups with adjustment for baseline. Results: Three trials involving 145 patients were identified; 93 patients received the NO donor, GTN, and 52 control. As compared with placebo, GTN significantly reduced SBP (WMD -9.80 mmHg, p< 0.001), DBP (WMD -4.43 mmHg, p<0.001), MAP (WMD -6.41 mmHg, p< 0.001), MBP (WMD -7.33 mmHg,p<0.001), PP (WMD -6.11 mmHg, p<0.001 ) and PPI (WMD -0.03, p=0.04 ). 3 GTN increased HR (WMD +3.87 bpm, p<0.001) and non-significantly lowered RPP (WMD -323 mmHg.bpm, p=0.14). Conclusion: The NO donor GTN reduces BP, PP and other derivatives in acute and subacute stroke whilst increasing heart rate.

Fluorine-18-labeled estrogens, progestins and corticosteroids for receptor-based imaging of breast tumors and target areas of the brain

Estrogens can be labeled with the positron-emitting radionuclide fluorine-18 (t$\sb{1/2}$ = 110 min) by fluoride ion (n-Bu$\sb4$N$\sp{18}$F) displacement of a 16$\beta$-trifluoromethanesulfonate (triflate) derivative of the corresponding estrone 3-triflate, and purification by HPLC. That sequence has been used to synthesize the 11$\beta$-methoxy 1 and 11$\beta$-ethyl 2 analogues of the breast tumor imaging agent, 16$\alpha$-($\sp{18}$F) fluoro-17$\beta$-estradiol (FES). Tissue distribution studies of 1 and 2 in immature female rats show high selectivity for target tissue (T, uterus) vs non-target (NT, muscle and lung), with T/NT ratios being 43 and 17 at one hour after injection for 1 and 2, respectively. The parent estrogen FES has previously been shown to display an intermediate value for tissue selectivity.

In vitro studies of drug transformations: application to forensic toxicology

The forensic toxicologist faces challenges in the detection of drugs and poisons in biological samples due to transformations which occur both during life and after death. For example, changes can result from drug metabolism during life or from the use of formalin solution for post mortem embalming purposes. The former requires the identification of drug metabolites and the latter the identification of chemical reaction products in order to know which substances had been administered. The work described in this thesis was aimed at providing ways of tackling these challenges and was divided into two parts. Part 1 investigated the use of in vitro drug metabolism by human liver microsomes (HLM) to obtain information on drug metabolites and Part 2 investigated the chemical reactions of drugs and a carbamate pesticide with formalin solution and formalin-blood. The initial aim of part I was to develop an in vitro metabolism method using HLM, based on a literature review of previous studies of this type. MDMA was chosen as a model compound to develop the HLM method because its metabolism was known and standards of its metabolites were commercially available. In addition, a sensitive and selective method was developed for the identification and quantitation of hydrophilic phase I drug metabolites using LC/MS/MS with a conventional reverse-phase (C18) column. In order to obtain suitable retention factors for polar drug metabolites on this column, acetyl derivatives were evaluated for converting the metabolites to more lipophilic compounds and an optimal separation system was developed. Acetate derivatives were found to be stable in the HPLC mobile phase and to provide good chromatographic separation of the target analytes. In vitro metabolism of MDMA and, subsequently, of other drugs involved incubation of 4 µg drug substance in pH 7.4 buffer with an NADPH generating system (NGS) at 37oC for 90 min with addition of more NGS after 30 min. The reaction was stopped at 90 min by the addition of acetonitrile before extraction of the metabolites. Acetate derivatives of MDMA metabolites were identified by LC/MS/MS using multiple reaction monitoring (MRM). Three phase I metabolites (both major and minor metabolites) of MDMA were detected in HLM samples. 3,4-dihydroxy-methamphetamine and 4-hydroxy-3-methoxymethamphetamine were found to be major metabolites of MDMA whereas 3,4-methylenedioxyamphetamine was found to be a minor metabolite. Subsequently, ten MDMA positive urines were analysed to compare the metabolite patterns with those produced by HLM. An LC/MS method for MDMA and its metabolites in urine samples was developed and validated. The method demonstrated good linearity, accuracy and precision and insignificant matrix effects, with limits of quantitation of 0.025 µg/ml. Moreover, derivatives of MDMA and its metabolites were quantified in all 10 positive human urine samples. The urine metabolite pattern was found to be similar to that from HLM. The second aim of Part 1 was to use the HLM system to study the metabolism of some new psychoactive substances, whose misuse worldwide has necessitated the development of analytical methods for these drugs in biological specimens. Methylone and butylone were selected as representative cathinones and para-methoxyamphetamine (PMA) was chosen as a representative ring-substituted amphetamine, because of the involvement of these drugs in recent drug-related deaths, because of a relative lack of information on their metabolism, and because reference standards of their metabolites were not commercially available. An LC/MS/MS method for the analysis of methylone, butylone, PMA and their metabolites was developed. Three phase I metabolites of methylone and butylone were detected in HLM samples. Ketone reduction to β-OH metabolites and demethylenation to dihydroxy-metabolites were found to be major phase I metabolic pathways of butylone and methylone whereas N-demethylation to nor-methylone and nor-butylone were found to be minor pathways. Also, demethylation to para-hydroxyamphetamine was found to be a major phase I metabolic pathway of PMA whereas β-hydroxylation to β-OH-PMA was found to be a minor pathway. Formaldehyde is used for embalming, to reduce decomposition and preserve cadavers, especially in tropical countries such as Thailand. Drugs present in the body can be exposed to formaldehyde resulting in decreasing concentrations of the original compounds and production of new substances. The aim of part II of the study was to evaluate the in vitro reactions of formaldehyde with selected drug groups including amphetamines (amphetamine, methamphetamine and MDMA), benzodiazepines (alprazolam and diazepam), opiates (morphine, hydromorphone, codeine and hydrocodone) and with a carbamate insecticide (carbosulfan). The study would identify degradation products to serve as markers for the parent compounds when these were no longer detectable. Drugs standards were spiked in 10% formalin solution and 10% formalin blood. Water and whole blood without formalin were used for controls. Samples were analysed by LC/MS/MS at different times from the start, over periods of up to 30 days. Amphetamine, methamphetamine and MDMA were found to rapidly convert to methamphetamine, DMA and MDDMA respectively, in both formalin solution and formalin blood, confirming the Eschweiler-Clarke reaction between amine-containing compounds and formaldehyde. Alprazolam was found to be unstable whereas diazepam was found to be stable in both formalin solution and water. Both were found to hydrolyse in formalin solution and to give open-ring alprazolam and open-ring diazepam. Other alprazolam conversion products attached to paraformaldehyde were detected in both formalin solution and formalin blood. Morphine and codeine were found to be more stable than hydromorphone and hydrocodone in formalin solution. Conversion products of hydromorphone and hydrocodone attached to paraformaldehyde were tentatively identified in formalin solution. Moreover, hydrocodone and hydromorphone rapidly decreased within 24 h in formalin blood and could not be detected after 7 days. Carbosulfan was found to be unstable in formalin solution and was rapidly hydrolysed within 24 h, whereas in water it was stable up to 48 h. Carbofuran was the major degradation product, plus smaller amounts of other products, 3-ketocarbofuran and 3-hydrocarbofuran. By contrast, carbosulfan slowly hydrolysed in formalin-blood and was still detected after 15 days. It was concluded that HLM provide a useful tool for human drug metabolism studies when ethical considerations preclude their controlled administration to humans. The use of chemical derivatisation for hydrophilic compounds such as polar drug metabolites for analysis by LC/MS/MS with a conventional C18 column is effective and inexpensive, and suitable for routine use in the identification and quantitation of drugs and their metabolites. The detection of parent drugs and their metabolites or conversion and decomposition products is potentially very useful for the interpretation of cases in forensic toxicology, especially when the original compounds cannot be observed.

Degradation of compounds present in cork boiling water by gamma radiation

Cork boiling water is an aqueous and complex dark liquor with high concentration of phenolic compounds such as phenolic acids and tannins [1, 2], which are considered biorecalcitrants [2]. Ionizing radiation has been widely studied as an alternative technology for the degradation of organic contaminants without the addition of any other (e.g.: Fenton technologies). The aim of this work was to identify the compounds present in cork boiling water and further evaluate the resulting stable degradation products after gamma irradiation. The irradiation experiments of standard solutions were carried out at room temperature using a Co-60 experimental equipment. The applied absorbed doses were 20 and 50 kGy at a dose rate of 1.5 kGy/h, determined by routine dosimeters [3]. The identification of radiolytic products was carried out by HPLC-DAD-ESI/MS. The phenolic compounds were identified by comparing their retention times and UV–vis and mass spectra with those obtained from standard compounds, when available, as well as by comparing the obtained information with available data reported in the literature. Concerning the obtained results and the literature review, the main cork wastewater components are: quinic, gallic, protocatechuic, vanillic, syringic and ellagic acids. Based on this, we used protocatechuic, vanillic and syringic acids as model compounds to study their degradation by gamma radiation in order to identify the corresponding radiolytic products. Standard aqueous solutions were irradiated and the derivatives of each model compound are represented in figure 1. The obtained results seem to demonstrate that the derivatives of the parent compounds could also be phenolic acids, since it was observed the loss of 44 u (CO2) from the [M-H]- ions. Gallic and protocatechuic acids are identified as derivatives of vanillic and syringic acids, and gallic acid as a protocatechuic acid derivative. Compound 5 ([M-H]- at m/z 169) was tentatively identified as 2,4,6-trihydroxybenzoic acid, since its fragmentation pattern (m/z 151, 125 and 107) is similar to that previously reported in literature [4]. The structure of compound 7 was proposed based on the molecular ion and its fragmentation and compound 6 remains unknown.

New Catalysts for the Renewable Production of Monomers for Bioplastics

Terephthalic acid (PTA) is one of the monomers used for the synthesis of the polyester, polyethylene terephthalate (PET), that is used for the large-scale manufacture of synthetic fibers and plastic bottles. PTA is largely produced from the liquid-phase oxidation of petroleum-derived p-xylene (PX). However, there are now ongoing worldwide efforts exploring alternative routes for producing PTA from renewable, biomass resources.

In this thesis, I present a new route to PTA starting from the biomass-derived platform chemical, 5-hydroxymethylfurfural (HMF). This route utilizes new, selective Diels-Alder-dehydration reactions involving ethylene and is advantageous over the previously proposed Diels-Alder-dehydration route to PTA from HMF via 2,5-dimethylfuran (DMF) since the H₂ reduction of HMF to DMF is avoided. Specifically, oxidized derivatives of HMF are reacted as is, or after etherification-esterification with methanol, with ethylene over solid Lewis acid catalysts that do not contain strong Brønsted acids in order to synthesize intermediates of PTA and its equally important diester, dimethyl terephthalate (DMT). The partially oxidized HMF, 5-(hydroxymethyl)furoic acid (HMFA) is reacted with high pressure ethylene over a pure-silica molecular sieve catalyst containing framework tin (Sn-Beta) to produce the Diels-Alder-dehydration product, 4-(hydroxymethyl)benzoic acid (HMBA), with ~30% selectivity at ~20% yield. If HMFA is protected with methanol to form methyl 5-(methoxymethyl)furan-2-carboxylate (MMFC), MMFC can react with ethylene in the presence of a pure-silica molecular sieve containing framework zirconium (Zr-Beta) to produce methyl 4-(methoxymethyl)benzenecarboxylate (MMBC) with >70% selectivity at >20% yield. HMBA and MMBC can then be oxidized to produce PTA and DMT, respectively. When Lewis acid containing mesoporous silica (MCM-41) and amorphous silica, or Brønsted acid containing zeolites (Al-Beta), are used as catalysts, a significant decrease in selectivity/yield of the Diels-Alder-dehydration product is observed.

An investigation to elucidate the reaction network and side products in the conversion of MMFC to MMBC was performed, and the main side products are found to be methyl 4-formylcyclohexa-1,3-diene-1-carboxylate and the ethylene Diels-Alder adduct of this cyclohexadiene. These products presumably form by a different dehydration pathway of the MMFC/ethylene Diels-Alder adduct and should be included when determining the overall selectivity to PTA or DMT since, like MMBC, these compounds are precursors to PTA or DMT.

Fundamental physical and chemical information on the ethylene Diels-Alder-dehydration reactions catalyzed by the Lewis acid-containing molecular sieves was obtained. Madon-Boudart experiments using Zr-Beta as catalyst show that the reaction rates are limited by chemical kinetics only (physical transport limitations are not present), all the Zr⁴⁺ centers are incorporated into the framework of the molecular sieve, and the whole molecular sieve crystal is accessible for catalysis. Apparent activation energies using Zr-Beta are low, suggesting that the overall activation energy of the system may be determined by a collection of terms and is not the true activation energy of a single chemical step.

BALD EAGLES (HALIAEETUS LEUCOCEPHALUS) AS INDICATORS OF GREAT LAKES ECOSYSTEM HEALTH

Environmental indicators have been proposed as a means to assess ecological integrity, monitoring both chemical and biological stressors. In this study, we used nestling bald eagles as indicators to quantify direct or indirect tertiary-level contaminant exposure. The spatial and temporal trends of polychlorinated biphenyl (PCB) congeners were evaluated in nestling plasma from 1999–2014. Two hexa-chlorinated congeners, PCB-138 and 153, were detected with the highest frequency and greatest concentrations throughout Michigan. Less-chlorinated congeners such as PCB-52 and 66 however, comprised a greater percentage of total PCB concentrations in nestlings proximate to urbanized areas, such as along the shorelines of Lake Erie. Toxic equivalents were greatest in the samples collected from nestlings located on Lake Erie, followed by the other Great Lakes spatial regions. Nestling plasma samples were also used to measure concentrations of the most heavily-used group of flame retardants, brominated diphenyl ethers (BDEs), and three groups of alternative flame retardants, non-BDE Brominated Flame Retardants (NBFRS), Dechloranes, and organophosphate esters (OPs). BDE-47, 99 and 100 contributed the greatest to total BDE concentrations. Concentrations of structurally similar NBFRs found in this study and recent atmospheric studies indicate that they are largely used as replacements to previously used BDE mixtures. A variety of Dechloranes, or derivatives of Mirex and Dechlorane Plus, were measured. Although, measured at lesser concentrations, environmental behavior of these compounds may be similar to mirex and warrant future research in aquatic species. Concentrations of OPs in nestling plasma were two to three orders of magnitude greater than all other groups of flame retardants. In addition to chemical indicators, bald eagles have also been proposed as indicators to identify ecological stressors using population measures that are tied to the fitness of individuals and populations. Using mortality as a population vitality rate, vehicle collisions were found to be the main source of mortality with a greater incidence for females during white-tailed deer (Odocoileus virginianus) hunting months and spring snow-melt. Lead poisoning was the second greatest source of mortality, with sources likely due to unretrieved hunter-killed, white-tailed deer carcasses, and possibly exacerbated by density-dependent effects due to the growing population in Michigan.

Synthesis and antibacterial study of some s-substituted aliphatic analogues of 2-mercapto-5-(1-(4-toluenesulfonyl) piperidin-4-yl)-1,3,4-oxadiazole

Purpose: To synthesize a series of analogues of 1,3,4-oxadiazole and to evaluate their antibacterial activity. Methods: Ethyl piperidin-4-carboxylate (1) was mixed with 4-toluenesulfonyl chloride (2) in benignant conditions to yield ethyl 1-(4-toluenesulfonyl)piperidin-4-carboxylate (3) and then 1-(4- toluenesulfonyl)piperidin-4-carbohydrazide (4). Intermolecular cyclization of 4 into 2-mercapto-5-(1-(4- toluenesulfonyl) piperidin-4-yl)-1,3,4-oxadiazole (5) was obtained on reflux with CS2 in the presence of KOH. Molecule 5 was stirred with alkyl halides, 6a-i, in DMF in the presence of LiH to synthesize the final compounds, 7a-i. The structures of these molecules were elucidated by Fourier transform infra-red (FTIR) spectroscopy, proton nuclear magnetic resonance (1H-NMR) and electron impact mass spectrometry (EI-MS). Antibacterial activity was evaluated against five bacterial strains, namely, Salmonella typhi, Escherichia coli, Pseudomonas aeruginosa, Staphylococcus aureus and Bacillus subtilis, with ciprofloxacin used as standard antibacterial agent. Results: Out of nine synthesized derivatives, compound 7a was the most active against three bacterial strains, S. typhi, E. coli and P. aeruginosa, with minimum inhibitory concentration (MIC) of 9.11 ± 0.40, 9.89 ± 0.45 and 9.14 ± 0.72 μM, respectively, compared with 7.45 ± 0.58, 7.16 ± 0.58 and 7.14 ± 0.18 μM, respectively, for the reference standard (ciprofloxacin). Similarly, compounds 7a - 7c showed relatively good antibacterial activity against B. subtilis strain while compound 7e - 7g revealed good results against S. typhi bacterial strain. Conclusion: The results indicate that S-substituted derivatives of the parent compound are more effective antibacterial agents than the parent compound, even with minor differences in substituents

Preliminary structure-activity relationship studies on some novel s-substituted aliphatic analogues of 5-{1-[(4- chlorophenyl) sulfonyl]-3-piperidinyl}-1, 3, 4-oxadiazol-2-yl sulfide

Purpose: To study the structure-activity relationships of synthetic multifunctional sulfides through evaluation of lipoxygenase and anti-bacterial activities. Methods: S-substituted derivatives of the parent compound 5-(1-(4-chlorophenylsulfonyl) piperidin-3- yl)-1, 3, 4-oxadiazole-2-thiol were synthesized through reaction with different saturated and unsaturated alkyl halides in DMF medium, with NaH catalyst. Spectral characterization of each derivative was carried out with respect to IR, 1H - NMR, 13C - NMR and EI - MS. The lipoxygenase inhibitory and antibacterial activities of the derivatives were determined using standard procedures. Results: Compound 5e exhibited higher lipoxygenase inhibitory potential than the standard (Baicalein®), with % inhibition of 94.71 ± 0.45 and IC50 of 20.72 ± 0.34 μmoles/L. Compound 5b showed significant antibacterial potential against all the bacterial strains with % inhibition ranging from 62.04 ± 2.78, 69.49 ± 0.41, 63.38 ± 1.97 and 59.70 ± 3.70 to 78.32 ± 0.41, while MIC ranged from 8.18 ± 2.00, 10.60 ± 1.83, 10.84 ± 3.00, 9.81 ± 1.86 and 11.73 ± 5.00 μmoles/L for S. typhi, E. coli, P. aeruginosa, B. subtilis and S. aureus, respectively. Compounds 5d, 5e and 5g showed good antibacterial activity against S. typhi and B. subtilis bacterial strains. Conclusion: The results suggest that compound 5e bearing n-pentyl group is a potent lipoxygenase inhibitor, while compound 5b with n-propyl substitution is a strong antibacterial agent. In addition, compounds 5d, 5e and 5g bearing n-butyl, n-pentyl and n-octyl groups, respectively, are good antibacterial agents against S. typhi and B. subtilis.

Systems of coupled clamped beams equations with full nonlinear terms: Existence and location results

This work gives sufficient conditions for the solvability of the fourth order coupled system┊ u⁽⁴⁾(t)=f(t,u(t),u′(t),u′′(t),u′′′(t),v(t),v′(t),v′′(t),v′′′(t)) v⁽⁴⁾(t)=h(t,u(t),u′(t),u′′(t),u′′′(t),v(t),v′(t),v′′(t),v′′′(t)) with f,h: [0,1]×ℝ⁸→ℝ some L¹- Carathéodory functions, and the boundary conditions {┊ u(0)=u′(0)=u′′(0)=u′′(1)=0 v(0)=v′(0)=v′′(0)=v′′(1)=0. To the best of our knowledge, it is the first time in the literature where two beam equations are considered with full nonlinearities, that is, with dependence on all derivatives of u and v. An application to the study of the bending of two elastic coupled campled beams is considered.

Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

Adaptive online gradient descent

We study the rates of growth of the regret in online convex optimization. First, we show that a simple extension of the algorithm of Hazan et al eliminates the need for a priori knowledge of the lower bound on the second derivatives of the observed functions. We then provide an algorithm, Adaptive Online Gradient Descent, which interpolates between the results of Zinkevich for linear functions and of Hazan et al for strongly convex functions, achieving intermediate rates between [square root T] and [log T]. Furthermore, we show strong optimality of the algorithm. Finally, we provide an extension of our results to general norms.

Effective shear modulus approach for two dimensional solids and plate bending problems by meshless point collocation method

For the analysis of material nonlinearity, an effective shear modulus approach based on the strain control method is proposed in this paper by using point collocation method. Hencky’s total deformation theory is used to evaluate the effective shear modulus, Young’s modulus and Poisson’s ratio, which are treated as spatial field variables. These effective properties are obtained by the strain controlled projection method in an iterative manner. To evaluate the second order derivatives of shape function at the field point, the radial basis function (RBF) in the local support domain is used. Several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method and comparisons have been made with analytical solutions and the finite element method (ABAQUS).

A second-order accurate numerical approximation for the Riesz space fractional advection-dispersion equation

In this paper, we consider a space Riesz fractional advection-dispersion equation. The equation is obtained from the standard advection-diffusion equation by replacing the ¯rst-order and second-order space derivatives by the Riesz fractional derivatives of order β 1 Є (0; 1) and β2 Є(1; 2], respectively. Riesz fractional advection and dispersion terms are approximated by using two fractional centered difference schemes, respectively. A new weighted Riesz fractional ¯nite difference approximation scheme is proposed. When the weighting factor Ѳ = 1/2, a second- order accurate numerical approximation scheme for the Riesz fractional advection-dispersion equation is obtained. Stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.