Algorithmic Differentiation of Numerical Methods: Second-order Adjoint Solvers for Parameterized Systems of Nonlinear Equations

AbstractAdjoint mode algorithmic (also know as automatic) differentiation (AD) transforms implementations of multivariate vector functions as computer programs into first-order adjoint code. Its reapplication or combinations with tangent mode AD yields higher-order adjoint code. Second derivatives play an important role in nonlinear programming. For example, second-order (Newton-type) nonlinear optimization methods promise faster convergence in the neighborhood of the minimum through taking into account second derivative information. The adjoint mode is of particular interest in large-scale gradient-based nonlinear optimization due to the independence of its computational cost on the number of free variables. Part of the objective function may be given implicitly as the solution of a system of n parameterized nonlinear equations. If the system parameters depend on the free variables of the objective, then second derivatives of the nonlinear system's solution with respect to those parameters are required. The local computational overhead as well as the additional memory requirement for the computation of second-order adjoints of the solution vector with respect to the parameters by AD depends on the number of iterations performed by the nonlinear solver. This dependence can be eliminated by taking a symbolic approach to the differentiation of the nonlinear system

Cerebrospinal Fluid Tau Protein Levels and F-18-Fluorodeoxyglucose Positron Emission Tomography in the Differential Diagnosis of Alzheimer's Disease

Aims: In this study, we aimed to compare cerebrospinal fluid (CSF) levels of total tau (t-tau), phosphorylated tau (p-tau(181)) and positron emission tomography with F-18-fluorodeoxyglucose (FDG-PET) in the differential diagnosis of Alzheimer's disease (AD) under clinical conditions. Method: In a cross-sectional, blinded, single-center study, we examined a sample of 75 unselected memory clinic patients with clinical diagnoses of dementia of Alzheimer type (DAT; n = 24), amnestic mild cognitive impairment (MCI; n = 16), other dementias (n = 13) and nondemented controls (n = 22). Discriminative accuracy, sensitivity and specificity were calculated and compared using ROC analyses. Results: p-tau(181) and FDG-PET were comparable in separating DAT from controls (sensitivity: 67 vs. 79%; specificity: 91% for both) and patients with other dementias (sensitivity: 71 vs. 79%; specificity: 100% for both). The sensitivity of p-tau 181 in differentiating MCI patients from controls was significantly (p < 0.05) superior to that of FDG-PET (75 vs. 44%) at a comparably high specificity (82 vs. 91%); t-tau measures were less accurate in all analyses. Conclusions: FDG-PET and CSF p-tau(181) levels are able to discriminate DAT in heterogeneous and unselected samples with a high accuracy. CSF p-tau(181) might be somewhat superior for a sensitive detection of patients with MCI. Copyright (C) 2010 S. Karger AG, Base

The law of purchase and sale

The Consumer Protection Act 68 of 2008 (‘CPA’; ‘the Act’) is discussed in detail in the chapter on The General Principles of Contract. The discussion in the present chapter focuses mainly on the influence of the Act on the law of purchase and sale. In this discussion, liberal use has been made of Evert van Eeden’s A Guide to the Consumer Protection Act (2009), NJ Melville’s The Consumer Protection Act Made Easy (2010), and W Jacobs, P Stoop and R van Niekerk’s ‘Fundamental consumer rights under the Consumer Protection Act 68 of 2008: A critical overview and analysis’ (2010) 13 Potchefstroom Electronic Law Journal 302 (available at ).http://www.jutalaw.co.za/catalogue/itemdisplay.jsp?item_id=3592am201

The law of purchase and sale

No abstract available.http://reference.sabinet.co.za/sa_epublication/ju_assalam201

The law of purchase and sale

The implementation of the Consumer Protection Act 68 of 2008 was postponed from 24 October 2010 to 31 March 2011. The impact of this Act on the Law of Purchase and Sale will be assessed in the 2011 Annual Survey.http://www.jutalaw.co.za/catalogue/itemdisplay.jsp?item_id=3592nf201

The alienation of land can take place orally : fact or fiction?

Section 2(1) of the Alienation of Land Act 68 of 1981 remains a bone of contention that emerges, too often, in case law. Although the application of this Act entails more than the mere purchase and sale of immovable property, its application is meticulously limited to the sale, exchange or donation of a unit, a right to claim transfer of land, an undivided share in land and for the purposes of chapters I and II of the Act, also any interest in land. The aim of this article is to illustrate that if an alienation of land does not fall strictly within the ambit of the above boundaries, the formality requirements of section 2(1) have no bearing on the validity of the agreement. For this purpose, all recent cases dealing with this topic are analysed to illustrate that the alienation of land can be achieved without complying with the stringent formality prerequisites of section 2(1).http://www.lexisnexis.co.zaam2016Mercantile La

The law of purchase and sale

Section 2(1) of the Alienation of Land Act 68 of 1981 once again raised its ugly head in Osborne and Another v West Dunes Properties 176 (Pty) Ltd and Others 2013 (6) SA 105 (WCC).http://reference.sabinet.co.za/sa_epublication/ju_assalam2016Mercantile La

The Corondimas principle is still alive and well :. Diggers Development (Pty) Ltd v City of Matlosana : Case no 824/2010. [2011] ZASCA 247

It is a trite principle of the law of contract that a condition is an uncertain future event upon which either the commencement of the duty to perform or the contract’s continued existence is made dependent. The effect of a suspensive condition can be summarised as follows (see Otto and Prozesky-Kuschke “General principles of the law of contract” in Nagel (ed) Commercial law (2011) 114–115): A condition is suspensive when the right or duty to performance in terms of a contract is suspended or postponed pending (and thus made dependent upon) the occurrence or non-occurrence of an uncertain future event specified in the contract. Pending the fulfilment of the condition, a contractual relationship indeed exists between the parties. One of the consequences of the existence of a contractual relationship is that neither party can withdraw from the contract and that any additional duty must be performed. Upon fulfilment of the condition, the parties are entitled to performance and obliged to perform. Before fulfilment, performance may not be claimed. If the condition is not fulfilled, the contract is terminated and neither party has to perform. In addition, either party is entitled to the return of anything already performed (see also Hutchison and Pretorius (eds) The law of contract in South Africa (2009) 247–248; Van der Merwe et al Contract general principles (2007) 289–293; Van Rensburg et al “Contract” 5(1) LAWSA (2004) paras 436–437; Kerr The Principles of the law of contract (2002) 446–449 and Christie The law of contract in South Africa (2011) 145 and authorities cited).http://www.lexisnexis.co.zaam201

Expanding the scope of latent defects and the protection afforded by " voetstoots" clauses? Developments in the case law

No abstract available.http://www.lexisnexis.co.za2015-05-31hb201