This article investigates a function f(x), constructed from the Nikol'skii class in S2. The estimation obtained will show that the Riesz mean of the spectral expansions is unable to be strengthened due to absence of localization caused by a singualrity at a definite point f(x), on the sphere. � 2017 Author(s).

Purpose: The purpose of this paper is to examine the extent of risk shifting behavior in bonds and sukuk. The examination is significant, as economists and scholars identify risk shifting as the primary cause of the global financial crisis. Yet, the dangers of this debt-financing feature are largely ignored � one needs to only witness the record growth of global debt even after the global financial crisis. Design/methodology/approach: To identify the signs of risk shifting existence in the corporations, this paper compares each corporation�s operating risk before and after issuing debt. Operating risk or risk of a firm�s activities is measured using the volatility of the operating earnings or coefficient variation of earning before interest, tax, depreciation and amortization (EBITDA). Using EBITDA as the variable offers one distinct advantage to using asset volatility as previous research has � EBITDA can be extracted directly from firms� accounting data and is not model-specific. Findings: Risk shifting can be found in not only the bond system but also the debt-based sukuk system � a noteworthy finding because sukuk, supposedly in a different class from bonds, have been criticized in some quarters for their apparent similarity to bonds. On the other hand, this study thus shows that equity feature, when it is embedded in bonds (as in convertible bonds) or when a financial instrument is based purely on equity (as in equity-based sukuk), the incentive to shift the risk can be mitigated. Research limitations/implications: Global awareness of the dangers of debt should be increased as a means of reducing the amount of debt outstanding globally. Although some regulators suggest that sukuk replace debt, they must also be aware that imitative sukuk pose the same threat to efforts to avoid debt. In short, efforts to ensure future financial stability cannot address only debts or bonds but must also address those types of sukuk that mirror bonds in their operation. In the wake of the global financial crisis, amid the frantic search for ways of protecting against future financial shocks, this analysis aims to help create future stability by encouraging market players to avoid debt-based activities. Originality/value: This paper differs from the previous literature in two important ways, viewing risk shifting behavior not only in relation to debt or bonds but also when set against debt-based sukuk, which has been subjected to similar criticism. Indeed, to the extent that debts and bonds encourage risk shifting behavior and threaten the entire financial system, so, too, can imitation sukuk or debt-based sukuk. Second, this paper is unique in exploring the ability of equity features to curb equity holders� incentive to engage in risk shifting behavior. Such an examination is necessary for the wake of the global financial crisis, for researchers and economists now agree that risk shifting must be a controlled behavior � and that one way of controlling risk shifting is by implementing the risk sharing feature of equity-based financing into the financial system. � 2018, Emerald Publishing Limited.

The study of chaotic motion in periodic self-excited oscillators are an area of interest in science and engineering. In the current research, a numerical solution in backward difference form is proposed for solving these chaotic motions in periodic- self excited oscillators. Study conducted in this article focuses on chaotic motions in the form of Duffing-Van Der Pol Oscillators. A backward difference formulation in predictor-corrector (PeCe) mode is introduced for solving these Duffing-Van Der Pol directly. Numerical simulations provided will show the accuracy of the PeCe backward difference formulation.

Purpose: The purpose of this paper is to examine incentives for risk shifting in debt- and equity-based contracts based on the critiques of the similarities between sukuk and bonds. Design/methodology/approach: This paper uses a theoretical and mathematical model to investigate whether incentives for risk taking exist in: debt contracts; and equity contracts. Findings: Based on this theoretical model, it argues that risk shifting behaviour exists in debt contracts only because debt naturally gives rise to risk shifting behaviour when the transaction takes place. In contrast, equity contracts, by their very nature, involve sharing transactional risk and returns and are thus thought to make risk shifting behaviour undesirable. Nonetheless, previous researchers have found that equity-based financing also might carry risk shifting incentives. Even so, this paper argues that the amount of capital provided and the underlying assets must be considered, especially in the event of default. Through mathematical modelling, this element of equity financing can make risk shifting unattractive, thus making equity financing more distinct than debt financing. Research limitations/implications: Global awareness of the dangers of debt should be increased as a means of reducing the amount of debt outstanding globally. Although some regulators suggest that sukuk replaces debt, they must also be aware that imitative sukuk poses the same threat to efforts to avoid debt. In short, efforts to ensure future financial stability cannot address only debts or bonds but must also address those types of sukuk that mirrors bonds in their operation. In the wake of the global financial crisis, amid the frantic search for ways of protecting against future financial shocks, this analysis aims to help create future stability by encouraging market players to avoid debt-based activities and promoting equity-based instruments. Practical implications: This paper�s findings are relevant for countries that feature more than one type of financial market (e.g. Islamic and conventional) because risk shifting behaviour can degrade economic and financial stability. Originality/value: This paper differs from the previous literature in two important ways, viewing risk shifting behaviour not only in relation to debt or bonds but also when set against debt-based sukuk, which has been subjected to similar criticism. Indeed, to the extent that debts and bonds encourage risk shifting behaviour and threaten the entire financial system, so, too, can imitation sukuk or debt-based sukuk. Second, this paper is unique in exploring the ability of equity features to curb equityholders� incentive to engage in risk shifting behaviour. Such an examination is necessary for the wake of the global financial crisis, for researchers and economists now agree that risk shifting must be controlled. � 2018, Emerald Publishing Limited.

A two-point block backward difference technique is established for solving nonlinear Riccati differential equations directly. The proposed method is coded using a variable order step size (VOS) algorithm. The advantage of the two-point block method is its programmability to implement parallel programming techniques. Combination of the block method and VOS algorithm allows for a significant reduction of computation cost in comparison to conventional methods. With an added advantage of the recursive relationship between integration coefficients of different orders, the proposed two-point block method provides efficient computation without loss of accuracy. � 2018 Author(s).

Nonlinear phenomena in science and engineering such as a periodically forced oscillator with nonlinear elasticity are often modeled by the Duffing oscillator (Duffing equation). The Duffling oscillator is a type of nonlinear higher order differential equation. In this research, a numerical approximation for solving the Duffing oscillator directly is introduced using a variable order stepsize (VOS) algorithm coupled with a backward difference formulation. By selecting the appropriate restrictions, the VOS algorithm provides a cost efficient computational code without affecting its accuracy. Numerical results have demonstrated the advantages of a variable order stepsize algorithm over conventional methods in terms of total steps and accuracy. � Published under licence by IOP Publishing Ltd.