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The purpose of this paper is to develop an alternative numerical approach for describing fractional diffusion in Cartesian and non‐Cartesian domains using a Monte Carlo random walk scheme. The resulting domain shifting scheme provides a numerical solution for multi‐dimensional steady state, source free diffusion problems with fluxes expressed in terms of Caputo fractional derivatives. This class of problems takes account of non‐locality in transport, expressed through parameters representing both the extent and direction of the non‐locality.

The method described here follows a similar approach to random walk methods previously developed for normal (local) diffusion. The key differences from standard methods are: first, the random shifting of the domain about the point of interest with, second, shift steps selected from non‐symmetric, power‐law tailed, Lévy probability distribution functions.

The domain shifting scheme is verified by comparing predictive solutions to known one‐dimensional and two‐dimensional analytical solutions for fractional diffusion problems. The scheme is also applied to a problem of fractional diffusion in a non‐Cartesian annulus domain. In contrast to the axisymmetric, steady state solution for normal diffusion, a non‐axisymmetric solution results.

This is the first random walk scheme to utilize the concept of allowing the domain to undergo the random walk about a point of interest. Domain shifting scheme solutions of fractional diffusion in non‐Cartesian domains provide an invaluable tool to direct the development of more sophisticated grid based finite element inspired fractional diffusion schemes.

The purpose of this study/paper is the generalized ontological law of monotheism (unity of knowledge) and its functioning in the financial world system is summarized and contrasted with the recent conception of “shari’ah-compliance”. Thereby, some specific rulings of shari’ah-compliance in Islamic finance are critically annulled. The principal problem of the inability of shari’ah-compliance in the formalism of rate-setting and debt cancellation is pointed out in analytical ways. The alternative valuation models in the light of the Tawhidi ontological law are formalized. Many important issues are examined in analytical and Tawhidi authentic ways of Islamic law contra to shari’ah-compliance.

The epistemological approach commencing from the Tawhidi ontological law is used as the premise of developing analytical formalism to counter the irrelevant rulings done by the field of shari’ah compliance. Thereby, endogenous moral and ethical foundations are studied in deriving analytical finance models of asset valuation, rate-determination and debt cancellation.

Substantive analytical results are derived for intellection in the area of the primal ontological law of Tawhid that negates many of the rulings framed up in shari’ah-compliance area of Islamic law. These results can guide financial academia, practitioners and policymakers.

The paper can be expanded subsequently to the area of analytical Islamic finance in general by further investigating the Modigliani and Miller theorem on optimal debt-equity structure of corporate finance. An introduction to this study is provided in this paper as a starting point of dealing with the debt problem of shari’ah-compliance.

The paper presents important guidance as input for the rulings of shari’ah-compliance idea held by shari’ah advisory boards and similar institutions presently operating at the financial level.

The paper presents a subtle transformation of the social and financial order in the light of the Tawhidi ontological law quite differently from the way that shari’ah-compliance envisions.

The theoretical and projected applied perspectives in analytical finance presented in this paper provide a methodological worldview for all areas of social finance with ethical consciousness. Such analytical approach is much needed today in the reconstruction of global finance in the scale of ethics and away from the sole focus on capital market efficiency.

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

This paper conceives of Hayek's overall project as presenting a theory of sociocognition, explication of which has a two-fold purpose: (1) to locate Hayek within the non-Cartesian tradition of cognitive science, and (2) to show how Hayek's philosophical psychology infuses his social theory.

Topology optimization is a method used for developing optimized geometric designs by distributing material pixels in a given design space that maximizes a chosen quantity of interest (QoI) subject to constraints. The purpose of this study is to develop a problem-agnostic automatic differentiation (AD) framework to compute sensitivities of the QoI required for density distribution-based topology optimization in an unstructured co-located cell-centered finite volume framework. Using this AD framework, the authors develop and demonstrate the topology optimization procedure for multi-dimensional steady-state heat conduction problems.

Topology optimization is performed using the well-established solid isotropic material with penalization approach. The method of moving asymptotes, a gradient-based optimization algorithm, is used to perform the optimization. The sensitivities of the QoI with respect to design variables, required for optimization algorithm, are computed using a discrete adjoint method with a novel AD library named residual automatic partial differentiator (Rapid).

Topologies that maximize or minimize relevant quantities of interest in heat conduction applications are presented. The efficacy of the technique is demonstrated using a variety of realistic heat transfer applications in both two and three dimensions, in conjugate heat transfer problems with finite conductivity ratios and in non-rectangular/non-cuboidal domains.

In contrast to most published work which has either used finite element methods or Cartesian finite volume methods for transport applications, the topology optimization procedure is developed in a general unstructured finite volume framework. This permits topology optimization for flow and heat transfer applications in complex design domains such as those encountered in industry. In addition, the Rapid library is designed to provide a problem-agnostic pathway to automatically compute all required derivatives to machine accuracy. This obviates the necessity to write new code for finding sensitivities when new physics are added or new cost functions are considered and permits general-purpose implementations of topology optimization for complex industrial applications.

The purpose of this paper concerns the dimensions of relevance in information retrieval systems and their completeness in new retrieval contexts such as mobile search. Geography as a factor in relevance is little understood and information seeking is assumed to take place in indoor environments. Yet the rise of information seeking on the move using mobile devices implies the need to better understand the kind of situational relevance operating in this kind of context.

The paper outlines and explores a geographic information seeking process in which geographic information needs (conditioned by needs and tasks, in context) drive the acquisition and use of geographic information objects, which in turn influence geographic behaviour in the environment. Geographic relevance is defined as “a relation between a geographic information need” (like an attention span) and “the spatio‐temporal expression of the geographic information objects needed to satisfy it” (like an area of influence). Some empirical examples are given to indicate the theoretical and practical application of this work.

The paper sets out definitions of geographical information needs based on cognitive and geographic criteria, and proposes four canonical cases, which might be theorised as anomalous states of geographic knowledge (ASGK). The paper argues that geographic relevance is best defined as a spatio‐temporally extended relation between information need (an “attention” span) and geographic information object (a zone of “influence”), and it defines four domains of geographic relevance. Finally a model of geographic relevance is suggested in which attention and influence are modelled as map layers whose intersection can define the nature of the relation.

Geographic relevance is a new field of research that has so far been poorly defined and little researched. This paper sets out new principles for the study of geographic information behaviour.

This paper presents a curvilinearly‐established finite‐difference time‐domain methodology for the enhanced 3D analysis of electromagnetic and acoustic propagation in generalised electromagnetic compatibility devices, junctions or bent ducts. Based on an exact multimodal decomposition and a higher‐order differencing topology, the new technique successfully treats complex systems of varying cross‐section and guarantees the consistent evaluation of their scattering parameters or resonance frequencies. To subdue the non‐separable modes at the structures' interfaces, a convergent grid approach is developed, while the tough case of abrupt excitations is also studied. Thus, the proposed algorithm attains significant accuracy and savings, as numerically verified by various practical problems.

A systematic, non‐orthogonal FDTD algorithm for the unified and fully dual construction of curvilinear PMLs in 3‐D lossy electromagnetic and advective acoustic problems, is presented in this paper. Postulating a consistent mathematical formulation, the novel methodology introduces a set of general vector parametric equations that describe wave propagation in both media and facilitate the effective treatment of the remarkably complex, arbitrarily‐aligned (non‐uniform) source or mean flow terms, particularly at low frequencies. The discretization procedure is performed via accurate higher‐order FDTD topological concepts, which along with a well‐posed variable transformation, suppress the undesired lattice dispersion and anisotropy errors. Hence, due to these additional degrees of design freedom and their optimal establishment, the new stable PMLs (split‐field or Maxwellian) accomplish a critical attenuation of the evanescent, vorticity or elastic wave families by carefully accounting for every loss mechanism. Numerical investigation reveals the superiority of the proposed technique in terms of various open‐region, waveguide and ducted‐domain simulations.

The main purpose of this paper is to develop two efficient and accurate numerical analytical methods for engineering computation of natural sloshing frequencies and modes i the case of truncated circular conical tanks.

The numerical‐analytical methods are based on a Ritz Treftz variational scheme with two distinct analytical harmonic functional bases.

Comparative numerical analysis detects the limit of applicability of variational methods in terms of the semi‐apex angle and the ratio between radii of the mean free surface and the circular bottom. The limits are caused by different analytical properties of the employed functional bases. However, parallel use of two or more bases makes it possible to give an accurate approximation of the lower natural frequencies for relevant tanks. For V‐shaped tanks, dependencies of the lowest natural frequency versus the semi‐apex angle and the liquid depth are described.

The methods provide the natural sloshing frequencies for V‐shaped tanks that are valuable for designing elevated containers in seismic areas. Approximate natural modes can be used in derivations of nonlinear modal systems, which describe a resonant coupling with structural vibrations.

Although variational methods have been widely used for computing the natural sloshing frequencies, this paper presents their application for truncated conical tanks for the first time. An original point is the use of two distinct functional bases.