# Mathematical Formulations (mathematical + formulations)

Distribution by Scientific Domains

## Selected Abstracts

### Impact of Biomass-Decay Terms on the Simulation of Pulsed Bioremediation

GROUND WATER, Issue 2 2000
Olaf A. Cirpka
A numerical model is used for simulating the stimulation of biomass growth by injection of alternating pulses of a primary substrate and oxygen. We consider that the substrate sorbs, whereas oxygen does not undergo mass transfer, and mixing of the reacting compounds is dominated by the chromatographic effect. Different mathematical formulations for biomass growth and decay are compared. In models considering biomass decay, a minimal time of joint exposure to both reactants can be determined. This leads to a multimodal distribution of the biomass after multiple injection cycles. In multidimensional heterogeneous domains, the location of the biomass peaks is determined by the advective arrival time. The biomass is much more homogeneously distributed when biomass decay is neglected, because under this condition there is no constraint by a minimal joint exposure time. For the case of oxygen-dependent biomass decay, an injection scheme using shorter pulses of higher oxygen concentrations is shown to be superior to a scheme with equally long pulses of oxygen and the substrate. [source]

### Modelling strain localization in granular materials using micropolar theory: mathematical formulations

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 15 2006
Mustafa I. Alsaleh
Abstract It has been known that classical continuum mechanics laws fail to describe strain localization in granular materials due to the mathematical ill-posedness and mesh dependency. Therefore, a non-local theory with internal length scales is needed to overcome such problems. The micropolar and high-order gradient theories can be considered as good examples to characterize the strain localization in granular materials. The fact that internal length scales are needed requires micromechanical models or laws; however, the classical constitutive models can be enhanced through the stress invariants to incorporate the Micropolar effects. In this paper, Lade's single hardening model is enhanced to account for the couple stress and Cosserat rotation and the internal length scales are incorporated accordingly. The enhanced Lade's model and its material properties are discussed in detail; then the finite element formulations in the Updated Lagrangian Frame (UL) are used. The finite element formulations were implemented into a user element subroutine for ABAQUS (UEL) and the solution method is discussed in the companion paper. The model was found to predict the strain localization in granular materials with low dependency on the finite element mesh size. The shear band was found to reflect on a certain angle when it hit a rigid boundary. Applications for the model on plane strain specimens tested in the laboratory are discussed in the companion paper. Copyright © 2006 John Wiley & Sons, Ltd. [source]

### Using the theory of constraints thinking processes to complement system dynamics' causal loop diagrams in developing fundamental solutions

INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH, Issue 1 2006
Victoria J. Mabin
Abstract Standard OR/MS methods mainly focus on ,hard' aspects of problems represented by quantitative or mathematical formulations. Organisational settings typically pose challenges in the use of such methods, in that they may be inadequate to capture the ,softer' issues surrounding human behaviours, organisational practices and policies. Over the last 20 years, a number of ,soft OR' tools, methods and methodologies have emerged as a means of addressing such challenges, and in this paper, we selectively examine how such methodologies can complement hard and other soft methods. In particular, we examine the potential contribution of the theory of constraints (TOC) and system dynamics (SD) to multi-methodological intervention. We begin by discussing their philosophical underpinnings in relation to other OR/MS methods, and by exploring how such an understanding can provide a theoretical basis for mixing methodologies and for their complementary use. Then, using a case suitable for classroom discussion, the paper provides insights into how the systemic qualities of selected TOC methods and tools may be harnessed in multi-methodological intervention by identifying the communality and complementarity of TOC and other hard and soft OR/MS approaches to problem solving, in particular, the causal loop diagramming method of SD. [source]

### Animal response to nested self-similar patches: a test with woolly bears

OIKOS, Issue 5 2009
Mark F. McClure
To gain insight into how animals respond to resource patchiness at different spatial scales, we envision their responses in environments comprised of nested, self-similar patches. In these environments, all resources reside within the smallest patches, and resource density declines as a constant exponent of patch size. Accordingly, we use simple mathematical formulations to describe a self-similar environment and a null model of how animals should respond to this environment if they do not perceive resource distribution. We then argue that animals that can perceive resource distribution should partition space by reducing the relative time searching between patches as patch size increases. On an experimental landscape, we found that woolly bear caterpillars Grammia geneura could partition space in this manner, but the range of patch sizes over which they did so tended to increase with resource aggregation. Nevertheless, scaling efficiency (i.e. the scaling of search time versus the scaling or resource density) was similar in all distributions when averaged over all patch sizes. These disparate patterns with similar outcomes resulted from differences in caterpillars' abilities to discriminate spatially among patches of different sizes via their movement pathways, and differences in their use of speed to detect resource items. Our work is relevant to the characterization of resource availability from an animal's perspective, and to the linking of optimal foraging theory to the modeling of search behavior. [source]

### Real-Time Work Schedule Adjustment Decisions: An Investigation and Evaluation

PRODUCTION AND OPERATIONS MANAGEMENT, Issue 4 2004
Daesik Hur
Service managers often find that available worker capacity does not match with actual demand during a given day. They then must attempt to modify the planned work schedule to improve service and increase profitability. This study, which defines such a setting as the real-time work schedule adjustment decision, proposes mathematical formulations of the real-time adjustment and develops efficient heuristic approaches for this decision. The study evaluates the relative effectiveness of these heuristics versus experienced service managers, investigates the effect of the degree of schedule adjustment on profitability, and assesses the effect of demand forecast update errors on the performance of the schedule adjustment efforts. First, the results indicate that the computer based heuristics achieve higher profit improvement than experienced managers. Second, there is a trade-off between schedule stability and profitability so that more extensive schedule revisions (efficiency first heuristics) generally result in higher profitability. However, the incremental return on schedule changes is diminishing. Third, we find that active adjustments of work schedules are beneficial as long as the direction of demand change is accurately identified. [source]