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This paper investigates spatial data on the unit sphere. Traditionally, isotropic Gaussian random fields are considered as the underlying mathematical model of the cosmic microwave background (CMB) data. We discuss the generalized multifractional Brownian motion and its pointwise Hölder exponent on the sphere. The multifractional approach is used to investigate the CMB data from the Planck mission. These data consist of CMB radiation measurements at narrow angles of the sky sphere. The results obtained suggest that the estimated Hölder exponents for different CMB regions do change from location to location. Therefore, the CMB temperature intensities are multifractional. The methodology developed is used to suggest two approaches for detecting regions with anomalies in the cleaned CMB maps.
We discuss a bi-objective two-stage assignment problem (BiTSAP) that aims at minimizing two objective functions: one comprising a nonlinear cost function defined explicitly in terms of assignment variables and the other a total completion time. A two-stage assignment problem deals with the optimal allocation of n jobs to n agents in two stages, where $n_1$ out of n jobs are primary jobs which constitute Stage-1 and the rest of the jobs are secondary jobs constituting Stage-2. The paper proposes an algorithm that seeks an optimal solution for a BiTSAP in terms of various efficient time-cost pairs. An algorithm for ranking all feasible assignments of a two-stage assignment problem in order of increasing total completion time is also presented. Theoretical justification and numerical illustrations are included to support the proposed algorithms.
We consider fully three-dimensional time-dependent outflow from a source into a surrounding fluid of different density. The source is distributed over a sphere of finite radius. The nonlinear problem is formulated using a spectral approach in which two streamfunctions and the density are represented as a Fourier-type series with time-dependent coefficients that must be calculated. Linearized theories are also discussed and an approximate stability condition for early stages in the outflow is derived. Nonlinear solutions are presented and different outflow shapes adopted by the fluid interface are investigated.
We explain some key challenges when dealing with a single- or multi-objective optimization problem in practice. To overcome these challenges, we present a mathematical program that optimizes the Nash social welfare function. We refer to this mathematical program as the Nash social welfare program (NSWP). An interesting property of the NSWP is that it can be constructed for any single- or multi-objective optimization problem. We show that solving the NSWP could result in more desirable solutions in practice than its single- or multi-objective counterpart. We also discuss several promising approaches that could be employed to solve the NSWP in practice.
This chapter aims to summarize current knowledge regarding the fluid dynamics of wind in canopies and to emphasize aspects that are the most relevant in the context of forest fires. We describe the main characteristics of wind flows in the lower part of the boundary layer, starting from the main features in homogeneous canopies, including velocity and turbulence profiles and characteristics of turbulent structures. Then we address two specific cases of heterogeneous canopies, the clearing-to-forest and the forest-to-clearing transitions, which have been extensively studied. The next section is dedicated to wind flow modeling and how such modeling is used in fire models. Finally, special focus is placed on wind measurement in the context of fire experiments. In this chapter, the feedbacks of fire on wind, as well as atmospheric stability, are not addressed. More information on these topics can be found in Chapters 4 and 8, respectively.
Coupled fire–atmosphere feedback is essential for modeling wildland fire spread, especially extreme fire phenomena. In this chapter, the suite of current and emerging tools capable of modeling this complexity is examined; these tools now dominate fundamental wildland fire research and are starting to be applied to fire operations, training, and planning. Some of the barriers to progress and challenges to validating these tools highlighted in this chapter suggest more emphasis on three areas: a scale-dependent and purposeful approach to comparing model results with appropriate observations, recognizing the limitations of each; the quantification of the errors and under-specifications in fuel properties and the impact of each; and assessing large-scale simulations and directing observations to address priority research gaps, from a position informed by the vast catalog of atmospheric scientific research.
This chapter describes the interactions between three-dimensional fuel metrics, intrinsic fuel properties, plant functional traits, and physical characteristics of fuels that inform a new understanding of fire and vegetation feedbacks. The integration of these themes introduces a new synthetic model of fire–vegetation feedbacks. Interrelated concepts of fire, fluid flow, functional traits, and computational fluid dynamics fire behavior models are discussed within the synthetic model framework.
To “spread like wildfire” is a phrase used to describe something that propagates unexpectedly, rapidly, and incessantly. Much of the unpredictable behavior of a wildfire stems from processes including the heat released from the combustion zone (flames), the structure and condition of fuels, the wind field and turbulence driving the fire, and terrain. However, the chemical make-up of the biomass fuel that powers a wildfire also provides a source for the capricious nature of combustion and the behavior of wildfires. This chapter provides a brief overview of the chemistry of biomass fuels and the chemical processes by which such fuels combust and release the energy that enables the fire to become self-sustaining. It then looks in some detail at the mechanisms through which the combustion chemistry driving the heat release from the fuels is influenced by the environment surrounding the combustion zone. In the worst instances these mechanisms can result in fire behavior that causes widespread death and destruction often over a very short period of time. In the best instances they enable fire to be used as a reliable tool for reducing the hazard present in the wild landscapes of our countryside and surrounding our homes.
We present a discussion of the structure of line fires, a canonical configuration in wildland fire research. This configuration allows detailed studies of the effects of wind and sloped terrain on heat transfer and fire spread mechanisms at flame scale. We emphasize in the discussion the existence of two limiting flame regimes in line fires: the plume-dominated regime, in which the flame is detached from the ground, and the wind or slope-driven regime, in which the flame is attached to that surface. These two regimes correspond to dramatically different flame structures, flow patterns, modes of heat transfer, and flame spread mechanisms. The transition between the two flame regimes is discussed in terms of critical values of Byram's convection number or slope angle. We limit our discussion to a simplified configuration corresponding to gas-fueled flames. Hence the heat release rate of the flame is controlled and the flame is non-spreading; difficulties associated with real wildland fuel are left out of the discussion. The structure of the line fires is discussed through results from high-resolution simulations of laboratory-scale flames based on a large eddy simulation (LES) approach. Additional insight is also obtained through a scaling analysis based on an integral model.
This chapter reviews the current state of knowledge on steps in the process of generating a spot fire. Firebrand formation and data on the size and shape of firebrands generated from wildfires are described. Various elements of firebrand lofting and transport modeling, including firebrand aerodynamics, the fire plume characteristics, models for the ambient wind field, model coupling, experimental results, and a discussion on the sensitivity of the model predictions to the inevitable uncertainty in the model input parameters, are examined. Recent work on the physics of firebrand deposition and spot fire generation is noted.
This chapter presents a synopsis of some of the latest developments in our understanding of pyroconvective interactions, their links to fire geometrym and their role in driving dynamic fire behavior and extreme wildfire development. We highlight the need to augment traditional quasi-steady wildfire modeling paradigms with more sophisticated approaches that combine highly-instrumented, larger-scale experimental studies with state-of-the-art computational modeling. We identify the need to take maximum advantage of technical advances in remote sensing technology to provide new ways of observing extreme fire events.
This chapter describes the fundamental mechanisms of energy transport in and near the flaming front. Convective and radiative processes that generate ignition and subsequent fire spread, the transport of heat in different forms, through and around fuels, both horizontally and vertically, as well as energy measurement considerations are discussed.
This paper looks at adapting the method of Medvedev and Scaillet for pricing short-term American options to evaluate short-term convertible bonds. However unlike their method, we provide explicit formulae for the coefficients of our series solution. This means that we do not need to solve complicated recursive systems, and can efficiently provide fast solutions. We also compare the method with numerical solutions, and find that it performs extremely well, giving accurate bond prices as well as accurate optimal conversion prices.
Wildland fires are among the most complicated environmental phenomena to model. Fire behavior models are commonly used to predict the direction and rate of spread of wildland fires based on fire history, fuel, and environmental conditions; however, more sophisticated computational fluid dynamic models are now being developed. This quantitative analysis of fire as a fluid dynamic phenomenon embedded in a highly turbulent flow is beginning to reveal the combined interactions of the vegetative structure, combustion-driven convective effects, and atmospheric boundary layer processes. This book provides an overview of the developments in modeling wildland fire dynamics and the key dynamical processes involved. Mathematical and dynamical principles are presented, and the complex phenomena that arise in wildland fire are discussed. Providing a state-of-the-art survey, it is a useful reference for scientists, researchers, and graduate students interested in wildland fire behavior from a broad range of fields.