Frequency domain of weakly translation invariant frame MRAs

Frequency domain of bandlimited frame multiresolution analyses (MRAs) plays a key role when derived framelets are applied into narrow-band signal processing and data analysis. In this study, we give a characterization of frequency domain of weakly translation invariant frame scaling functions [Formula: see text] with frequency domain [Formula: see text]. Based on it, we further study convex and ball-shaped frequency domains. If frequency domain of bandlimited frame scaling function [Formula: see text] is convex and completely symmetric about the origin, then it must be weakly invariant and [Formula: see text]. If [Formula: see text] has a ball-shaped frequency domain, the ball radius must be bounded by [Formula: see text]. These frequency domain characters are owned uniquely by frame scaling functions and not by orthogonal scaling functions.

Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects

The scale dependence of the effective anti-plane shear modulus response in microstructures with statistical ergodicity and spatial wide-sense stationarity is investigated. In particular, Cauchy and Dagum autocorrelation functions which can decouple the fractal and the Hurst effects are used to describe the random shear modulus fields. The resulting stochastic boundary value problems (BVPs) are set up in line with the Hill–Mandel condition of elastostatics for different sizes of statistical volume elements (SVEs). These BVPs are solved using a physics-based cellular automaton (CA) method that is applicable for anti-plane elasticity to study the scaling of SVEs towards a representative volume element (RVE). This progression from SVE to RVE is described through a scaling function, which is best approximated by the same form as the Cauchy and Dagum autocorrelation functions. The scaling function is obtained by fitting the scaling data from simulations conducted over a large number of random field realizations. The numerical simulation results show that the scaling function is strongly dependent on the fractal dimension D, the Hurst parameter H, and the mesoscale δ, and is weakly dependent on the autocorrelation function. Specifically, it is found that a larger D and a smaller H results in a higher rate of convergence towards an RVE with respect to δ.

Universal finite-size scaling function for coarsening in the Potts model with conserved dynamics

We study kinetics of phase segregation in multicomponent mixtures via Monte Carlo simulations of the q-state Potts model, in two spatial dimensions, for 2 ≤ q ≤ 20. The associated growth of domains in finite boxes, irrespective of q and temperature, can be described by a single universal finite-size scaling function, with only the introduction of a nonuniversal metric factor in the scaling variable. Our results show that although the scaling function is independent of the type of transition, the q-dependence of the metric factor hints to a crossover at q = 5 where the type of transition in the model changes from second to first order.

Optimization in the construction of cardinal and symmetric wavelets on the line

We present an optimization approach to wavelet architecture that hinges on the Zak transform to formulate the construction as a minimization problem. The transform warrants parametrization of the quadrature mirror filter in terms of the possible integer sample values of the scaling function and the associated wavelet. The parameters are predicated to satisfy constraints derived from the conditions of regularity, compact support and orthonormality. This approach allows for the construction of nearly cardinal scaling functions when an objective function that measures deviation from cardinality is minimized. A similar objective function based on a measure of symmetry is also proposed to facilitate the construction of nearly symmetric scaling functions on the line.

ON THE CHARACTERIZATION OF SCALING FUNCTIONS ON NON-ARCHEMEDEAN FIELDS

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. This gap was filled by Gabardo and Nashed [11] by establishing a constructive algorithm based on the theory of spectral pairs for constructing non-uniform wavelet basis in \(L^2(\mathbb R)\). In this setting, the associated translation set \(\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z\) is no longer a discrete subgroup of \(\mathbb R\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we characterize the scaling function for non-uniform multiresolution analysis on local fields of positive characteristic (LFPC). Some properties of wavelet scaling function associated with non-uniform multiresolution analysis (NUMRA) on LFPC are also established.

Assessing Model Uncertainty for the Scaling Function Inversion of Potential Fields

We use the very fast simulated annealing algorithm to invert the scaling function along selected ridges, lying in a vertical section formed by upward continuing gravity data to a set of altitudes. The scaling function is formed by the ratio of the field derivative by the field itself and it is evaluated along the lines formed by the zeroes of the horizontal field derivative at a set of altitudes. We also use the same algorithm to invert gravity anomalies only at the measurement altitude. Our goal is analyzing the different models obtained through the two different inversions and evaluating the relative uncertainties. One main difference is that the scaling function inversion is independent on density and the unknowns are the geometrical parameters of the source. The gravity data are instead inverted for the source geometry and the density simultaneously. A priori information used for both the inversions is that the source has a known depth to the top. We examine the results over the synthetic examples of a salt dome structure generated by Talwani’s approach and real gravity datasets over the Mors salt dome and the Decorah (USA) basin. For all these cases, the scaling function inversion yielded models with a better sensitivity to specific features of the sources, such as the tilt of the body, and reduced uncertainty. We finally analyzed the density, which is one of the unknowns for the gravity inversion and it is estimated from the geometric model for the scaling function inversion. The histograms over the density estimated at many iterations show a very concentrated distribution for the scaling function, while the density contrast retrieved by the gravity inversion, according to the fundamental ambiguity density/volume, is widely dispersed, this making difficult to assess its best estimate.

Thermodynamic Derivation of Scaling at the Liquid–Vapor Critical Point

With the use of thermodynamics and general equilibrium conditions only, we study the entropy of a fluid in the vicinity of the critical point of the liquid–vapor phase transition. By assuming a general form for the coexistence curve in the vicinity of the critical point, we show that the functional dependence of the entropy as a function of energy and particle densities necessarily obeys the scaling form hypothesized by Widom. Our analysis allows for a discussion of the properties of the corresponding scaling function, with the interesting prediction that the critical isotherm has the same functional dependence, between the energy and the number of particles densities, as the coexistence curve. In addition to the derivation of the expected equalities of the critical exponents, the conditions that lead to scaling also imply that, while the specific heat at constant volume can diverge at the critical point, the isothermal compressibility must do so.

Cost minimization of algorithms for scheduling parallel, single-threaded, heterogeneous, speed-scalable processors

We synthesize online scheduling algorithms to optimally assign a set of arriving heterogeneous tasks to heterogeneous speed-scalable processors under the single threaded computing architecture. By using dynamic speed-scaling, where each processor's speed is able to dynamically change within hardware and software processing constraints, the goal of our algorithms is to minimize the total financial cost (in dollars) of response time and energy consumption (TCRTEC) of the tasks. In our work, the processors are heterogeneous in that they may differ in their hardware specifications with respect to maximum processing rate, power function parameters and energy sources. Tasks are heterogeneous in terms of computation volume, memory and minimum processing requirements. We also consider that the unit price of response time for each task is heterogeneous because the user may be willing to pay higher/lower unit prices for certain tasks, thereby increasing/decreasing their optimum processing rates. We model the overhead loading time incurred when a task is loaded by a given processor prior to its execution and assume it to be heterogeneous as well. Under the single threaded, single buffered computing architecture, we synthesize the SBDPP algorithm and its two other versions. Its first two versions allow the user to specify the unit price of energy and response time for executing each arriving task. The algorithm's second version extends the functionality of the first by allowing the user or the OS of the computing device to further modify a task's unit price of time or energy in order to achieve a linearly controlled operation point that lies somewhere in the economy-performance mode continuum of a task's execution. The algorithm's third version operates exclusively on the latter. We briefly extend the algorithm and its versions to consider migration, where an unfinished task is paused and resumed on another processor. The SBDPP algorithm is qualitatively compared against its two other versions. The SBDPP dispatcher is analytically shown to perform better than the well known Round Robin dispatcher in terms of the TCRTEC performance metric. Through simulations we deduce a relationship between the arrival rate of tasks, number of processors and response time of tasks. Under the Single threaded, multi-buffered computing architecture we have four contributions that constitute the SMBSPP algorithm. First, we propose a novel task dispatching strategy for assigning the tasks to the processors. Second, we propose a novel preemptive service discipline called Smallest remaining Computation Volume Per unit Price of response Time (SCVPPT) to schedule the tasks on the assigned processor. Third, we propose a dynamic speed-scaling function that explicitly determines the optimum processing rate of each task. Most of the simulations consider both stochastic and deterministic traffic conditions. Our simulation results show that SCVPPT outperforms the two known service disciplines, Shortest Remaining Processing Time (SRPT) and the First Come First Serve (FCFS), in terms of minimizing the TCRTEC performance metric. The results also show that the algorithm's dispatcher drastically outperforms the well known Round Robin dispatcher with cost savings exceeding 100% even when the processors are mildly heterogeneous. Finally, analytical and simulation results show that our speed scaling function performs better than a comparable speed scaling function in current literature. Under a fixed budget of energy, we synthesize the SMBAD algorithm which uses the micro-economic laws of Supply and Demand (LSD) to heuristically adjust the unit price of energy in order to extend battery life and execute more than 50% of tasks on a single processor (under the single threaded, multi buffered computing architecture). By extending all our multiprocessor algorithms to factor independent (battery) energy sources that is associated with each processor, we analytically show that load balancing effects are induced on hetergeneous parallel processors. This happens when the unit price of energy is adjusted by the battery level of each processor in accordance with LSD. Furthermore, we show that a variation of this load balancing effect also occurs when the heterogeneous processors use a single battery as long as they operate at unconstrained processing rates.

Cost minimization of algorithms for scheduling parallel, single-threaded, heterogeneous, speed-scalable processors

We synthesize online scheduling algorithms to optimally assign a set of arriving heterogeneous tasks to heterogeneous speed-scalable processors under the single threaded computing architecture. By using dynamic speed-scaling, where each processor's speed is able to dynamically change within hardware and software processing constraints, the goal of our algorithms is to minimize the total financial cost (in dollars) of response time and energy consumption (TCRTEC) of the tasks. In our work, the processors are heterogeneous in that they may differ in their hardware specifications with respect to maximum processing rate, power function parameters and energy sources. Tasks are heterogeneous in terms of computation volume, memory and minimum processing requirements. We also consider that the unit price of response time for each task is heterogeneous because the user may be willing to pay higher/lower unit prices for certain tasks, thereby increasing/decreasing their optimum processing rates. We model the overhead loading time incurred when a task is loaded by a given processor prior to its execution and assume it to be heterogeneous as well. Under the single threaded, single buffered computing architecture, we synthesize the SBDPP algorithm and its two other versions. Its first two versions allow the user to specify the unit price of energy and response time for executing each arriving task. The algorithm's second version extends the functionality of the first by allowing the user or the OS of the computing device to further modify a task's unit price of time or energy in order to achieve a linearly controlled operation point that lies somewhere in the economy-performance mode continuum of a task's execution. The algorithm's third version operates exclusively on the latter. We briefly extend the algorithm and its versions to consider migration, where an unfinished task is paused and resumed on another processor. The SBDPP algorithm is qualitatively compared against its two other versions. The SBDPP dispatcher is analytically shown to perform better than the well known Round Robin dispatcher in terms of the TCRTEC performance metric. Through simulations we deduce a relationship between the arrival rate of tasks, number of processors and response time of tasks. Under the Single threaded, multi-buffered computing architecture we have four contributions that constitute the SMBSPP algorithm. First, we propose a novel task dispatching strategy for assigning the tasks to the processors. Second, we propose a novel preemptive service discipline called Smallest remaining Computation Volume Per unit Price of response Time (SCVPPT) to schedule the tasks on the assigned processor. Third, we propose a dynamic speed-scaling function that explicitly determines the optimum processing rate of each task. Most of the simulations consider both stochastic and deterministic traffic conditions. Our simulation results show that SCVPPT outperforms the two known service disciplines, Shortest Remaining Processing Time (SRPT) and the First Come First Serve (FCFS), in terms of minimizing the TCRTEC performance metric. The results also show that the algorithm's dispatcher drastically outperforms the well known Round Robin dispatcher with cost savings exceeding 100% even when the processors are mildly heterogeneous. Finally, analytical and simulation results show that our speed scaling function performs better than a comparable speed scaling function in current literature. Under a fixed budget of energy, we synthesize the SMBAD algorithm which uses the micro-economic laws of Supply and Demand (LSD) to heuristically adjust the unit price of energy in order to extend battery life and execute more than 50% of tasks on a single processor (under the single threaded, multi buffered computing architecture). By extending all our multiprocessor algorithms to factor independent (battery) energy sources that is associated with each processor, we analytically show that load balancing effects are induced on hetergeneous parallel processors. This happens when the unit price of energy is adjusted by the battery level of each processor in accordance with LSD. Furthermore, we show that a variation of this load balancing effect also occurs when the heterogeneous processors use a single battery as long as they operate at unconstrained processing rates.