Deterministic Set Research Articles

Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers

We investigate exponential sums over those numbers ${\leqslant}x$ all of whose prime factors are ${\leqslant}y$. We prove fairly good minor arc estimates, valid whenever $\log ^{3}x\leqslant y\leqslant x^{1/3}$. Then we prove sharp upper bounds for the $p$th moment of (possibly weighted) sums, for any real $p>2$ and $\log ^{C(p)}x\leqslant y\leqslant x$. Our proof develops an argument of Bourgain, showing that this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of $a+b=c$ in $y$-smooth integers less than $x$ whenever $\log ^{C}x\leqslant y\leqslant x$. Previously this was only known assuming the generalised Riemann hypothesis. Combining them with transference machinery of Green, we prove Roth’s theorem for subsets of the $y$-smooth numbers whenever $\log ^{C}x\leqslant y\leqslant x$. This provides a deterministic set, of size ${\approx}x^{1-c}$, inside which Roth’s theorem holds.

Set Automata

We consider the model of deterministic set automata which are basically deterministic finite automata equipped with a set as an additional storage medium. The basic operations on the set are the insertion of elements, the removing of elements, and the test whether an element is in the set. We investigate the computational power of deterministic set automata and compare the language class accepted with the context-free languages and classes of languages accepted by queue automata. As result the incomparability to all classes considered is obtained. Furthermore, we examine the closure properties under several operations. Then we show that deterministic set automata may be an interesting model from a practical point of view by proving that their regularity problem as well as the problems of emptiness, finiteness, infiniteness, and universality are decidable. Finally, the descriptional complexity of deterministic and nondeterministic set automata is investigated. A conversion procedure that turns a deterministic set automaton accepting a regular language into a deterministic finite automaton is developed which leads to a double exponential upper bound. This bound is proved to be tight in the order of magnitude by presenting also a double exponential lower bound. In contrast to these recursive bounds we obtain non-recursive trade-offs when nondeterministic set automata are considered.

Reconstruction of Sparse Wavelet Signals From Partial Fourier Measurements

In this paper, we show that high-dimensional sparse wavelet signals of finite levels can be constructed from their partial Fourier measurements on a deterministic sampling set with cardinality about a multiple of signal sparsity.

Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems

The fast basin of an attractor of an iterated function system (IFS) is the set of points in the domain of the IFS whose orbits under the associated semigroup intersect the attractor. Fast basins can have non-integer dimension and comprise a class of deterministic fractal sets. The relationship between the basin and the fast basin of a point-fibred attractor is analyzed. To better understand the topology and geometry of fast basins, and because of analogies with analytic continuation, branched fractal manifolds are introduced. A branched fractal manifold is a metric space constructed from the extended code space of a point-fibred attractor, by identifying some addresses. Typically, a branched fractal manifold is a union of a nondenumerable collection of nonhomeomorphic objects, isometric copies of generalized fractal blowups of the attractor.

DNA tetrominoes: the construction of DNA nanostructures using self-organised heterogeneous deoxyribonucleic acids shapes.

The unique programmability of nucleic acids offers alternative in constructing excitable and functional nanostructures. This work introduces an autonomous protocol to construct DNA Tetris shapes (L-Shape, B-Shape, T-Shape and I-Shape) using modular DNA blocks. The protocol exploits the rich number of sequence combinations available from the nucleic acid alphabets, thus allowing for diversity to be applied in designing various DNA nanostructures. Instead of a deterministic set of sequences corresponding to a particular design, the protocol promotes a large pool of DNA shapes that can assemble to conform to any desired structures. By utilising evolutionary programming in the design stage, DNA blocks are subjected to processes such as sequence insertion, deletion and base shifting in order to enrich the diversity of the resulting shapes based on a set of cascading filters. The optimisation algorithm allows mutation to be exerted indefinitely on the candidate sequences until these sequences complied with all the four fitness criteria. Generated candidates from the protocol are in agreement with the filter cascades and thermodynamic simulation. Further validation using gel electrophoresis indicated the formation of the designed shapes. Thus, supporting the plausibility of constructing DNA nanostructures in a more hierarchical, modular, and interchangeable manner.

Transient Behaviors in Noise-Induced Bifurcations Captured by Generalized Cell Mapping Method with an Evolving Probabilistic Vector

In this paper, an idea of evolving probabilistic vector (EPV) is introduced into the Generalized Cell Mapping (GCM) method to replace the classical fix-sized probabilistic vector in order to efficiently capture the transient behaviors in noise-induced bifurcations, by which an initial localized probability distribution around a deterministic attracting set of a nonlinear dynamical system may expand abruptly or escape with a jump as the noise intensity increases and exceeds some critical values. A Mathieu–Duffing oscillator under excitation of both additive and multiplicative noise is studied as an example of application to show the validity of the proposed method and the interesting phenomena in noise-induced explosive and dangerous bifurcations of the oscillator that are characterized respectively by an abrupt enlargement and a sudden fast jump of the response probability distribution. The insight into the roles of deterministic global structure and noise as well as their interplay is gained.

Prediction of elastic properties using seismic prestack inversion and neural network analysis

The use of artificial intelligence algorithms to solve geophysical problems is a recent development. Neural network analysis is one of these algorithms. It uses the information from multiple wells and seismic data to train a neural network to predict properties away from the well control. Neural network analysis can significantly improve the seismic inversion result when the outputs of the inversion are used as external attributes in addition to regular seismic attributes for training the network. We found that integration of prestack inversion and neural network analysis can improve the characterization of a late Pliocene gas sandstone reservoir. For inversion, the input angle stacks was conditioned to match the theoretical amplitude-variation-with-offset response. The inversion was performed using a deterministic wavelet set. Neural network analysis was then used to enhance the [Formula: see text], [Formula: see text], and density volumes from the inversion. The improvement was confirmed by comparisons with logs from a blind well.

Dimension Conservation for Self-Similar Sets and Fractal Percolation

We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let |$K$| be a self-similar subset of |$\mathbb {R}^2$| with Hausdorff dimension |$\dim _H K >1$| such that the rotational components of the underlying similarities generate the full rotation group. Then, for all |$\epsilon >0$|⁠, writing |$\pi _\theta $| for projection onto the |$L_\theta $| in direction |$\theta $|⁠, the Hausdorff dimensions of the sections satisfy |$\dim _H (K\cap \pi _\theta ^{-1}x)> \dim _H K - 1 - \epsilon $| for a set of |$x \in L_\theta $| of positive Lebesgue measure for all directions |$\theta $| except for those in a set of Hausdorff dimension 0. For a class of self-similar sets, we obtain a similar conclusion for all directions, but with lower box dimension replacing Hausdorff dimensions of sections. We obtain similar inequalities for the dimensions of sections of Mandelbrot percolation sets.

Spectral stochastic modeling of uncertainties in nonlinear diffusion problems of moisture transfer in wood

This paper deals with the stochastic numerical analysis of moisture transfer in wood with the random diffusion coefficient after heating of wood (when temperature is already constant). The simulation is based on the unsteady-state nonlinear (the model respects the dependence of diffusion coefficients on moisture and constant temperature) diffusion of moisture with respect to the orthotropic nature of wood. The spectral solution of this problem is based on discretization the resulting random field (moisture) in the stochastic dimension by the orthogonal polynomials (generalized polynomial chaos algorithm). A Galerkin projection is applied in the stochastic dimension to obtain the deterministic set of partial differential equations that is solved by finite element method.The main purpose of this paper is to demonstrate that the stochastic spectral method based on polynomial chaos expansion can be more efficient in modeling uncertainties associated with moisture transfer in wood than Monte Carlo method mainly when considering a small number of random variables. This spectral approach has a big advantage over the Monte Carlo method (statistical approach) in terms of computer time. Numerical example of diffusion of moisture in convective drying of wood is given and there is shown that the results (mean and the standard deviation) obtained with the stochastic spectral method are in good agreement with the results of the Monte Carlo simulations.

Robust Optimization Based Optimal DG Placement in Microgrids

This paper proposes a novel Microgrid (MG) planning methodology to decide optimal locations, sizes and mix of dispatchable and intermittent distributed generators (DGs). The long-term costs in the proposed planning model include investment, operation and maintenance (O&M), fuel and emission costs of DGs while the revenue includes payment by MG loads and utility grid. The problem is formulated as a mixed-integer program (MIP) considering the probabilistic nature of DG outputs and load consumption, wherein the costs are minimized and profits are maximized. The model is transformed to be a two-stage robust optimization problem. A column and constraint generation (CCG) framework is used to solve the problem. Compared with conventional MG planning approaches, the proposed model is more practical in that it fully considers the system uncertainties and only requires a deterministic uncertainty set, rather than a probability distribution of uncertain data which is difficult to obtain. Case studies of a MG with wind turbines, photovoltaic generators (PVs) and microturbines (MTs) demonstrate the effectiveness of the proposed methodology.

Scan power-aware deterministic test scheme using a low-transition linear decompressor

Growing test data volume and excessive testing power are both serious challenges in the testing of very large-scale integrated circuits. This article presents a scan power-aware deterministic test method based on a new linear decompressor which is composed of a traditional linear decompressor, k-input AND gates and T flip-flops. This decompression architecture can generate the low-transition deterministic test set for a circuit under test. When applying the test patterns generated by the linear decompressor, only a few transitions occur in the scan chains, and hence the switching activity during testing decreases significantly. Entire test flow compatible with the design is also presented. Experimental results on several large International Symposium on Circuits and Systems’89 and International Test Conference’99 benchmark circuits demonstrate that the proposed methodology can reduce test power significantly while providing a high compression ratio with limited hardware overhead.

Efficient general AGH-unification

General E-unification is an important tool in cryptographic protocol analysis, where the equational theory E represents properties of the cryptographic algorithm, and uninterpreted function symbols represent other functions. The property of a homomorphism over an Abelian group is common in encryption algorithms such as RSA. The general E-unification problem in this theory is NP-complete, and existing algorithms are highly nondeterministic. We give a mostly deterministic set of inference rules for solving general E-unification modulo a homomorphism over an Abelian group, and prove that it is sound, complete and terminating. These inference rules have been implemented in Maude, and will be incorporated into the Maude-NRL Protocol Analyzer (Maude-NPA).

Bounding WTP distributions to reflect the ‘actual’ consideration set

In this paper we extend the independent availability logit and combined latent class mixed logit models to accommodate respondents with different consideration sets due to their cost thresholds and cut-offs. Pertinent features of our model are that it is estimated in WTP-space and that the class-specific WTP distributions are truncated to be within the bounds deduced from the cost level(s) within each class-specific consideration set. Our analysis shows that our approach is well suited at uncovering the heterogeneity in the cost thresholds and cut-offs used by respondents. This is shown to help build a richer insight into respondent׳s behaviour as well as raise a number of concerns about the appropriateness of assuming deterministic choice sets. We discuss the implications of our results for welfare analysis.

The effect of aggregation on extremes from asymptotically independent light-tailed risks

Portfolio risk diversification is a well established concept in finance. While aggregation of several risky assets generally reduces the overall investment risk due to diversification, the effectiveness of diversification depends on the stochastic behaviour of the returns of the assets comprising the portfolio. The paper proposes a new approach to quantifying the effect of portfolio tail diversification by looking at the asymptotic behaviour of the ratio of the largest loss on the portfolio to the sum of the largest losses on the individual investments held on the stand-alone basis. It is assumed that independent and identically distributed random vectors from the underlying distribution, describing the behaviour of returns on the assets in the portfolio, can be scaled to converge onto a deterministic limit set. This property is satisfied by a number of distributions commonly used in finance. Several analytical examples are given to illustrate the proposed asymptotic diversification index as a measure of the effect of risk aggregation on extremes as well as to quantify the impact of dimension on the diversification and as a tool in optimal portfolio selection.

Two‐stage adjustable robust optimisation for unit commitment under uncertainty

A new adjustable robust optimisation formulation to solve unit commitment (UC) problems under uncertainty is proposed. Unlike the stochastic programming or chance-constrained approaches, the robust formulation does not require information on the exact probability distributions. This method protects power systems against all possible scenarios within a deterministic uncertainty set. The degree of conservatism can be conveniently controlled by the parameters of the uncertainty set. The formulation is divided into two stages, where here-and-now UC decisions are determined in the first stage, and the second stage makes wait-and-see dispatch decisions. The second-stage problems are reformulated into the equivalent form, so that a practical cutting-plane algorithm can be used to solve the proposed two-stage problems in an iterative manner. Numerical studies show that the robust formulation offers more consistent performance than the stochastic method when the underlying distribution of demand or generating unit reliability varies.

Multivariate Discrete Least-Squares Approximations with a New Type of Collocation Grid

In this work, we discuss the problem of approximating a multivariate function by discrete least-squares projection onto a polynomial space using a specially designed deterministic point set. The independent variables of the function are assumed to be random variables, stemming from the motivating application of uncertainty quantification. Our deterministic points are inspired by a theorem due to André Weil. We first work with the Chebyshev measure and consider the approximation in Chebyshev polynomial spaces. We prove the stability and an optimal convergence estimate, provided the number of points scales quadratically with the dimension of the polynomial space. A possible application for quantifying epistemic uncertainties is then discussed. We show that the point set asymptotically equidistributes to the product-Chebyshev measure, allowing us to propose a weighted least-squares framework and extending our method to more general polynomial approximations. Numerical examples are given to confirm the theoretical results. It is shown that the performance of our deterministic points is similar to that of randomly generated points. However, our construction, being deterministic, does not suffer from probabilistic qualifiers on convergence results (e.g., convergence “with high probability'').

Equivalent upscaled hydro-mechanical properties of a damaged and fractured claystone around a gallery (Meuse/Haute-Marne Underground Research Laboratory)

Abstract In this work, we present calculations and analyses of equivalent continuum (upscaled) coefficients describing the damaged, fissured and fractured claystone around an underground gallery. We focus here on mechanical and coupled hydro-mechanical properties of the damaged claystone (the upscaled Darcy permeability of the same claystone was studied in a previous paper focused on hydraulics without mechanical deformations). Concerning the geometric structure of the damaged clay stone around the cylindrical excavation, we use a hybrid 3D geometric model of fissuring and fracturing, comprising (a) a set of 10 000 statistical fissures with radially inhomogeneous statistics (size, thickness and density increasing near the wall), and (b) a deterministic set of large curved ‘chevrons’ fractures, periodically spaced along the axis of the drift according to a 3D chevron pattern. The hydro-mechanical coefficients calculated here are second- and fourth-rank tensors, which are displayed using ellipsoids. For simplicity, we also calculate equivalent isotropic coefficients extracted from these tensors: Young's modulus ( E ), bulk modulus ( K ), Lamé shear modulus (μ), Poisson's ratio (ν), Biot coefficient ( B , stress–pressure coupling) and Biot modulus ( M , pressure–fluid production coupling). All of these coefficients are affected by the degree of damage and fracturing, which increases near the wall of the gallery. Both 3D and ‘2D transverse’ distributions are analysed, on grids of 3D cubic voxels and 2D pixels, respectively. Global coefficients upscaled over the entire damaged and fractured zone are also analysed. Other types of averages are presented, for example, upscaled values over a cylindrical annular shell at various radial distances from the gallery wall. The relation to the degree of fracturing is discussed, including for instance the effect of fracturing on bulk and shear stiffnesses, and on the hydro-mechanical coupling coefficients of the damaged claystone.

Artificial immune simulation for improved forecasting of electricity consumption with random variations

This paper presents an integrated algorithm for forecasting annual electrical energy consumption based on Artificial Immune System (AIS), Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and computer simulation. Computer simulation is developed to generate random variables for annual electricity consumptions in selected countries. Most recent studies are concerned with deterministic data sets which could enhance relative error. However, this study utilizes fitted random variables as input data to decrease the relative error. Mean Absolute Percentage Error (MAPE) is used for evaluating the results and selecting the best forecasting model. To show the applicability of the proposed algorithm, the annual electricity consumptions for 16 countries from 1980 to 2006 are considered and the proposed algorithm is applied to the corresponding historical data. Three considered meta-heuristics (i.e. AIS, GA, and PSO) are compared with each other in estimation of electricity consumption in the selected countries. The comparison is made based on MAPE for the test period data. For the selected countries, AIS method with the Clonal Selection Algorithm (CLONALG) shows satisfactory results when applied with simulated data and has been selected as the preferred method. This is the first study that uses an integrated AIS-simulation for improved forecasting of electricity consumption with random variations.

Minimax filtering for sequential aggregation: Application to ensemble forecast of ozone analyses

This paper presents a new algorithm for sequential aggregation of an ensemble of forecasts. At any forecasting step, the aggregation consists of (1) computing new weights for the ensemble members represented by different numerical models and (2) forecasting with a weighted linear combination of the ensemble members. We assume that the time evolution of the weights is described by a linear equation with uncertain parameters and apply a minimax filter (and also Kalman filter, for comparison) in order to estimate the vector of weights given “observations”. The “observation” equation for the filter compares the aggregated forecast with the analysis determined in a data assimilation cycle together with its variance. The minimax approach allows one to work with flexible uncertainty description: deterministic bounding sets for uncertain parameters in weight's equation, and error covariance matrices for the “observational” errors. Our key contribution is an uncertainty estimate of the aggregated forecast, for which we introduce an evaluation test. The performance of the method is assessed for the forecast of ground‐level ozone daily peaks over Europe, for the year 2001. Compared to forecasts generated by classical data assimilation, the root mean square error is decreased by 16% for prediction of the analyses and by 20% for prediction of the observations.

Construction of fractal nanostructures based on Kepler-Shubnikov nets

A system of information codes for deterministic fractal lattices and sets of multifractal curves is proposed. An iterative modular design was used to obtain a series of deterministic fractal lattices with generators in the form of fragments of 2D structures and a series of multifractal curves (based on some Kepler-Shubnikov nets) having Cantor set properties. The main characteristics of fractal structures and their lacunar spectra are determined. A hierarchical principle is formulated for modules of regular fractal structures.