Cyclic Lattices, Ideal Lattices and Bounds for the Smoothing Parameter

<div>Cyclic lattices and ideal lattices were introduced by Micciancio in \cite{D2}, Lyubashevsky and Micciancio in \cite{L1} respectively, which play an efficient role in Ajtai's construction of a collision resistant Hash function (see \cite{M1} and \cite{M2}) and in Gentry's construction of fully homomorphic encryption (see \cite{G}). Let $R=Z[x]/\langle \phi(x)\rangle$ be a quotient ring of the integer coefficients polynomials ring, Lyubashevsky and Micciancio regarded an ideal lattice as the correspondence of an ideal of $R$, but they neither explain how to extend this definition to whole Euclidean space $\mathbb{R}^n$, nor exhibit the relationship of cyclic lattices and ideal lattices.</div><div>In this paper, we regard the cyclic lattices and ideal lattices as the correspondences of finitely generated $R$-modules, so that we may show that ideal lattices are actually a special subclass of cyclic lattices, namely, cyclic integer lattices. In fact, there is a one to one correspondence between cyclic lattices in $\mathbb{R}^n$ and finitely generated $R$-modules (see Theorem \ref{th4} below). On the other hand, since $R$ is a Noether ring, each ideal of $R$ is a finitely generated $R$-module, so it is natural and reasonable to regard ideal lattices as a special subclass of cyclic lattices (see corollary \ref{co3.4} below). It is worth noting that we use more general rotation matrix here, so our definition and results on cyclic lattices and ideal lattices are more general forms. As application, we provide cyclic lattice with an explicit and countable upper bound for the smoothing parameter (see Theorem \ref{th5} below). It is an open problem that is the shortest vector problem on cyclic lattice NP-hard? (see \cite{D2}). Our results may be viewed as a substantial progress in this direction.</div>

Cyclic Lattices, Ideal Lattices and Bounds for the Smoothing Parameter

<div>Cyclic lattices and ideal lattices were introduced by Micciancio in \cite{D2}, Lyubashevsky and Micciancio in \cite{L1} respectively, which play an efficient role in Ajtai's construction of a collision resistant Hash function (see \cite{M1} and \cite{M2}) and in Gentry's construction of fully homomorphic encryption (see \cite{G}). Let $R=Z[x]/\langle \phi(x)\rangle$ be a quotient ring of the integer coefficients polynomials ring, Lyubashevsky and Micciancio regarded an ideal lattice as the correspondence of an ideal of $R$, but they neither explain how to extend this definition to whole Euclidean space $\mathbb{R}^n$, nor exhibit the relationship of cyclic lattices and ideal lattices.</div><div>In this paper, we regard the cyclic lattices and ideal lattices as the correspondences of finitely generated $R$-modules, so that we may show that ideal lattices are actually a special subclass of cyclic lattices, namely, cyclic integer lattices. In fact, there is a one to one correspondence between cyclic lattices in $\mathbb{R}^n$ and finitely generated $R$-modules (see Theorem \ref{th4} below). On the other hand, since $R$ is a Noether ring, each ideal of $R$ is a finitely generated $R$-module, so it is natural and reasonable to regard ideal lattices as a special subclass of cyclic lattices (see corollary \ref{co3.4} below). It is worth noting that we use more general rotation matrix here, so our definition and results on cyclic lattices and ideal lattices are more general forms. As application, we provide cyclic lattice with an explicit and countable upper bound for the smoothing parameter (see Theorem \ref{th5} below). It is an open problem that is the shortest vector problem on cyclic lattice NP-hard? (see \cite{D2}). Our results may be viewed as a substantial progress in this direction.</div>

The Generalised Plug-in Algorithm for the Diffeomorphism Kernel Estimate

The optimal value of the smoothing parameter of the Kernel estimator can be obtained by the well known Plug-in algorithm. The optimality is realised in the sense of Mean Integrated Square Error (MISE). In this paper, we propose to generalise this algorithm to the case of the difficult problem of the estimation of a distribution which has a bounded support. The proposed algorithm consists in searching the optimal smoothing parameter by iterations from the expression of MISE of the kernel-diffeomorphism estimator. By some simulations applied to some distribution having a support bounded and semi bounded, we show that the support of the pdf estimator respects the one of the theoretical distribution. We also prove that the proposed method minimizes the Gibbs phenomenon.

PEMILIHAN PARAMETER OPTIMUM MENGGUNAKAN EXPONENTIAL SMOOTHING DENGAN METODE GOLDEN SECTION UNTUK PERAMALAN JUMLAH TITIK PANAS DI KALIMANTAN TIMUR

The exponential smoothing method is one method that can be used to predict time series data by smoothing the data. In this study, the method used was exponential smoothing with one smoothing parameter from Brown. The data used is the number of hotspots in East Kalimantan from January 2019 to September 2019. The purpose of this study is to obtain the optimum smoothing parameter values for exponential smoothing from the results of the optimization process using the golden section method to minimize the MAPE value, to obtain forecasting results for each method in exponential smoothing for the number of hotspots in East Kalimantan from October to December 2019, and obtain a good exponential smoothing method to predict data on the number of hotspots in East Kalimantan. From this analysis, the researchers chose the methods used were DES and TES. The optimum smoothing parameter obtained at DES was 0,558430 and TES was 0,376352. The results of forecasting the number of hotspots obtained in DES in October were 2.142, November was 2.707, and December was 3.271 with a MAPE value of 95%. The TES method forecasting results were obtained in October as many as 2.193, November as much as 2.975, and December as many as 3.852 with a MAPE value of 108%. Based on the comparison of the MAPE values in the two methods, the DES method is better than the TES for calculating the predicted value of the number of hotspots in East Kalimantan, although the two methods are not yet suitable for handling this case.

Devising a self-adjusting zero-order Brown’s model for predicting irreversible processes and phenomena

A self-adjusting zero-order Brown’s model has been devised. This model makes it possible to predict with high accuracy not only fires in the premises but also irreversible processes and phenomena of a random and chaotic nature under actual conditions. The essence of the self-adjusting model is that, based on Kalman’s approach, it is proposed to set the smoothing parameter for each time moment. Such a parameter is determined depending on the resulting current forecast error, taking into consideration the real and unknown dynamics of the studied series and noise. That does not require the selection of the smoothing parameter characteristic of known models. In addition, the proposed Brown’s model, unlike the known modifications, does not require setting a dynamics model of the level of the examined time series. The self-adjusting model provides negligible errors and efficiency of the forecast. The operability of the devised model was checked using an example of the experimental time series for the current measure of the recurrence of the increments of the state of the air medium in the laboratory chamber during alcohol combustion. As quantitative indicators of the quality of the forecast error, the current values for the square and absolute values were considered. It has been established that the current square of the forecast error is more than six orders of magnitude smaller compared to the case of a fixed smoothing parameter from a beyond-the-limit set. However, the current square of the forecast error for abrupt changes in the dynamics of the series level is half that of the fixed parameter of the beyond-the-limit set. It is noted that the results confirm the feasibility of the proposed self-adjusting Brown’s model

Bootstrap Selector for the Smoothing Parameter of Beran’s Estimator

This work proposes a resampling technique to approximate the smoothing parameter of Beran’s estimator. It is based on resampling by the smoothed bootstrap and minimising the bootstrap approximation of the mean integrated squared error to find the bootstrap bandwidth. The behaviour of this method has been tested by simulation on several models. Bootstrap confidence intervals are also addressed in this research and their performance is analysed in the simulation study.

A Study on Learning Parameters in Application of Radial Basis Function Neural Network Model to Rotor Blade Design Approximation

Meta-model sre generally applied to approximate multi-objective optimization, reliability analysis, reliability based design optimization, etc., not only in order to improve the efficiencies of numerical calculation and convergence, but also to facilitate the analysis of design sensitivity. The radial basis function neural network (RBFNN) is the meta-model employing hidden layer of radial units and output layer of linear units, and characterized by relatively fast training, generalization and compact type of networks. It is important to minimize some scattered noisy data to approximate the design space to prevent local minima in the gradient based optimization or the reliability analysis using the RBFNN. Since the noisy data must be smoothed out in order for the RBFNN to be applied as the meta-model to any actual structural design problem, the smoothing parameter must be properly determined. This study aims to identify the effect of various learning parameters including the spline smoothing parameter on the RBFNN performance regarding the design approximation. An actual rotor blade design problem was considered to investigate the characteristics of RBFNN approximation with respect to the range of spline smoothing parameter, the number of training data, and the number of hidden layers. In the RBFNN approximation of the rotor blade design, design sensitivity characteristics such as main effects were also evaluated including the performance analysis according to the variation of learning parameters. From the evaluation results of learning parameters in the rotor blade design, it was found that the number of training data had larger influence on the RBFNN meta-model accuracy than the spline smoothing parameter while the number of hidden layers had little effect on the performances of RBFNN meta-model.

Short-term fire forecast based on air state gain recurrence and zero-order brown model

Possibilities of parameterization of the zero-order Brown model for indoor air forecasting based on the current measure of air state gain recurrence are considered. The key to the zero-order parametric Brown forecasting model is the selection of the smoothing parameter, which characterizes forecast adaptability to the current air state gain recurrence measure. It is shown that for effective short-term indoor fire forecast, the Brown model parameter must be selected from the out-of-limit set defined by 1 and 2. The out-of-limit set for the Brown model parameter is an area of effective fire forecasting based on the measure of current indoor air state gain recurrence. Errors of fire forecast based on the parameterized zero-order Brown model in the case of the classical and out-of-limit sets of the model parameters are investigated using the example of ignition of various materials in a laboratory chamber. As quantitative indicators of forecast quality, the absolute and mean forecast errors exponentially smoothed with a parameter of 0.4 are investigated. It was found that for alcohol, the smoothed absolute and mean forecast errors for the classical smoothing parameter in the no-ignition interval do not exceed 20 %. At the same time, for the out-of-limit case, the indicated forecast errors are, on average, an order of magnitude smaller. Similar ratios for forecast errors remain in paper, wood and textile ignition. However, for the transition zone corresponding to the time of material ignition, a sharp decrease in the current measure of chamber air state gain recurrence is observed. It was found that for this zone, the smoothed absolute forecast error for alcohol is about 2 % if the model parameter is selected from the classical set. If the model parameter is selected from the out-of-limit set, the forecast error is about 0.2 %. The results generally demonstrate significant advantages of using the zero-order Brown parametric model with out-of-limit model parameters for indoor fire forecasting

Stress minimization for lattice structures. Part I: Micro-structure design

Lattice structures are periodic porous bodies which are becoming popular since they are a good compromise between rigidity and weight and can be built by additive manufacturing techniques. Their optimization has recently attracted some attention, based on the homogenization method, mostly for compliance minimization. The goal of our two-part work is to extend lattice optimization to stress minimization problems two-dimensionally. The present first part is devoted to the choice of a parametrized periodicity cell that will be used for structural optimization in the second part of our work. In order to avoid stress concentration, we propose a square cell microstructure with a super-ellipsoidal hole instead of the standard rectangular hole often used for compliance minimization. This type of cell is parametrized two-dimensionally by one orientation angle, two semi-axis and a corner smoothing parameter. We first analyse their influence on the stress amplification factor by performing some numerical experiments. Second, we compute the optimal corner smoothing parameter for each possible microstructure and macroscopic stress. Then, we average (with specific weights) the optimal smoothing exponent with respect to the macroscopic stress. Finally, to validate the results, we compare our optimal super-ellipsoidal hole with the Vigdergauz microstructure which is known to be optimal for stress minimization in some special cases. This article is part of the theme issue ‘Topics in mathematical design of complex materials’.