A panorama of positivity. II: Fixed dimension

2020 ◽  
pp. 109-150
Author(s):  
Alexander Belton ◽  
Dominique Guillot ◽  
Apoorva Khare ◽  
Mihai Putinar
Keyword(s):  
Fixed Dimension

scholarly journals GBUO: “The Good, the Bad, and the Ugly” Optimizer

2021 ◽  
Vol 11 (5) ◽  
pp. 2042
Author(s):  
Hadi Givi ◽  
Mohammad Dehghani ◽  
Zeinab Montazeri ◽  
Ruben Morales-Menendez ◽  
Ricardo A. Ramirez-Mendoza ◽  
...  
Keyword(s):  
High Dimension ◽  
Essential Role ◽  
Optimization Problems ◽  
Optimization Algorithms ◽  
Stochastic Search ◽  
Objective Functions ◽  
Science And Engineering ◽  
Fixed Dimension ◽  
Dimension Functions ◽  
Simulation Results

Optimization problems in various fields of science and engineering should be solved using appropriate methods. Stochastic search-based optimization algorithms are a widely used approach for solving optimization problems. In this paper, a new optimization algorithm called “the good, the bad, and the ugly” optimizer (GBUO) is introduced, based on the effect of three members of the population on the population updates. In the proposed GBUO, the algorithm population moves towards the good member and avoids the bad member. In the proposed algorithm, a new member called ugly member is also introduced, which plays an essential role in updating the population. In a challenging move, the ugly member leads the population to situations contrary to society’s movement. GBUO is mathematically modeled, and its equations are presented. GBUO is implemented on a set of twenty-three standard objective functions to evaluate the proposed optimizer’s performance for solving optimization problems. The mentioned standard objective functions can be classified into three groups: unimodal, multimodal with high-dimension, and multimodal with fixed dimension functions. There was a further analysis carried-out for eight well-known optimization algorithms. The simulation results show that the proposed algorithm has a good performance in solving different optimization problems models and is superior to the mentioned optimization algorithms.

scholarly journals Integer Polynomial Optimization in Fixed Dimension

2006 ◽  
Vol 31 (1) ◽  
pp. 147-153 ◽  
Author(s):  
Jesús A. De Loera ◽  
Raymond Hemmecke ◽  
Matthias Köppe ◽  
Robert Weismantel
Keyword(s):  
Polynomial Optimization ◽  
Fixed Dimension ◽  
Integer Polynomial

scholarly journals Kac Polynomials for Canonical Algebras

2017 ◽  
Vol 2019 (13) ◽  
pp. 3981-4003
Author(s):  
Pierre-Guy Plamondon ◽  
Olivier Schiffmann
Keyword(s):  
Finite Field ◽  
Dimension Vector ◽  
Indecomposable Representations ◽  
Moduli Stacks ◽  
Fixed Dimension ◽  
Canonical Algebra

Abstract We prove that the number of geometrically indecomposable representations of fixed dimension vector $\mathbf{d}$ of a canonical algebra $C$ defined over a finite field $\mathbb{F}_q$ is given by a polynomial in $q$ (depending on $C$ and $\mathbf{d}$). We prove a similar result for squid algebras. Finally, we express the volume of the moduli stacks of representations of these algebras of a fixed dimension vector in terms of the corresponding Kac polynomials.

scholarly journals A universal framework for featurization of atomistic systems

2021 ◽  
Author(s):  
Xiangyun Lei ◽  
Andrew Medford
Keyword(s):  
Machine Learning ◽  
Molecular Dynamics ◽  
Force Fields ◽  
Reactive Systems ◽  
Quantum Molecular Dynamics ◽  
Learning Models ◽  
Fixed Dimension ◽  
Comparable Performance ◽  
Invaluable Tool ◽  
Dynamics Simulations

Abstract Molecular dynamics simulations are an invaluable tool in numerous scientific fields. However, the ubiquitous classical force fields cannot describe reactive systems, and quantum molecular dynamics are too computationally demanding to treat large systems or long timescales. Reactive force fields based on physics or machine learning can be used to bridge the gap in time and length scales, but these force fields require substantial effort to construct and are highly specific to a given chemical composition and application. A significant limitation of machine learning models is the use of element-specific features, leading to models that scale poorly with the number of elements. This work introduces the Gaussian multipole (GMP) featurization scheme that utilizes physically-relevant multipole expansions of the electron density around atoms to yield feature vectors that interpolate between element types and have a fixed dimension regardless of the number of elements present. We combine GMP with neural networks to directly compare it to the widely used Behler-Parinello symmetry functions for the MD17 dataset, revealing that it exhibits improved accuracy and computational efficiency. Further, we demonstrate that GMP-based models can achieve chemical accuracy for the QM9 dataset, and their accuracy remains reasonable even when extrapolating to new elements. Finally, we test GMP-based models for the Open Catalysis Project (OCP) dataset, revealing comparable performance to graph convolutional deep learning models. The results indicate that this featurization scheme fills a critical gap in the construction of efficient and transferable machine-learned force fields.

scholarly journals Forecasting Software Using Laplacian AR Model based on Bootstrap-Reversible Jump MCMC: Application on Stock Price Data

Webology ◽  
2021 ◽  
Vol 18 (Special Issue 04) ◽  
pp. 1045-1055
Author(s):  
Sup arman ◽  
Yahya Hairun ◽  
Idrus Alhaddad ◽  
Tedy Machmud ◽  
Hery Suharna ◽  
...  
Keyword(s):  
Stock Price ◽  
Complex Structure ◽  
Arma Model ◽  
Bayesian Estimator ◽  
Ar Model ◽  
Reversible Jump ◽  
Mcmc Algorithm ◽  
Reversible Jump Mcmc ◽  
Variable Dimension ◽  
Fixed Dimension

The application of the Bootstrap-Metropolis-Hastings algorithm is limited to fixed dimension models. In various fields, data often has a variable dimension model. The Laplacian autoregressive (AR) model includes a variable dimension model so that the Bootstrap-Metropolis-Hasting algorithm cannot be applied. This article aims to develop a Bootstrap reversible jump Markov Chain Monte Carlo (MCMC) algorithm to estimate the Laplacian AR model. The parameters of the Laplacian AR model were estimated using a Bayesian approach. The posterior distribution has a complex structure so that the Bayesian estimator cannot be calculated analytically. The Bootstrap-reversible jump MCMC algorithm was applied to calculate the Bayes estimator. This study provides a procedure for estimating the parameters of the Laplacian AR model. Algorithm performance was tested using simulation studies. Furthermore, the algorithm is applied to the finance sector to predict stock price on the stock market. In general, this study can be useful for decision makers in predicting future events. The novelty of this study is the triangulation between the bootstrap algorithm and the reversible jump MCMC algorithm. The Bootstrap-reversible jump MCMC algorithm is useful especially when the data is large and the data has a variable dimension model. The study can be extended to the Laplacian Autoregressive Moving Average (ARMA) model.

scholarly journals On minimal log discrepancies and kollár components

2021 ◽  
pp. 1-20
Author(s):  
Joaquín Moraga
Keyword(s):  
Blow Up ◽  
Chain Condition ◽  
Fano Varieties ◽  
Finite Sets ◽  
Fixed Dimension ◽  
Log Canonical ◽  
Finite Set ◽  
Canonical Singularities ◽  
Ascending Chain Condition

Abstract In this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$ -dimensional $a$ -log canonical singularities with standard coefficients, which admit an $\epsilon$ -plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on $d,\,a$ and $\epsilon$ . This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

scholarly journals Connected choice and the Brouwer fixed point theorem

2019 ◽  
Vol 19 (01) ◽  
pp. 1950004 ◽  
Author(s):  
Vasco Brattka ◽  
Stéphane Le Roux ◽  
Joseph S. Miller ◽  
Arno Pauly
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Unit Cube ◽  
Closed Set ◽  
Brouwer Fixed Point Theorem ◽  
Fixed Dimension ◽  
Lipschitz Constants ◽  
Weak König’S Lemma ◽  
König’S Lemma

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak Kőnig’s Lemma. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.

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