COMBINED NEWTON’S THIRD-ORDER CONVERGENCE METHOD FOR MINIMIZE ONE VARIABLE FUNCTIONS

Contex. The article deals with the actual problem of numerical optimization of slowly computed unimodal functions of one variable. The analysis of existing methods of minimization of the first and second orders of convergence, which showed that these methods can be used to quickly solve these problems for functions, the values of which can be obtained without difficulty. For slowly computed functions, these methods give slow algorithms; therefore, the problem of developing fast methods for minimizing such functions is urgent. Objective. Development of a combined third-order Newtonian method of convergence to minimize predominantly slowly computed unimodal functions, as well as the development of a database, including smooth, monotonic and partially constant functions, to test the method and compare its effectiveness with other known methods. Method. A technique and an algorithm for solving the problem of fast minimization of a unimodal function of one variable by a combined numerical Newtonian method of the third order of convergence presented. The method is capable of recognizing strictly unimodal, monotonic and constant functions, as well as functions with partial or complete sections of a flat minimum. Results. The results of comparison of the proposed method with other methods, including the fast Brent method, presented. 6954 problems were solved using the combined Newtonian method, while the method turned out to be faster than other methods in 95.5% of problems, Brent’s method worked faster in only 4.5% of problems. In general, the analysis of the calculation results showed that the combined method worked 1.64 times faster than the Brent method. Conclusions. A combined third-order Newtonian method of convergence proposed for minimizing predominantly slowly computed unimodal functions of one variable. A database of problems developed, including smooth, monotone and partially constant functions, to test the method and compare its effectiveness with other known methods. It is shown that the proposed method, in comparison with other methods, including the fast Brent method, has a higher performance.

A Novel Technique to Determine Concentration-Dependent Solvent Dispersion in Vapex

Vapex (vapor extraction of heavy oil and bitumen) is a promising recovery technology because it consumes low energy, and is very environmentally-friendly. The dispersion of solvents into heavy oil and bitumen is a crucial transport property governing Vapex. The accurate determination of solvent dispersion in Vapex is essential to effectively predict the amount and time scale of oil recovery as well to optimize the field operations. In this work, a novel technique is developed to experimentally determine the concentration-dependent dispersion coefficient of a solvent in Vapex process. The principles of variational calculus are utilized in conjunction with a mass transfer model of the experimental Vapex process. A computational algorithm is developed to optimally compute solvent dispersion as a function of its concentration in heavy oil. The developed technique is applied to Vapex utilizing propane as a solvent. The results show that dispersion of propane is a unimodal function of its concentration in bitumen.

A Novel Technique to Determine Concentration-Dependent Solvent Dispersion in Vapex

Vapex (vapor extraction of heavy oil and bitumen) is a promising recovery technology because it consumes low energy, and is very environmentally-friendly. The dispersion of solvents into heavy oil and bitumen is a crucial transport property governing Vapex. The accurate determination of solvent dispersion in Vapex is essential to effectively predict the amount and time scale of oil recovery as well to optimize the field operations. In this work, a novel technique is developed to experimentally determine the concentration-dependent dispersion coefficient of a solvent in Vapex process. The principles of variational calculus are utilized in conjunction with a mass transfer model of the experimental Vapex process. A computational algorithm is developed to optimally compute solvent dispersion as a function of its concentration in heavy oil. The developed technique is applied to Vapex utilizing propane as a solvent. The results show that dispersion of propane is a unimodal function of its concentration in bitumen.

Linear Bayesian equilibrium in insider trading with a random time under partial observations

<abstract><p>In this paper, the insider trading model of Xiao and Zhou (<italic>Acta Mathematicae Applicatae, 2021</italic>) is further studied, in which market makers receive partial information about a static risky asset and an insider stops trading at a random time. With the help of dynamic programming principle, we obtain a unique linear Bayesian equilibrium consisting of insider's trading intensity and market liquidity parameter, instead of none Bayesian equilibrium as before. It shows that (i) as time goes by, both trading intensity and market depth increase exponentially, while residual information decreases exponentially; (ii) with average trading time increasing, trading intensity decrease, but both residual information and insider's expected profit increase, while market depth is a unimodal function with a unique minimum with respect to average trading time; (iii) the less information observed by market makers, the weaker trading intensity and market depth are, but the more both expect profit and residual information are, which is in accord with our economic intuition.</p></abstract>

Toward a Unified Framework for Cognitive Maps

In this study, we integrated neural encoding and decoding into a unified framework for spatial information processing in the brain. Specifically, the neural representations of self-location in the hippocampus (HPC) and entorhinal cortex (EC) play crucial roles in spatial navigation. Intriguingly, these neural representations in these neighboring brain areas show stark differences. Whereas the place cells in the HPC fire as a unimodal function of spatial location, the grid cells in the EC show periodic tuning curves with different periods for different subpopulations (called modules). By combining an encoding model for this modular neural representation and a realistic decoding model based on belief propagation, we investigated the manner in which self-location is encoded by neurons in the EC and then decoded by downstream neurons in the HPC. Through the results of numerical simulations, we first show the positive synergy effects of the modular structure in the EC. The modular structure introduces more coupling between heterogeneous modules with different periodicities, which provides increased error-correcting capabilities. This is also demonstrated through a comparison of the beliefs produced for decoding two- and four-module codes. Whereas the former resulted in a complete decoding failure, the latter correctly recovered the self-location even from the same inputs. Further analysis of belief propagation during decoding revealed complex dynamics in information updates due to interactions among multiple modules having diverse scales. Therefore, the proposed unified framework allows one to investigate the overall flow of spatial information, closing the loop of encoding and decoding self-location in the brain.

Fibonacci Algorithm for Determining Minimum Value of a Unimodal Function using MATLAB Programmed Computing Software

Fibonacci number sequencing process is a number generating procedure. One among the several applications of such sequencing process is to search the interval where the certain Fibonacci numbers are either at the boundary or at the position related to the Fibonacci numbered sub interval generating expressions. In this paper the author used MATLAB programed computing software to determine the local minimum of a unimodal function that’s domain is not necessarily a continuous number interval and searched the values of the function using a suitable algorithm named Fibonacci algorithm for determining minimum value of a unimodal function.

Decision strategies in reward-based crowdfunding: the role of crowdfunding platforms

Purpose The purpose of this paper is to investigate the influence of the crowdfunding platforms on reward-based crowdfunding projects. This study offers guidance for the creator on how to choose among platforms and how to make optimal product and pricing decisions. Design/methodology/approach Usually, crowdfunding platforms are able to help creators to lower unit costs and charge platform fees in return. In this paper, the reduction of the unit cost and the platform fee are selected for determining the competitive strength (CS) of a platform. Then, the CS affecting the creators and the backers of the projects is analyzed. Findings In the basic model, when the product quality level is exogenous, the optimal price increases in the product quality level and decreases in the difficulty level of the project, while the corresponding expected profit is a unimodal function of the product quality level and the difficulty level. In the endogenous case, the optimal price is exactly twice the unit cost. With the influence of platforms, platforms with higher CS tend to help the creator to lower the prices and to achieve higher profitability. Moreover, platforms with higher CS usually help the creator to offer higher quality products and to charge higher prices. Research limitations/implications The opportunity cost is zero in this paper. In reality, backers arrive at the project in different order. Usually, earlier backers bare more opportunity cost and risk. Originality/value To the best of authors’ knowledge, this paper is the first one to offer suggestions for creators on how to choose among crowdfunding platforms. The study provides theoretical guidance on product and pricing decisions on an analytical side.

The burning question: does fire affect habitat selection and forage preference of the black rhinoceros Diceros bicornis in East African savannahs?

The conservation of threatened species requires information on how management activities influence habitat quality. The Critically Endangered black rhinoceros Diceros bicornis is restricted to savannahs representing c. 5% of its historical range. Fire is used extensively in savannahs but little is known about how rhinos respond to burning. Our aim was to understand rhino responses to fire by studying habitat selection and foraging at multiple scales. We used resource selection functions and locations of 31 rhinos during 2014–2016 to study rhino habitat use in Serengeti National Park, Tanzania. Rhino selectivity was quantified by comparing forage consumption to plant species availability in randomly sampled vegetation plots; rhino diets were subsequently verified through DNA metabarcoding analysis of faecal samples. Rhino habitat use was a unimodal function of fire history, with highly occupied sites having fire frequencies of < 0.6 fires/year and maximum occupancy occurring at a fire frequency of 0.1 fires/year. Foraging stations had characteristic plant communities, with 17 species associated with rhino foraging. Rhinos were associated with, and disproportionately consumed, woody plants, forbs and legumes, all of which decreased in abundance with increasing fire frequency. In contrast to common management practices, multiple lines of evidence suggest that the current fire regime in the Serengeti negatively influences rhino habitat use and foraging and that frequent fire limits access of rhinos to preferred forage. We outline a conceptual model to guide managers and conservationists in the use of fire under variable habitat conditions.

OPTIMIZATION OF UNIMODAL FUNCTIONS USING PARABOLIC PREDICTOR search

A combined parabolic predictor search is proposed for the conditional minimization of the unimodal function using the predictive-based selective application of phases of extremum search by golden section search and parabolic search. The formula for calculating the value of parabolic predictor function is given, with its help it is possible to work out the forecast and tactics of extremum search of the minimized function. Predictor includes forecasting extremeness, monotony and constancy of function on a segment of uncertainty. Identification forecast for a direct function is described, using which allows to find a solution in three calculations. The assertion is made that if three successive computations of a function give points with similar ordinates, then abscissa of each point can be a solution of the problem. The procedure of identifying non-direct monotonic functions is described. It is shown that the reliability of monotonicity forecast can be determined by five calculations of the function. There has been described the procedure of using phases of parabolic method, which can be performed at favorable prediction of detecting the internal extremum of function. It has been stated that carrying out these phases, even with favorable forecast, can be considered inexpedient for cases when it is recognized that the problem is weakly sensitive or insensitive to the parabolic forecast. Block diagrams of algorithms implementing the method are given. It is shown that, compared to golden section search, the predictor has 3-5 times faster response for smooth functions and is comparable by this criterion to Brent method. The predictor achieves the greatest speed when minimizing monotonic functions. The method works somewhat slower than golden section search, however, it is much faster than Brent method when searching for the minimum of piecewise, flat, planar and other functions of a similar nature for which approximation of parabola does not give the expected effect. In comparison with Brent method, parabolic predictor has 1.5-4 times more speed in solving problems of such type.