Von Neumann Algorithm Research Articles

A dynamic input-output model of the Tyumen Oblast south's development based on the von Neumann algorithm

Subject. This article deals with the problem of constructing a real trajectory of the region's development and comparing it with the stated one. Objectives. The article aims to develop an algorithm for calculating and analyzing a dynamic input-output model of the south of the Tyumen Oblast, including social and infrastructural subsystems, and households as an endogenous variable-aggregate. Methods. For the study, we used the methods of construction and analysis of dynamic models by W. Leontief and J. von Neumann. Indicators of balanced growth trajectories were estimated using the Perron–Frobenius theorem. Results. The article proposes an algorithm for the formation of an investment variable. The matrix of direct costs for each year and on average for the periods 2017–2019 and 2017–2020 was identified. Based on the available statistical data, von Neumann highways and model development trajectories were constructed. Conclusions. The shown mechanism for a possible comparison of actual and planned optimal development can help organize digital monitoring of the region's development and analyze deviations from the optimal forecast.

DNA synthesis for true random number generation

The volume of securely encrypted data transmission required by today’s network complexity of people, transactions and interactions increases continuously. To guarantee security of encryption and decryption schemes for exchanging sensitive information, large volumes of true random numbers are required. Here we present a method to exploit the stochastic nature of chemistry by synthesizing DNA strands composed of random nucleotides. We compare three commercial random DNA syntheses giving a measure for robustness and synthesis distribution of nucleotides and show that using DNA for random number generation, we can obtain 7 million GB of randomness from one synthesis run, which can be read out using state-of-the-art sequencing technologies at rates of ca. 300 kB/s. Using the von Neumann algorithm for data compression, we remove bias introduced from human or technological sources and assess randomness using NIST’s statistical test suite.

ASYMPTOTIC ANALYSIS OF PERES’ ALGORITHM FOR RANDOM NUMBER GENERATION

von Neumann [(1951). Various techniques used in connection with random digits. National Bureau of Standards Applied Math Series 12: 36–38] introduced a simple algorithm for generating independent unbiased random bits by tossing a (possibly) biased coin with unknown bias. While his algorithm fails to attain the entropy bound, Peres [(1992). Iterating von Neumann's procedure for extracting random bits. The Annals of Statistics 20(1): 590–597] showed that the entropy bound can be attained asymptotically by iterating von Neumann's algorithm. Let $b(n,p)$ denote the expected number of unbiased bits generated when Peres’ algorithm is applied to an input sequence consisting of the outcomes of $n$ tosses of the coin with bias $p$. With $p=1/2$, the coin is unbiased and the input sequence consists of $n$ unbiased bits, so that $n-b(n,1/2)$ may be referred to as the cost incurred by Peres’ algorithm when not knowing $p=1/2$. We show that $\lim _{n\to \infty }\log [n-b(n,1/2)]/\log n =\theta =\log [(1+\sqrt {5})/2]$ (where $\log$ is the logarithm to base $2$), which together with limited numerical results suggests that $n-b(n,1/2)$ may be a regularly varying sequence of index $\theta$. (A positive sequence $\{L(n)\}$ is said to be regularly varying of index $\theta$ if $\lim _{n\to \infty }L(\lfloor \lambda n\rfloor )/L(n)=\lambda ^\theta$ for all $\lambda > 0$, where $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) Some open problems on the asymptotic behavior of $nh(p)-b(n,p)$ are briefly discussed where $h(p)=-p\log p- (1-p)\log (1-p)$ denotes the Shannon entropy of a random bit with bias $p$.

Examples with decreasing largest inscribed ball for deterministic rescaling algorithms

Recently, Peña and Soheili presented a deterministic rescaling perceptron algorithm and proved that it solves a feasible perceptron problem in perceptron update steps, where ρ is the radius of the largest inscribed ball. The original non-deterministic rescaling perceptron algorithm of Dunagan and Vempala is based on systematic increase of ρ, while the proof of Peña and Soheili is based on the increase of the volume of a so-called cap. In this note we present a perceptron example to show that ρ may decrease after one rescaling step in this deterministic rescaling method. Furthermore, inspired by our previous work on the duality relationship between the perceptron and the von Neumann algorithms, we propose a deterministic rescaling von Neumann algorithm which is derived directly from the deterministic rescaling perceptron algorithm. Though the complexity of this algorithm is not established yet, we show by constructing a von Neumann example that ρ does not increase monotonically for the deterministic rescaling von Neumann algorithm. Thus, no complexity proof for the rescaling von Neumann algorithm can be based on increasing ρ monotonically. The von Neumann example serves as the foundation of the perceptron example. Finally, we present computational experiments with the deterministic rescaling von Neumann algorithm. The results show that the performance of the rescaling algorithm is significantly better than the performance of the original von Neumann algorithm.

A Comparison Among Simple Algorithms for Linear Programming

This paper presents a comparison between a family of simple algorithms for linear programming and the optimal pair adjustment algorithm. The optimal pair adjustment algorithm improvements the convergence of von Neumann's algorithm which is very attractive because of its simplicity. However, it is not practical to solve linear programming problems to optimality, since its convergence is slow. The family of simple algorithms results from the generalization of the optimal pair adjustment algorithm, including a parameter on the number of chosen columns instead of just a pair of them. Such generalization preserves the simple algorithms nice features. Significant improvements over the optimal pair adjustment algorithm were demonstrated through numerical experiments on a set of linear programming problems.

A Comparison Among Simple Algorithms for Linear Programming

This paper presents a comparison between a family of simple algorithms for linear programming and the optimal pair adjustment algorithm. The optimal pair adjustment algorithm improvements the convergence of von Neumann's algorithm which is very attractive because of its simplicity. However, it is not practical to solve linear programming problems to optimality, since its convergence is slow. The family of simple algorithms results from the generalization of the optimal pair adjustment algorithm, including a parameter on the number of chosen columns instead of just a pair of them. Such generalization preserves the simple algorithms nice features. Significant improvements over the optimal pair adjustment algorithm were demonstrated through numerical experiments on a set of linear programming problems.

On the von Neumann and Frank--Wolfe Algorithms with Away Steps

The von Neumann algorithm is a simple coordinate-descent algorithm to determine whether the origin belongs to a polytope generated by a finite set of points. When the origin is in the interior of the polytope, the algorithm generates a sequence of points in the polytope that converges linearly to zero. The algorithm's rate of convergence depends on the radius of the largest ball around the origin contained in the polytope. We show that under the weaker condition that the origin is in the polytope, possibly on its boundary, a variant of the von Neumann algorithm that includes away steps generates a sequence of points in the polytope that converges linearly to zero. The new algorithm's rate of convergence depends on a certain geometric parameter of the polytope that extends the above radius but is always positive. Our linear convergence result and geometric insights also extend to a variant of the Frank--Wolfe algorithm with away steps for minimizing a convex quadratic function over a polytope.

BCVEGPY2.2: A newly upgraded version for hadronic production of the meson [formula omitted] and its excited states

A newly upgraded version of the BCVEGPY, a generator for hadronic production of the meson $B_c$ and its excited states, is available. In comparison with the previous one [C.H. Chang, J.X. Wang and X.G. Wu, Comput. Phys. Commun. {\bf 175}, 624 (2006)], the new version is to apply an improved hit-and-miss technology to generating the un-weighted events much more efficiently under various simulation environments. The codes for production of $2S$-wave $B_c$ states are also given here.

Convergence Analysis of Markov Chain Monte Carlo Linear Solvers Using Ulam--von Neumann Algorithm

The convergence of Markov chain--based Monte Carlo linear solvers using the Ulam--von Neumann algorithm for a linear system of the form $x = Hx + b$ is investigated in this paper. We analyze the convergence of the Monte Carlo solver based on the original Ulam--von Neumann algorithm under the conditions that $\|H\|<1$ as well as $\rho(H)<1$, where $\rho(H)$ is the spectral radius of $H$. We find that although the Monte Carlo solver is based on sampling the Neumann series, the convergence of Neumann series is not a sufficient condition for the convergence of the Monte Carlo solver. Actually, properties of $H$ are not the only factors determining the convergence of the Monte Carlo solver; the underlying transition probability matrix plays an important role. An improper selection of the transition matrix may result in divergence even though the condition $\|H\|<1$ holds. However, if the condition $\|H\|<1$ is satisfied, we show that there always exist certain transition matrices that guarantee convergence of ...

GENXICC2.1: An improved version of GENXICC for hadronic production of doubly heavy baryons

We present an improved version of GENXICC, which is a generator for hadronic production of the doubly heavy baryons $\Xi_{cc}$, $\Xi_{bc}$ and $\Xi_{bb}$ and has been raised by C.H. Chang, J.X. Wang and X.G. Wu [Comput. Phys. Commun. 177 (2007) 467; Comput. Phys. Commun. 181 (2010) 1144]. In comparison with the previous GENXICC versions, we update the program in order to generate the unweighted baryon events more effectively under various simulation environments, whose distributions are now generated according to the probability proportional to the integrand. One Les Houches Event (LHE) common block has been added to produce a standard LHE data file that contains useful information of the doubly heavy baryon and its accompanying partons. Such LHE data can be conveniently imported into PYTHIA to do further hadronization and decay simulation, especially, the color-flow problem can be solved with PYTHIA8.0.

A family of linear programming algorithms based on an algorithm by von Neumann

In this article, we present a family of algorithms for linear programming based on an algorithm proposed by von Neumann. The von Neumann algorithm is very attractive due to its simplicity, but is not practical for solving most linear programs to optimality due to its slow convergence. Our algorithms were developed with the objective of improving the practical convergence of the von Neumann algorithm while maintaining its attractive features. We present results from computational experiments on a set of linear programming problems that show significant improvements over the von Neumann algorithm.

How to turn loaded dice into fair coins

We present a new technique for simulating fair coin flips using a biased, stationary source of randomness. Sequences of random numbers are of pervasive importance in cryptography and vital to many other computing applications. Many sources of randomness, such as radioactive or quantum-mechanical sources, possess the property of stationarity. In other words, they produce independent outputs over fixed probability distributions. The output of such sources may be viewed as the result of rolling a biased or loaded die. While a biased die may be a good source of entropy, many applications require input in the form of unbiased bits, rather than biased ones. For this reason, von Neumann (1951) presented a now well-known and extensively investigated technique for using a biased coin to simulate a fair coin. We describe a new generalization of von Neumann's algorithm distinguished by its high level of practicality and amenability to analysis. In contrast to previous efforts, we are able to prove our algorithm optimally efficient, in the sense that it simulates the maximum possible number of fair coin flips for a given number of die rolls. In fact, we are able to prove that in an asymptotic sense our algorithm extracts the full entropy of its input. Moreover, we demonstrate experimentally that our algorithm achieves a high level of computational and output efficiency in a practical setting.

A computational note on von Neumann's algorithm for determining optimal strategy

AbstractThis article is concerned with von Neumann's algorithm for solving a matrix game. Some of the numerical comparisons, made between this algorithm and three others from the literature, have been abstracted for discussion. In general, the performance of von Neumann's algorithm is inferior to other methods for solving matrix games; Dantzig's Simplex Method converges in fewer iterations; and Brown's Fictitious Play requires a much simpler arithmetic process and fewer iterations for a similar convergence.