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Published: 7 July 2024

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1

Eslami, Z., G. B. Khosrovshahi, and B. Tayfeh-Rezaie. "On halvings of the - design." Journal of Statistical Planning and Inference 86, no. 2 (May 2000): 411–19. http://dx.doi.org/10.1016/s0378-3758(99)00121-4.

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2

Laue, Reinhard. "Halvings on small point sets." Journal of Combinatorial Designs 7, no. 4 (1999): 233–41. http://dx.doi.org/10.1002/(sici)1520-6610(1999)7:4<233::aid-jcd1>3.0.co;2-e.

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3

Alcock, Nat, and Dan Miles. "AN EARLY FIFTEENTH-CENTURY WARWICKSHIRE CRUCK HOUSE USING JOGGLED HALVINGS." Vernacular Architecture 43, no. 1 (October 2012): 19–27. http://dx.doi.org/10.1179/0305547712z.0000000003.

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4

Besley, Timothy, and Robin Burgess. "Halving Global Poverty." Journal of Economic Perspectives 17, no. 3 (August 1, 2003): 3–22. http://dx.doi.org/10.1257/089533003769204335.

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The Millennium Development Goals—global targets that the world's leaders set at the Millennium Summit in September 2000—are an ambitious agenda for reducing poverty. As a central plank, these goals include halving the proportion of people living below a dollar a day from around 30 percent of the developing world’s population in 1990 to 15 percent by 2015—a reduction in the absolute number of poor of around one billion. This paper examines what economic research can tell us about how to fulfill these goals. It begins by discussing poverty trends on a global scale—where the poor are located in the world and how their numbers have been changing over time. It then discusses the relationship of economic growth and income distribution to poverty reduction. Finally, it suggests an evidence-based agenda for poverty reduction in the developing world.

5

Peto, R., A. D. Lopez, and O. F. Norheim. "Halving premature death." Science 345, no. 6202 (September 11, 2014): 1272. http://dx.doi.org/10.1126/science.1259971.

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6

Kasperson, Jeanne X., and Robert W. Kates. "Introduction: halving hunger." Food Policy 15, no. 4 (August 1990): 274–76. http://dx.doi.org/10.1016/0306-9192(90)90068-b.

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7

Fron?ek, Dalibor, and Mariusz Meszka. "Halving transversal designs." Journal of Combinatorial Designs 8, no. 2 (2000): 83–99. http://dx.doi.org/10.1002/(sici)1520-6610(2000)8:2<83::aid-jcd2>3.0.co;2-v.

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8

Draaisma, Douwe. "Neuroscience: Halving it all." Nature 518, no. 7539 (February 2015): 298–99. http://dx.doi.org/10.1038/518298a.

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9

Beygelzimer, Alina, and Stanisław Radziszowski. "On halving line arrangements." Discrete Mathematics 257, no. 2-3 (November 2002): 267–83. http://dx.doi.org/10.1016/s0012-365x(02)00430-2.

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10

Das, Pramod K., and Alexander Rosa. "Halving Steiner triple systems." Discrete Mathematics 109, no. 1-3 (November 1992): 59–67. http://dx.doi.org/10.1016/0012-365x(92)90278-n.

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11

Bierbrauer, J�rgen, and Yves Edel. "Halving PSL(2,q)." Journal of Geometry 64, no. 1-2 (March 1999): 51–54. http://dx.doi.org/10.1007/bf01229211.

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12

Khovanova, Tanya, and Dai Yang. "On halving-edges graphs." Involve, a Journal of Mathematics 11, no. 1 (January 1, 2018): 1–11. http://dx.doi.org/10.2140/involve.2018.11.1.

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13

Fujiwara, Yuichiro. "Halving Steiner 2-designs." Discrete Mathematics 307, no. 13 (June 2007): 1551–58. http://dx.doi.org/10.1016/j.disc.2006.09.005.

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14

Soper, Daniel S. "Hyperparameter Optimization Using Successive Halving with Greedy Cross Validation." Algorithms 16, no. 1 (December 27, 2022): 17. http://dx.doi.org/10.3390/a16010017.

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Training and evaluating the performance of many competing Artificial Intelligence (AI)/Machine Learning (ML) models can be very time-consuming and expensive. Furthermore, the costs associated with this hyperparameter optimization task grow exponentially when cross validation is used during the model selection process. Finding ways of quickly identifying high-performing models when conducting hyperparameter optimization with cross validation is hence an important problem in AI/ML research. Among the proposed methods of accelerating hyperparameter optimization, successive halving has emerged as a popular, state-of-the-art early stopping algorithm. Concurrently, recent work on cross validation has yielded a greedy cross validation algorithm that prioritizes the most promising candidate AI/ML models during the early stages of the model selection process. The current paper proposes a greedy successive halving algorithm in which greedy cross validation is integrated into successive halving. An extensive series of experiments is then conducted to evaluate the comparative performance of the proposed greedy successive halving algorithm. The results show that the quality of the AI/ML models selected by the greedy successive halving algorithm is statistically identical to those selected by standard successive halving, but that greedy successive halving is typically more than 3.5 times faster than standard successive halving.

15

Ramadhani, Muhammad Harits Zidni Khatib. "The Impact of Bitcoin Halving Day on Stock Market in Indonesia." Journal of International Conference Proceedings 5, no. 3 (September 29, 2022): 127–37. http://dx.doi.org/10.32535/jicp.v5i3.1800.

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This study aims to determine the impact of the bitcoin halving day phenomenon that occurred on May 11, 2020, on abnormal return and trading volume activity on listed firms of index LQ45 on the Indonesia Stock Exchange. At the bitcoin halving day, there was a jump in the price of bitcoin and other cryptocurrencies. Based on the event theory, the researcher wanted to find out whether capital market investors in Indonesia tend to switch investment instruments to the cryptocurrency market with this phenomenon. These data were taken weekly for four weeks before the halving and four weeks after the 5-month halving. Using the Paired sample T-test to determine whether there is a difference in abnormal return and trading volume. The results showed that there was a significant difference between stock price’s abnormal return and trading volume before the halving day and after the bitcoin halving day. The abnormal return did not increase, but the trading volume decreased, which is thought to be due to the halving day

16

Meynkhard, Artur. "Fair market value of bitcoin: halving effect." Investment Management and Financial Innovations 16, no. 4 (November 28, 2019): 72–85. http://dx.doi.org/10.21511/imfi.16(4).2019.07.

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The purpose of this article is to analyze the effect that halving has on the fair market value of bitcoins. The main hypothesis of the study is that the decline in the cost of miners’ remuneration for mining is a significant factor that affects the price of cryptocurrencies. The article examines the factors that regulate the issuing process. The significance of a limited supply of bitcoin is detailed in the article, as well as the mechanism for the implementation of the issue of new bitcoins. The study compares the historical inflation data of the US dollar and the projected data on the inflation of bitcoin. The article analyzes the main technical element of cryptocurrency – halving – when the miner’s reward is halved. This analysis includes the mathematical methods of statistical data processing. Research results show that reducing remuneration by half every four years leads to an increased market value of the cryptocurrency. This relationship is clearly illustrated by the Kendall rank correlation method. The results of the study can have a significant impact on the fundamental assessment of bitcoin and can also enable investors to assess any of the existing and operating cryptocurrencies according to this method.

17

Pietruszka, Michał, Łukasz Borchmann, and Filip Graliński. "Successive Halving Top-k Operator." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 18 (May 18, 2021): 15869–70. http://dx.doi.org/10.1609/aaai.v35i18.17931.

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We propose a differentiable successive halving method of relaxing the top-k operator, rendering gradient-based optimization possible. The need to perform softmax iteratively on the entire vector of scores is avoided using a tournament-style selection. As a result, a much better approximation of top-k and lower computational cost is achieved compared to the previous approach.

18

Ardila, Federico. "The Number of Halving Circles." American Mathematical Monthly 111, no. 7 (August 2004): 586. http://dx.doi.org/10.2307/4145159.

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19

Zheng, Chunfang, Qian Zhu, and David Sankoff. "Genome Halving with an Outgroup." Evolutionary Bioinformatics 2 (January 2006): 117693430600200. http://dx.doi.org/10.1177/117693430600200028.

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20

Cazenave, Tristan. "Sequential Halving Applied to Trees." IEEE Transactions on Computational Intelligence and AI in Games 7, no. 1 (March 2015): 102–5. http://dx.doi.org/10.1109/tciaig.2014.2317737.

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21

Taylor, Richard. "Halving deaths from cervical cancer." New South Wales Public Health Bulletin 14, no. 3 (2003): 55. http://dx.doi.org/10.1071/nb03017.

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22

Bomberg, Hagen, Franziska Meiser, Philipp Daume, Martin Bellgardt, Thomas Volk, Daniel I. Sessler, Heinrich V. Groesdonk, and Andreas Meiser. "Halving the Volume of AnaConDa." Anesthesia & Analgesia 129, no. 2 (August 2019): 371–79. http://dx.doi.org/10.1213/ane.0000000000003452.

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23

Ardila, Federico. "The Number of Halving Circles." American Mathematical Monthly 111, no. 7 (August 2004): 586–91. http://dx.doi.org/10.1080/00029890.2004.11920117.

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24

Sankoff, David, Chunfang Zheng, and Qian Zhu. "Polyploids, genome halving and phylogeny." Bioinformatics 23, no. 13 (July 1, 2007): i433—i439. http://dx.doi.org/10.1093/bioinformatics/btm169.

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25

Prautzsch, H., and Charles A. Micchelli. "Computing curves invariant under halving." Computer Aided Geometric Design 4, no. 1-2 (July 1987): 133–40. http://dx.doi.org/10.1016/0167-8396(87)90030-6.

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26

Csikós, B. "Halving surfaces in 3-manifolds." Periodica Mathematica Hungarica 20, no. 1 (March 1989): 51–56. http://dx.doi.org/10.1007/bf01849501.

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27

Fabus, Juraj, Iveta Kremenova, Natalia Stalmasekova, and Terezia Kvasnicova-Galovicova. "An Empirical Examination of Bitcoin’s Halving Effects: Assessing Cryptocurrency Sustainability within the Landscape of Financial Technologies." Journal of Risk and Financial Management 17, no. 6 (May 29, 2024): 229. http://dx.doi.org/10.3390/jrfm17060229.

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This article explores the significance of Bitcoin halving events within the cryptocurrency ecosystem and their impact on market dynamics. While the existing literature addresses the periods before and after Bitcoin halving, as well as financial bubbles, there is an absence of forecasting regarding Bitcoin price in the time after halving. To address this gap and provide predictions of Bitcoin price development, we conducted a rigorous analysis of past halving events in 2012, 2016, and 2020, focusing on Bitcoin price behaviour before and after each occurrence. What interests us is not only the change in the price level of Bitcoins (top and bottom), but also when this turn occurs. Through synthesizing data and trends from previous events, this article aims to uncover patterns and insights that illuminate the impact of Bitcoin halving on market dynamics and sustainability, movement of the price level, the peaks reached, and price troughs. Our approach involved employing methods such as RSI, MACD, and regression analysis. We looked for the relationship between the price of Bitcoin (top and bottom) and the number of days after the halving. We have uncovered a mathematical model, according to which the next peak will be reached 19 months (in November 2025) and the trough 31 months after Bitcoin halving 2024 (in November 2026). Looking towards the future, this study estimates predictions and expectations for the upcoming Bitcoin halving. These discoveries significantly enhance our understanding of Bitcoin’s trajectory and its implications for the finance cryptocurrency market. By offering novel insights into cryptocurrency market dynamics, this study contributes to advancing knowledge in the field and provides valuable information for cryptocurrency markets, investors, and stakeholders.

28

Georg, Paul Julius, Meret Emily Schmid, Sofia Zahia, Sebastian Probst, Simone Cazzaniga, Robert Hunger, and Simon Bossart. "Evaluation of a Semi-Automated Wound-Halving Algorithm for Split-Wound Design Studies: A Step towards Enhanced Wound-Healing Assessment." Journal of Clinical Medicine 13, no. 12 (June 20, 2024): 3599. http://dx.doi.org/10.3390/jcm13123599.

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Background: Chronic leg ulcers present a global challenge in healthcare, necessitating precise wound measurement for effective treatment evaluation. This study is the first to validate the “split-wound design” approach for wound studies using objective measures. We further improved this relatively new approach and combined it with a semi-automated wound measurement algorithm. Method: The algorithm is capable of plotting an objective halving line that is calculated by splitting the bounding box of the wound surface along the longest side. To evaluate this algorithm, we compared the accuracy of the subjective wound halving of manual operators of different backgrounds with the algorithm-generated halving line and the ground truth, in two separate rounds. Results: The median absolute deviation (MAD) from the ground truth of the manual wound halving was 2% and 3% in the first and second round, respectively. On the other hand, the algorithm-generated halving line showed a significantly lower deviation from the ground truth (MAD = 0.3%, p < 0.001). Conclusions: The data suggest that this wound-halving algorithm is suitable and reliable for conducting wound studies. This innovative combination of a semi-automated algorithm paired with a unique study design offers several advantages, including reduced patient recruitment needs, accelerated study planning, and cost savings, thereby expediting evidence generation in the field of wound care. Our findings highlight a promising path forward for improving wound research and clinical practice.

29

Perles, Micha A., Horst Martini, and Yaakov S. Kupitz. "Touching perfect matchings and halving lines." Ars Mathematica Contemporanea 15, no. 2 (July 10, 2018): 375–82. http://dx.doi.org/10.26493/1855-3974.1228.d7d.

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30

Lees, Kennedy R., and Paolo Milia. "Halving Effort in Acute Stroke Trials." Clinical and Experimental Hypertension 28, no. 3-4 (January 2006): 309–12. http://dx.doi.org/10.1080/10641960600549355.

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31

Fong, K., D. Hankerson, J. Lopez, and A. Menezes. "Field inversion and point halving revisited." IEEE Transactions on Computers 53, no. 8 (August 2004): 1047–59. http://dx.doi.org/10.1109/tc.2004.43.

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32

Ajoodani-Namini, S., and G. B. Khosrovashahi. "More on halving the complete designs." Discrete Mathematics 135, no. 1-3 (December 1994): 29–37. http://dx.doi.org/10.1016/0012-365x(93)e0096-m.

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33

Malthouse, Edward C., and Kalyan Raman. "The Geometric Law of Annual Halving." Journal of Interactive Marketing 27, no. 1 (February 2013): 28–35. http://dx.doi.org/10.1016/j.intmar.2012.08.001.

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34

Dey, T. K., and H. Edelsbrunner. "Counting triangle crossings and halving planes." Discrete & Computational Geometry 12, no. 3 (September 1994): 281–89. http://dx.doi.org/10.1007/bf02574381.

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35

Gidney, Craig. "Halving the cost of quantum addition." Quantum 2 (June 18, 2018): 74. http://dx.doi.org/10.22331/q-2018-06-18-74.

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We improve the number of T gates needed to perform an n-bit adder from 8n+O(1) to 4n+O(1). We do so via a "temporary logical-AND" construction which uses four T gates to store the logical-AND of two qubits into an ancilla and zero T gates to later erase the ancilla. This construction is equivalent to one by Jones, except that our framing makes it clear that the technique is far more widely applicable than previously realized. Temporary logical-ANDs can be applied to integer arithmetic, modular arithmetic, rotation synthesis, the quantum Fourier transform, Shor's algorithm, Grover oracles, and many other circuits. Because T gates dominate the cost of quantum computation based on the surface code, and temporary logical-ANDs are widely applicable, this represents a significant reduction in projected costs of quantum computation. In addition to our n-bit adder, we present an n-bit controlled adder circuit with T-count of 8n+O(1), a temporary adder that can be computed for the same cost as the normal adder but whose result can be kept until it is later uncomputed without using T gates, and discuss some other constructions whose T-count is improved by the temporary logical-AND.

36

Rontó, A., and M. Rontó. "Periodic successive approximations and interval halving." Miskolc Mathematical Notes 13, no. 2 (2012): 459. http://dx.doi.org/10.18514/mmn.2012.562.

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37

Martini, Horst, and Senlin Wu. "Halving circular arcs in normed planes." Periodica Mathematica Hungarica 57, no. 2 (December 2008): 207–15. http://dx.doi.org/10.1007/s10998-008-8207-6.

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38

Bárány, I., Z. Füredi, and L. Lovász. "On the number of halving planes." Combinatorica 10, no. 2 (June 1990): 175–83. http://dx.doi.org/10.1007/bf02123008.

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39

Martin, Megan, Cornelia A. Van Cott, and Qiyu Zhang. "The Beauty of Halving it All." Math Horizons 31, no. 2 (November 3, 2023): 14–17. http://dx.doi.org/10.1080/10724117.2023.2249357.

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40

Singla, Arun, Meenu Singla, and Mandeep Gupta. "Unpacking the Impact of Bitcoin Halving on the Crypto Market: Benefits and Limitations." Scientific Journal of Metaverse and Blockchain Technologies 1, no. 1 (2023): 43–50. http://dx.doi.org/10.36676/sjmbt.v1i1.06.

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Bitcoin halving events have become pivotal moments in the cryptocurrency market, generating widespread anticipation and speculation among investors, miners, and enthusiasts. Understanding the impact of these halving events requires a nuanced examination of their effects on supply dynamics, market sentiment, and the broader crypto ecosystem. Bitcoin halving events, while not directly affecting NFTs (Non-Fungible Tokens) and the metaverse, can have a notable impact on these ecosystems through interconnected market dynamics and investor sentiment. The relationship between Bitcoin, as the flagship cryptocurrency, and the broader crypto market often influences trends in specific sectors, including NFTs and the metaverse. As Bitcoin undergoes halving, the reduction in block rewards tends to generate heightened interest and speculation in the cryptocurrency space. This positive sentiment can create a favorable environment for various crypto assets, including those within the NFT and metaverse realms. Present research has focused on all these aspects.

41

Wącław, Joanna, Magdalena Zawada, Sylwia Czekalska, Bogdan Ochrem, and Tomasz Sacha. "Comparison of ABL1 and Gusb Reference Genes in the qRT-PCR Analysis of Halving Time and Early Molecular Response to TKI Therapy in Patients with Chronic Myeloid Leukemia." Blood 128, no. 22 (December 2, 2016): 5424. http://dx.doi.org/10.1182/blood.v128.22.5424.5424.

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Abstract Objectives and background: Early molecular response (EMR) to tyrosine kinase inhibitors (TKI) therapy defined as BCR-ABL1 transcript level ≤ 10% at 3 months is a well-established predictive factor in CML patients. Recently, several study groups presented the concept of BCR-ABL1 decline velocity or "halving time" as a new predictive factor for optimal TKI treatment response. Different reference genes were used in the quantitative RT-PCR (qRT-PCR) analysis to calculate halving time and EMR rate, potentially influencing results of these analyses. The aim of the study was to compare ABL1 and beta-glucuronidase (GUSB) as reference genes used for calculation of halving time and EMR rate in chronic phase CML patients (pts) treated with imatinib 400 mg/d in the clinical settings. Methods: A total of 52 chronic phase CML pts with blood samples collected at baseline and at 3 months of imatinib therapy were investigated. Transcript levels of BCR-ABL1 were determined by qRT-PCR with the use of ABL1 and GUSB as control genes for each patient. Halving time for BCR-ABL1 transcripts was calculated using baseline transcript levels, transcript levels at 3 months of treatment, and number of days between these two points according to methodology reported elswhere1. Using receiver operating characteristic (ROC) analysis, the optimal halving time thresholds for achievement of EMR for each control gene were determined. Optimal thresholds along the ROC curves were calculated using the Youden index. Results: 40 of 52 (76.9%) and 44 of 52 (84.6%) pts achieved EMR with BCR-ABL1 transcripts levels calculated with GUSB and ABL1 as control genes, respectively (p=0.55). In 4 pts there was no BCR-ABL1 transcript reduction from baseline at 3 month using GUSB as a control gene resulting with negative halving times. The GUSB based halving times of these 4 patients were imputed to the longest positive halving time (8530 days) calculated for the patient with the smallest BCR-ABL1 decline. Transcript reduction occurred in all pts when ABL1 was used as a control gene. ROC curve analysis resulted in the optimal halving time thresholds for achievement of EMR of 39 days (area under the curve [AUC ] 0.96, ;95% CI 0.9-1.0, ) and 31 days (AUC 0.95; 95% CI 0.87-1.0, ) for GUSB and ABL1 control gene, respectively. Discussion and Conclusions: Lack of linearity in qRT-PCR analysis observed in blood samples collected in CML patients at diagnosis, with high amount of BCR-ABL1 and wild type ABL1 transcripts could affect the accuracy of early kinetics decline of BCR-ABL1 transcript calculations when ABL1 is used as a reference gene. The analysis with the use of GUSB as a control gene is not biased by this phenomenon. In this study, a non - statistically significant differences in the rate of EMR achieved at 3 months of imatinib therapy and halving time were observed. Analyses performed in our study with ABL1 and GUSB as reference genes suggest that both GUSB and ABL1 are able to identify relevant cut-offs for EMR achievement and could be used for halving time calculation. Reference:Branford S, Yeung DT, Parker WT, et al. Prognosis for patients with CML and >10% BCR-ABL1 after 3 months of imatinib depends on the rate of BCR-ABL1 decline. Blood 2014 Jul 24;124(4):511-8 Disclosures Zawada: Novartis: Speakers Bureau. Czekalska:Novartis: Speakers Bureau. Sacha:Pfizer: Consultancy, Honoraria, Speakers Bureau; Incyte: Consultancy, Honoraria, Speakers Bureau; BMS: Consultancy, Honoraria, Speakers Bureau; Novartis: Consultancy, Honoraria, Speakers Bureau; Adamed: Consultancy, Honoraria.

42

Chan, Jireh Yi-Le, Seuk Wai Phoong, Seuk Yen Phoong, Wai Khuen Cheng, and Yen-Lin Chen. "The Bitcoin Halving Cycle Volatility Dynamics and Safe Haven-Hedge Properties: A MSGARCH Approach." Mathematics 11, no. 3 (January 30, 2023): 698. http://dx.doi.org/10.3390/math11030698.

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This paper introduces a unique perspective towards Bitcoin safe haven and hedge properties through the Bitcoin halving cycle. The Bitcoin halving cycle suggests that Bitcoin price movement follows specific sequences, and Bitcoin price movement is independent of other assets. This has significant implications for Bitcoin properties, encompassing its risk profile, volatility dynamics, safe haven properties, and hedge properties. Bitcoin’s institutional and industrial adoption gained traction in 2021, while recent studies suggest that gold lost its safe haven properties against the S&P500 in 2021 amid signs of funds flowing out of gold into Bitcoin. Amid multiple forces at play (COVID-19, halving cycle, institutional adoption), the potential existence of regime changes should be considered when examining volatility dynamics. Therefore, the objective of this study is twofold. The first objective is to examine gold and Bitcoin safe haven and hedge properties against three US stock indices before and after the stock market selloff in March 2020. The second objective is to examine the potential regime changes and the symmetric properties of the Bitcoin volatility profile during the halving cycle. The Markov Switching GARCH model was used in this study to elucidate regime changes in the GARCH volatility dynamics of Bitcoin and its halving cycle. Results show that gold did not exhibit safe haven and hedge properties against three US stock indices after the COVID-19 outbreak, while Bitcoin did not exhibit safe haven or hedge properties against the US stock market indices before or after the COVID-19 pandemic market crash. Furthermore, this study also found that the regime changes are associated with low and high volatility periods rather than specific stages of a Bitcoin halving cycle and are asymmetric. Bitcoin may yet exhibit safe haven and hedge properties as, at the time of writing, these properties may manifest through sustained adoption growth.

43

Jeong, DongEon, SangHeon Lee, and Jee-Hyong Lee. "Improving Successive Halving Algorithm by Considering the Learning Characteristics of Neural Network." Journal of Korean Institute of Intelligent Systems 29, no. 5 (October 31, 2019): 333–38. http://dx.doi.org/10.5391/jkiis.2019.29.5.333.

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44

Pasternack, Bernard S., Milton Sobel, and James Thomas. "Group-testing, halving procedures, and binary search∗." Communications in Statistics - Theory and Methods 16, no. 10 (January 1987): 2851–71. http://dx.doi.org/10.1080/03610928708829546.

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45

Ashworth, Sally. "Halving the load and doubling the benefits." Primary Teacher Update 2015, no. 42 (March 2, 2015): 17–19. http://dx.doi.org/10.12968/prtu.2015.1.42.17.

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46

Savard, Olivier Tremblay, Yves Gagnon, Denis Bertrand, and Nadia El-Mabrouk. "Genome Halving and Double Distance with Losses." Journal of Computational Biology 18, no. 9 (September 2011): 1185–99. http://dx.doi.org/10.1089/cmb.2011.0136.

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47

&NA;. "Halving the antibacterial course in otitis media." Inpharma Weekly &NA;, no. 810 (October 1991): 10. http://dx.doi.org/10.2165/00128413-199108100-00025.

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48

Boehmer-Christiansen, Sonja. "Factor Four: Doubling Wealth, Halving Resource Use." Energy & Environment 8, no. 4 (December 1997): 323–26. http://dx.doi.org/10.1177/0958305x9700800407.

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49

Robertson, James. "Factor four: Doubling wealth, halving resource use." Futures 29, no. 9 (November 1997): 873–74. http://dx.doi.org/10.1016/s0016-3287(97)00062-1.

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50

WARREN, ROBERT, and DAVID SANKOFF. "GENOME HALVING WITH DOUBLE CUT AND JOIN." Journal of Bioinformatics and Computational Biology 07, no. 02 (April 2009): 357–71. http://dx.doi.org/10.1142/s0219720009004102.

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The genome halving problem, previously solved by El-Mabrouk for inversions and reciprocal translocations, is here solved in a more general context allowing transpositions and block interchange as well, for genomes including multiple linear and circular chromosomes. We apply this to several datasets and compare the results to the previous algorithm.