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New Advances in Algebra, Ring Theory and Homological Algebra

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section " Algebra, Geometry and Topology ".

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 9733

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Dear colleagues,

The Development of Associative Algebra during the last century has resulted in the emergence of numerous theories or specialties that have given solution to many of the needs of the society we live in, increasingly developed from the technological point of view.

These needs fall into two broad groups: purely technological needs, and theoretical needs associated with developments in both applied algebra and other branches of mathematics. After all, it is not unreasonable to think that algebra is something like the "mathematics of mathematics".

There are many branches of algebra whose contributions solve problems posed by the scientific challenges arising from the advancement of technology. Two of them also stand out for their popularity in society: Cryptography and Coding Theory.

And from the theoretical point of view it is remarkable the momentum that some disciplines have had in the last 20 years. Thus, Homological Algebra has been given a big push with the emergence of the different classes of Gorenstein modules, and especially in recent years, of the relative Gorenstein modules. And the emergence of Hopf Algebras has made a huge impact on many branches of mathematics and physics. And of course, one cannot forget very active branches with immense applications at all times: Module Theory and Quivers Representation Theory.

Thus, we present this special issue of Mathematics as a tool to show recent and interesting results on the branches of Homological Algebra, Module Theory, Quivers Representation Theory, Hopf Algebras, Cryptography and Coding Theory.

Prof. Juan Ramón García Rozas Prof. Luis Oyonarte Alcalá Prof. Driss Bennis Guest Editors

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• Covers, envelopes and cotorsion pairs
• Relative Gorenstein modules and objects
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• (Sub)projectivity, (sub)injectivity, (sub)flatness domains and extensions
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The Year in Math

December 22, 2023

Video : In 2023, mathematicians improved bounds on Ramsey numbers, a central measure of order in graphs; found a new aperiodic monotile; and discovered a new upper bound to the size of sets without 3-term arithmetic progressions.

Christopher Webb Young, Kristina Armitage and Merrill Sherman/ Quanta Magazine

Introduction

Mathematical truths are often born of the conflict between order and disorder. Mathematicians discover patterns, and, to better understand the mysterious forces at play, they look for countervailing impulses that disrupt those patterns.

That tension came up repeatedly in our coverage this past year. We covered breakthroughs in graph theory, combinatorics, number theory and geometry — areas where patterns arise in unexpected ways, sometimes because of connections between seemingly distinct mathematical structures, and sometimes because of hidden intrinsic mechanisms uncovered by mathematicians in new proofs.

In a riveting interview with our senior writer Jordana Cepelewicz, Andrew Granville discussed how calculation and experimentation can, in sometimes forgotten ways, help mathematicians search for hidden patterns. He also spoke about changes in what it takes to convince other mathematicians that a result is true, and why he believes that examining the social nature of mathematics is essential to understanding what a proof is.

This was one of several conversations we published this past year about the nature of mathematical truth. Eugenia Cheng spoke with Joy of Why podcast host Steven Strogatz about category theory , a sort of “mathematics of mathematics” that can scare off other mathematicians with its level of abstraction. And Justin Moore spoke with Strogatz about the limits of the axioms — basic, obvious truths — of set theory and why there will always be important, unanswerable mathematical questions.

Though the bulk of our coverage fell squarely in the abstract realm, Minhyong Kim spoke with Kevin Hartnett about Mathematics for Humanity, an organization he founded to support mathematicians who want to use math to solve social challenges. And Mike Orcutt reported on how mathematics is used to ascertain the fairness of legislative district maps and to draw more equitable ones.

DVDP for Quanta Magazine

A Big Year in Graph Theory

If there is one area of math that was particularly fruitful in 2023, it’s graph theory. One of the biggest mathematical discoveries of the past year was the proof of a new, tighter upper bound to Ramsey numbers . These numbers measure the size that graphs must reach before inevitably containing objects called cliques. The discovery, announced in March, was the first advance of its type since 1935. It pertained to so-called symmetric Ramsey numbers. This was followed in June by a new result on the more general asymmetric case.

Both of these papers concerned what happens as graphs grow infinitely large. But Quanta also pondered the middle distance , looking at what mathematicians can prove about graphs that are too large to analyze using brute force, but smaller than the infinite, asymptotic limit.

We chronicled new results on how networks of connected oscillators come into synchrony and how graph theory connects to quantum field theory . We reported a new discovery about the possibilities of subdividing mathematical objects called vector spaces in a particular way into subsets called designs . And Patrick Honner, our Quantized Academy columnist, wrote about the way that local properties of graphs govern their global structure.

Quanta also published articles on two long-standing coloring problems. One explored the proof of the famous four-color theorem , which shows how four colors are enough to color any map on the plane so that no two adjacent regions have the same color. The other covered a new result on a less well-known but equally intriguing question, which asks how much of a plane can be colored in a way that ensures that no two points that are exactly one unit apart have the same color.

Samuel Velasco/ Quanta Magazine

Making Combinatorics Conjectures Count

Graph theory can be thought of as a branch of combinatorics — the mathematical study of counting. Counting what can happen with collections of nodes and edges is, in some sense, a special case of counting combinations more generally.

The year ended with a landmark proof by four prominent mathematicians of a longstanding conjecture that relates combinatorics to the algebraic structure of sets.

Back in February, two computer scientists, Zander Kelley and Raghu Meka, stunned mathematicians with news of an out-of-left-field breakthrough on an old combinatorics question: How many integers can you throw into a bucket while making sure that no three of them form an evenly spaced progression (like 3, 8 and 13 or 101, 201 and 301)? Kelley and Meka smashed a long-standing upper bound on the number of integers smaller than some cap N that could be put in the bucket without creating such a pattern.

The previous month, Kevin Hartnett reported on a paper from November 2022 by another outsider — a researcher at Google named Justin Gilmer who had left mathematics years before, but had never stopped thinking about a combinatorial problem called the union-closed conjecture. This conjecture concerns families of sets like {1}, {1, 2}, {2, 3, 4}, {1, 2, 3, 4}. This family is “union-closed” because if you combine any two sets in the family, the combination is also in the family. The conjecture says that if a family is union-closed, it must have at least one number that appears in at least half the sets. Gilmer used an argument drawn from information theory that relied on randomly choosing two sets from a union-closed family that met certain characteristics to prove a result that is an important step towards the full conjecture. His argument is yet another example of how randomness can be used as a tool to infer the existence of structure.

By contrast, an April article by Kevin Hartnett described an instance where intricate but simple structures surprisingly turn out to be possible. Bernardo Subercaseaux and Marijn Heule showed that it’s possible to fill an infinite grid with numbers in such a way that the distance between two occurrences of the same number must be greater than the number itself — using only the numbers between 1 and 15.

And longtime Quanta contributor Erica Klarreich wrote about the surprising prevalence of so-called intransitive dice . These are, for example, sets of three dice A, B and C in which A is likely to beat (roll a higher number than) B, B is likely to beat C and C is likely to beat A. A new paper showed that if you know only that die A beats die B and B beats C, that gives no information about whether A or C is likely to prevail in a head-to-head matchup.

Courtesy of Samuel Jinglian Li

New Connections in Number Theory

Perhaps more than in any other area of mathematics, number theorists can prove simple-sounding theorems using incredibly complicated technical constructions. This year, Quanta took readers on a tour of some of those constructions. We published an in-depth visual explainer of modular forms, which have been described as the “fifth fundamental operation” of math, along with addition, subtraction, multiplication and division. And we took readers on a historical tour of quadratic reciprocity , one of number theory’s most powerful tools. The modular-forms explainer was inspired by an article about so-called noncongruence modular forms — a less well-studied type of function that nevertheless has major implications for physics.

Max Levy, who wrote the quadratic-reciprocity explainer, got interested in the subject while reporting about a surprising summer discovery about patterns that circles can make. Levy recounted how two students working on a summer research project helped disprove a long-standing conjecture about how circles can be harmoniously nested, called the local-to-global conjecture. It was one of many developments this year that showcased the increasing utility of computational tools in mathematics. The students and their co-authors first found evidence that the conjecture was false by poring over computer-generated plots they’d created in an effort to see it at work.

Modular forms are closely related to elliptic curves — smooth functions of two variables where one variable is squared and the other cubed. (The functions also satisfy some particular mathematical constraints.) The relationship between the two was central to Andrew Wiles’ 1994 proof of Fermat’s Last Theorem. Hartnett wrote about advances in researchers’ understanding of that relationship for elliptic curves that are defined with variables drawn from imaginary quadratic fields — numbers of the form a + b $latex \sqrt{-5}$ where a and b are both rational numbers, or fractions.

He also wrote about a long-awaited magnum opus — a 451-page manuscript by the Fields medalist Akshay Venkatesh, together with Yiannis Sakellaridis and David Ben-Zvi, which elaborates further connections between objects related to modular forms and L -functions, an important type of infinite sum with a deep relationship to prime numbers.

Number theorists pay particular attention to prime numbers and the subtle and beautiful ways they’re distributed among the other integers. Intriguingly, if you consider them going out to infinity, it has long been known that the primes leave equal numbers of remainders when divided by some number — for instance, if you divide all the prime numbers by 5, you’ll get equal numbers of the remainders 1, 2, 3 and 4. But mathematicians keep striving to prove results about how quickly primes even out. In October, we reported on a new generation of mathematicians proving theorems about the ways in which primes are distributed.

We also introduced — and reintroduced — a fun mathematical game called Hyperjumps that explores the tension between structure and randomness by challenging players to create simple sequences of numbers using basic arithmetic.

Aperiodic Monotile Found After Long Search

It was also an exciting year in geometry. The most attention-getting result of the year was the discovery of a new kind of tile that covers the plane in a pattern that never repeats. A two-tile combination that does this has been known since the 1970s, but the single tile, discovered by a hobbyist named David Smith and announced in March, was a sensation. Fans used the simple design as a cookie cutter and sewed it into quilts. We followed our news coverage with a column explaining some of the underlying math and another giving a brief history of tiling .

Speaking of needles, it was also a year of progress on the Kakeya conjecture, which asks how small a volume of space an idealized needle can occupy while spinning in all directions. A new proof of a special case of the conjecture (called the “sticky” Kakeya conjecture) gives strong evidence that the more general conjecture is true.

The conjecture turns out to have implications not only for geometry, but also for harmonic analysis and the study of partial differential equations. A follow-up explainer examines those implications. And a Quantized Academy column takes readers through the conjecture’s underlying logic.

In other geometry news, a long-standing idea about maps between spheres of different dimensionality, called the telescope conjecture, was shown to be false . Particular types of contact structures (patterns of planes that satisfy certain mathematical properties) that had long been thought to be impossible turned out to exist .

We interviewed Emmy Murphy , a geometer who studies such contact structures. Murphy describes contact geometry (and its sibling, symplectic geometry) as existing in the middle of a spectrum of rigidity and flexibility. In rigid geometry, much depends, she said, on precise measurements, while flexible geometry tends to resemble algebra. But in between, she said, is where “visual thinking is more useful.”

In January, the mathematician Assaf Naor and the computer scientist Oded Regev proved the existence of so-called spherical cubes. These are objects whose surface area grows slowly — as does the surface area of spheres in higher dimensions — but which can completely fill space the way cubes can.

One of the most prominent geometers of the 20th century, Eugenio Calabi, died at age 100 on September 25. Jerry Kazdan, one of his longtime colleagues, said that Calabi would “ask interesting questions that no one else was thinking about.” Our obituary of Calabi explores those questions, focusing particularly on his best-known discovery, Calabi-Yau manifolds, which later became central to string theory in physics.

Harol Bustos for Quanta Magazine

It’s an Unstable World After All

Speaking of physics, we also published several new results about the mathematics of black holes, a favorite subject of contributing writer Steve Nadis. He wrote about a new paper that found an infinite number of different black hole shapes in higher dimensions, and another paper that clarifies the mathematics of the boundaries of black holes .

In April, we described how mathematicians are teaming up with physicists to understand new kinds of symmetries in quantum field theories.

Kathryn Mann and Thomas Barthelmé, along with Steven Frankel, published a series of papers characterizing dynamical systems called Anosov flows that balance chaos and stability. At any given point, the flows converge in one direction and diverge in another.

And in what might be the most unsettling math article of the year, we related news of a series of three papers by Marcel Guàrdia, Jacques Fejoz and Andrew Clarke showing that planetary orbits in a model solar system will always be unstable. The good news is that their model is quite unlike our solar system, although Clarke thinks similar instabilities may exist here as well.

But if they do, they’re not going to send any of the planets out of their orbits anytime soon, so you can look forward to another year of math coverage from Quanta in 2024.

Correction:   January 12, 2024 An earlier version of this story said that Justin Gilmer had proved the union-closed conjecture. That is incorrect; he proved a result that is an important step towards that conjecture, but not the full result. The text has been updated.

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On the zeros of partition functions with multi-spin interactions

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The area under the receiver operating characteristic curve (AUROC) of the test set is used throughout machine learning (ML) for assessing a model’s performance. However, when concordance is not the only ambition, this gives only a partial insight into performance, masking distribution shifts of model outputs and model instability.

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A New Formula for Pi Is Here. And It’s Pushing Scientific Boundaries.

This breakthrough method optimizes complex computations like never before.

• While building a simpler model for particle interactions, scientists made a sleek new pi.
• Representations of pi help scientists use values close to real life without storing a million digits.
• The making of the new pi involved using a series, which is a structured set of terms that either converge to one expression or diverge.

Because quantum mechanics looks at the tiniest particles, one at a time, even simple questions can have complex answers that require massive computing power. Rendering high-tech video games and movies like Avatar can take days or more, and that’s still not at the level of reality. In this new paper , published in the peer-reviewed journal Physical Review Letters , physicists Arnab Priya Saha and Aninda Sinha describe their new version of a quantum model that reduces complexity but maintains accuracy.

This is called optimization. Think of the way early internet video buffered in chunks of similar colors, or how classic animators painted static bodies with individual moving parts on top. Heck, think of how people cut the corners of squared-off walking paths until they make a dirt-path shortcut . We’re surrounded by optimization and optimizing behaviors.

As detailed in their paper, Saha and Sinha combined two existing ideas from math and science: the Feynman diagram of particle scattering and the Euler beta function for scattering in string theory. What results is a series—something represented in math by the Greek letter Σ surrounded by parameters.

Series can end up generalizing into overall equations or expressions, but they don’t have to. And while some series diverge—meaning that the terms continue to alternate away from each other—others converge on one approximate, concrete result. That’s where pi comes in. The digits of pi extend into infinity, and pi is itself an irrational number, meaning it can’t be truly represented by an integer fraction (the one we often learn in school, 22/7, is not very accurate by 2024 standards).

But it can be represented pretty quickly and well by a series. That’s because a series can continue to build out values well into the tiniest digits. If a mathematician compiles a series’ terms, they can use the resulting abstraction to do math that isn’t possible with an approximation of pi that’s cut off at 10 digits by a standard desk calculator. A sophisticated approximation enables the kind of nanoscopic particle work that inspired these scientists in the first place.

“In the early 1970s,” Sinha said in a statement from the Indian Institute of Science, “ scientists briefly examined this line of research, but quickly abandoned it since it was too complicated.”

But math analysis like this has come a long way since the 1970s. Today, Sinha and Saha are able to analyze an existing model and remodel it with altered terms. They’re able to build a sequence and see that it converges on the value of pi within far fewer terms than expected, making it easier for scientists to run the series and then use that for further work.

All of that requires decades of foundational work in the field and large bodies of work showing that certain mathematical moves work where other ones don’t. It’s a comment on the ongoing and collaborative nature of math theory , even when what results is a working model that might help scientists. Our ability to meaningfully approximate has grown in tandem with our ability to solve complex problems outright.

“Doing this kind of work, although it may not see an immediate application in daily life, gives the pure pleasure of doing theory for the sake of doing it,” Sinha said in the statement.

Caroline Delbert is a writer, avid reader, and contributing editor at Pop Mech. She's also an enthusiast of just about everything. Her favorite topics include nuclear energy, cosmology, math of everyday things, and the philosophy of it all. 

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Generative AI Can Harm Learning

59 Pages Posted: 18 Jul 2024

Hamsa Bastani

University of Pennsylvania - The Wharton School

Osbert Bastani

University of Pennsylvania - Department of Computer and Information Science

Özge Kabakcı

International Business School - Budapest (IBS)

Rei Mariman

Independent; Independent

Date Written: July 15, 2024

Generative artificial intelligence (AI) is poised to revolutionize how humans work, and has already demonstrated promise in significantly improving human productivity. However, a key remaining question is how generative AI affects learning , namely, how humans acquire new skills as they perform tasks. This kind of skill learning is critical to long-term productivity gains, especially in domains where generative AI is fallible and human experts must check its outputs. We study the impact of generative AI, specifically OpenAI's GPT-4, on human learning in the context of math classes at a high school. In a field experiment involving nearly a thousand students, we have deployed and evaluated two GPT based tutors, one that mimics a standard ChatGPT interface (called GPT Base) and one with prompts designed to safeguard learning (called GPT Tutor). These tutors comprise about 15% of the curriculum in each of three grades. Consistent with prior work, our results show that access to GPT-4 significantly improves performance (48% improvement for GPT Base and 127% for GPT Tutor). However, we additionally find that when access is subsequently taken away, students actually perform worse than those who never had access (17% reduction for GPT Base). That is, access to GPT-4 can harm educational outcomes. These negative learning effects are largely mitigated by the safeguards included in GPT Tutor. Our results suggest that students attempt to use GPT-4 as a "crutch" during practice problem sessions, and when successful, perform worse on their own. Thus, to maintain long-term productivity, we must be cautious when deploying generative AI to ensure humans continue to learn critical skills. * HB, OB, and AS contributed equally

Keywords: Generative AI, Human Capital Development, Education, Human-AI Collaboration, Large Language Models

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Mathematics at MIT is administratively divided into two categories: Pure Mathematics and Applied Mathematics. They comprise the following research areas:

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• Introduction
• Conclusions
• Article Information

5:2 MR indicates an intermittent fasting plan consisting of 2 nonconsecutive fasting days and 5 days of habitual intake per week and meal replacement diet.

A, Changes in hemoglobin A 1c (HbA 1c ) concentration from baseline to 16 weeks. The adjusted least-squares mean (LSM) changes were intermittent fasting plan consisting of 2 nonconsecutive fasting days and 5 days of habitual intake per week and meal replacement diet (5:2 MR), −1.9% (SE, 0.2%); metformin, −1.6% (SE, 0.2%); and empagliflozin, −1.5% (SE, 0.2%) (to convert to proportion of total hemoglobin, multiply by 0.01). The adjusted LSM difference between 5:2 MR and metformin was −0.3% (95% CI, −0.5% to 0.1%), and the adjusted LSM difference between 5:2 MR and empagliflozin was −0.4% (95% CI, −0.6% to 0.2%). B, Changes in HbA 1c concentration from baseline to 16 weeks in patients with overweight or obesity. C, Percentage of patients with HbA 1c concentrations of less than 7.0% or of less than 6.5% at week 16. D, Changes in fasting plasma glucose (FPG) concentrations from baseline to 16 weeks (to convert glucose to millimoles per liter, multiply by 0.0555). Error bars display SEs.

Patients were randomized to receive intermittent fasting plan consisting of 2 nonconsecutive fasting days and 5 days of habitual intake per week and meal replacement diet (5:2 MR) (n = 135), metformin (n = 134), or empagliflozin (n = 136). BMI indicates body mass index (calculated as weight in kilograms divided by height in meters squared); and LSM, least-squares mean.

Trial Protocol

eMethods. Multiple Imputation

eTable 1. Study Centers, Location, Ethics Committee Approvals

eTable 2. The Contents of Dietary and Exercise Guidance, and General Diabetes Education for All Participants

eTable 3. Nutrition Facts of Kang Zhijun™

eTable 4. Changes in Primary and Secondary Outcomes From Baseline to Week 16 (Intention-to-Treat Analysis)

eTable 5. Unadjusted Analyses of Outcomes From Baseline to Week 16 in the Three Groups

eTable 6. Primary and Secondary Outcomes From Baseline to Week 16 in Three Groups (Completers Analysis)

eTable 7. Primary and Secondary Outcomes From Baseline to Week 24 in Three Groups (Completers Analysis)

eFigure 1. Study Design

eFigure 2. The 5:2 MR Approach

eFigure 3. Changes in Body Weight

eFigure 4. Post-Hoc Subgroup Analysis of Weight Loss at Week 16 by Intention-to-Treat Analysis

eReference.

Data Sharing Statement

• Fasting, Meal Substitute Improved Blood Sugar More Than Diabetes Drugs JAMA Medical News in Brief July 12, 2024 Emily Harris

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Guo L , Xi Y , Jin W, et al. A 5:2 Intermittent Fasting Meal Replacement Diet and Glycemic Control for Adults With Diabetes : The EARLY Randomized Clinical Trial . JAMA Netw Open. 2024;7(6):e2416786. doi:10.1001/jamanetworkopen.2024.16786

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A 5:2 Intermittent Fasting Meal Replacement Diet and Glycemic Control for Adults With Diabetes : The EARLY Randomized Clinical Trial

• 1 Department of Endocrinology, Beijing Hospital, National Center of Gerontology, Institute of Geriatric Medicine, Chinese Academy of Medical Sciences, Beijing, China
• 2 Department of Endocrinology, The Third Affiliated Hospital of Jinzhou Medical University, Jinzhou, Liaoning, China
• 3 Department of Endocrinology, Nanyang Central Hospital, Nanyang, Henan, China
• 4 Department of Endocrinology, Henan Provincial People’s Hospital, Zhengzhou, Henan, China
• 5 Department of Endocrinology, The First Affiliated Hospital of Zhengzhou University, Zhengzhou, Henan, China
• 6 Department of Endocrinology, Hebei Provincial People’s Hospital, Shijiazhuang, Hebei, China
• 7 Department of Endocrinology, The Second Hospital of Hebei Medical University, Shijiazhuang, Hebei, China
• 8 Department of Endocrinology, Sir Run Run Hospital, Nanjing Medical University, Nanjing, China
• 9 Department of Endocrinology, The First Affiliated Hospital of Soochow University, Soochow, Jiangsu, China
• 10 Department of Clinical Nutrition, Beijing Tongren Hospital, Capital Medical University, Beijing, China
• Medical News in Brief Fasting, Meal Substitute Improved Blood Sugar More Than Diabetes Drugs Emily Harris JAMA

Question   What is the effect of a 16-week intermittent fasting plan consisting of 2 nonconsecutive fasting days and 5 days of habitual intake per week and meal replacement diet (5:2 MR) on the changes in hemoglobin A 1c level in Chinese adults with early type 2 diabetes?

Findings   In this randomized clinical trial of 405 adults, the 5:2 MR approach achieved better glycemic control at 16 weeks compared with metformin and empagliflozin.

Meaning   The 5:2 MR approach may serve as an effective initial lifestyle intervention instead of antidiabetic drugs for patients with type 2 diabetes.

Importance   An intermittent fasting plan consisting of 2 nonconsecutive fasting days and 5 days of habitual intake per week and meal replacement diet (5:2 MR) could provide additional benefits to patients with type 2 diabetes.

Objective   To evaluate the effect of the 5:2 MR on glycemic control among patients with early type 2 diabetes compared with metformin and empagliflozin.

Design, Setting, and Participants   The EARLY (Exploration of Treatment of Newly Diagnosed Overweight/Obese Type 2 Diabetes Mellitus) study is a randomized, open-label, active parallel-controlled clinical trial conducted between November 13, 2020, and December 29, 2022, in 9 centers across China. A total of 509 eligible patients underwent screening, out of which 405 were randomly assigned to 3 groups and included in the intention-to-treat analysis.

Interventions   Patients were randomly allocated in a 1:1:1 ratio to receive either metformin, empagliflozin, or 5:2 MR. The treatment was 16 weeks, with an 8-week follow-up.

Main Outcomes and Measures   The primary end point was the change in hemoglobin A 1c (HbA 1c ) level from baseline to 16 weeks. Secondary end points included changes in body weight, anthropometric measurements, and biochemical parameters.

Results   Of the 405 randomized participants (265 men [65.4%]; mean [SD] age, 45.5 [11.0] years; mean [SD] body mass index, 29.5 [4.1]; and mean [SD] HbA 1c level, 7.9% [0.6%]), 332 completed the 16-week treatment. From baseline to week 16, participants in the 5:2 MR group showed the greatest reduction in HbA 1c (least-squares mean [LSM], −1.9% [SE, 0.2%]), significantly greater than patients receiving metformin (LSM, −1.6% [SE, 0.2%]; adjusted LSM difference, −0.3% [95% CI, −0.4% to −0.1%]) and empagliflozin (LSM, −1.5% [SE, 0.2%]; adjusted LSM difference, −0.4% [95% CI, −0.6% to −0.2%]). At week 16, the mean weight loss in the 5:2 MR group (LSM, −9.7 kg [SE, 2.2 kg]) was greater than that in the metformin group (LSM, −5.5 kg [SE, 2.3 kg]) and empagliflozin group (LSM, −5.8 kg [SE, 2.3 kg]).

Conclusions and Relevance   This randomized clinical trial of Chinese adults with overweight or obesity and with early type 2 diabetes found that 5:2 MR could improve glycemic outcomes and weight loss in the short term compared with metformin or empagliflozin, making it a promising initial intervention and early management for type 2 diabetes.

Trial Registration   Chinese Clinical Trial Registry Identifier: ChiCTR2000040656

The latest data from the International Diabetes Federation in 2021 reveal that there are 537 million adults with diabetes globally, affecting approximately 1 in 10 adults. 1 China has the highest number of adults with diabetes in the world; from 2011 to 2021, the number increased from 90 million to 140.9 million, a 56.6% increase. 1 The prevalence of diabetes among Chinese adults is 12.4%. 2 According to China standards, 3 about half the population is either overweight (body mass index [BMI; calculated as weight in kilograms divided by height in meters squared] 24-27.9) or obese (BMI ≥28). 4

Overweight and obesity are significant risk factors for the development of type 2 diabetes. 5 - 7 Appropriate weight loss can improve glycemic control and reduce the dosage of antidiabetic drugs among patients with type 2 diabetes. 8 However, achieving weight loss is often challenging, necessitating the implementation of strategies such as meal replacement (MR) or dietary restriction.

Meal replacement is a prepackaged food or beverage that is substituted for 1 or more meals and provides energy. 9 The Look AHEAD study has demonstrated that, as part of a comprehensive lifestyle intervention, at 1 year MR effectively reduced hemoglobin A 1c (HbA 1c ) levels by 0.7% (to convert to proportion of total hemoglobin, multiply by 0.01) and achieved initial weight loss of 8.6% to 9.0% among patients with overweight or obesity and type 2 diabetes. 10 , 11 A systematic review including 23 studies and 7884 adults found that MR was associated with more weight loss (mean, −1.4 kg [95% CI −2.5 to −0.4 kg]) compared with other diets. 12 Important randomized clinical trials in White European indviduals (DiRECT), 13 Middle Eastern indviduals (DIADEM-I) 14 and South Asian individuals (STANDby) 15 have proved that MR can alleviate diabetes by lowering body weight.

As a dietary therapy, the 5:2 intermittent fasting diet involves 2 nonconsecutive fasting days (one-fourth the energy intake of habitual diet) and 5 days of habitual intake per week. 16 Individuals with obesity have successfully lost weight with this diet through both short-term and long-term interventions. 17 - 19 A single-center randomized clinical trial with a small sample size of 137 participants found that a 12-month 5:2 intermittent fasting diet significantly decreased HbA 1c levels among patients with overweight or obesity and type 2 diabetes, compared with a continuous energy restriction diet. 20

Combining the 5:2 intermittent fasting diet with MR (5:2 MR) could provide additional benefits to patients and is worthy of investigation. We aimed to investigate the efficacy of 16 weeks of 5:2 MR on HbA 1c changes among Chinese adults with overweight or obesity and early-stage type 2 diabetes.

The EARLY (Exploration of Treatment of Newly Diagnosed Overweight/Obese Type 2 Diabetes Mellitus) study is a randomized, open-label, active parallel-controlled clinical trial. The study protocol ( Supplement 1 ) was approved by the ethics committees of all participating centers (Beijing Hospital; the Third Affiliated Hospital of Jinzhou Medical University; Nanyang Central Hospital; Henan Provincial People’s Hospital; the First Affiliated Hospital of Zhengzhou University; Hebei Provincial People’s Hospital; the Second Hospital of Hebei Medical University; Sir Run Run Hospital, Nanjing Medical University; and the First Affiliated Hospital of Soochow University). The trial followed the International Conference on Harmonization Guidelines for Good Clinical Practice and the Declaration of Helsinki. 21 All patients provided written informed consent. This report adhered to the Consolidated Standards of Reporting Trials ( CONSORT ) reporting guideline.

We recruited adults with newly diagnosed (within 1 year) type 2 diabetes who had not used antidiabetic agents in the past 3 months, aged 18 to 65 years, with a BMI of 24 or more and an HbA 1c level of 7% to 9%. The recruitment was conducted concurrently at 9 hospitals across China (eTable 1 in Supplement 2 ) from November 13, 2020, to December 29, 2022. We excluded participants who had used weight-loss drugs or products within the past 3 months before enrollment, as well as pregnant or breastfeeding women ( Supplement 1 ).

Randomization was conducted using an interactive web response system. The randomization list of participants was generated by the stratified blocked randomization method using SAS software, version 9.4 (SAS Institute Inc), in which stratification was based on the center (block size of 9). Within each stratum, participants were randomized using a block randomization method, with a block size of 9, in a ratio of 1:1:1 to receive either metformin, empagliflozin, or 5:2 MR. Both the lists for participant and treatment allocation were inputted into the interactive web response system. At the study site, participants were administered treatment based on the randomization code and the corresponding treatment group obtained from the interactive web response system. Due to the nature of the intervention, blinding of participants and investigators was not feasible in this study. However, during the data analysis, the statisticians remained blinded to the study groupings.

The treatment period lasted for 16 weeks, followed by an 8-week follow-up (eFigure 1 in Supplement 2 ). All participants received dietary and exercise guidance as well as general diabetes education from nutritionists and research physicians in accordance with China Guideline 22 (eTable 2 in Supplement 2 ) every 4 weeks.

Patients in the 5:2 MR group consumed low-energy MR product A (Kang zhijun, Beijing MetabolicControl Technology Co Ltd; eTable 3 in Supplement 2 ). The 5:2 MR approach (eFigure 2 in Supplement 2 ) means that, within 1 week, there were 2 nonconsecutive days on which meals are replaced. On these 2 days, participants were required to consume 1 serving of Kang zhijun A instead of all 3 regular meals, with a daily energy intake of 500 kcal for women and 600 kcal for men. On the remaining 5 days, participants chose their own breakfast and lunch but had 1 serving of Kang zhijun B for dinner and were encouraged to monitor their calorie intake. Throughout the 16 weeks, dietary intake was recorded in a diary.

Patients took metformin (Shanghai Bristol-Myers Squibb), 0.5 g, twice a day. If the initial drug dosage was well tolerated, it was escalated to 2 g per day. Empagliflozin (Shanghai Boehringer Ingelheim), 10 mg, was administered once a day. During the study, patients were instructed to promptly contact the research center’s physician in case of severe hypoglycemia.

The primary outcome was the change in HbA 1c level from baseline to 16 weeks. Secondary outcomes included changes in weight (measured by InBody 770 [InBody]), BMI, waist circumference, hip circumference, waist to hip ratio, systolic and diastolic blood pressure, fasting plasma glucose (FPG) level, fasting insulin level, fasting C-peptide level, homeostasis model assessment of insulin resistance (HOMA-IR = FPG [mmol/L] × fasting insulin [μU/mL]/22.5), lipid profiles (total cholesterol, triglycerides, high-density lipoprotein cholesterol [HDL-C], and low-density lipoprotein cholesterollevels), and uric acid levels. The primary and secondary outcomes were reevaluated at the end of 8-week follow-up (week 24).

Adverse events were assessed throughout the study. Adverse events of particular interest included gastrointestinal reactions, urinary tract and reproductive system infections, hypoglycemia, and hyperglycemia. Laboratory testing was conducted at a central laboratory.

The sample size calculation was based on the SD of the change in HbA 1c level from a previous study, 23 with a 2-sided α of .05, β of 0.2, a minimum detectable between-group difference of 0.1%, and an anticipated SD of 0.2% based on pilot data analysis and a multiple pairwise comparison test using the Tukey-Kramer test. It was computed that each group required 108 participants using PASS 15 software (NCSS). Accounting for an expected 20% dropout rate, each group required 135 patients.

The primary outcome was analyzed following the intention-to-treat principle in the full analysis set, which included all randomized participants who received at least 1 dose of drugs or 5:2 MR. The safety outcome was analyzed in the safety analysis set, defined as participants randomized who received at least 1 dose of drugs or 5:2 MR and had safety assessment data collected at least once after the baseline.

The primary outcome was analyzed using the analysis of covariance model, which calculated the least-squares mean (LSM) and 95% CI to compare changes in HbA 1c level and key secondary outcomes among the 3 groups. The model adjusted for sex, age, height, weight, family history of diabetes and hypertension, physical activity, smoking, alcohol consumption, and baseline HbA 1c . Multiple imputation was used for missing values in the primary and key secondary outcomes (eMethods in Supplement 2 ). Post hoc subgroup analyses were conducted to explore the potential effect of baseline differences on HbA 1c and weight loss. Statistical analyses were performed using SPSS, version 24.0 software (SPSS Inc). All P values were from 2-sided tests and results were deemed statistically significant at P  < .05.

Of the 509 participants screened, 405 adults with type 2 diabetes (265 men [65.4%] and 140 women [34.6%]; mean [SD] age, 45.5 [11.0] years; mean [SD] BMI, 29.5 [4.1]; mean [SD] HbA 1c level, 7.9% [0.6%]) were randomly allocated. The patients’ baseline characteristics are presented in Table 1 . Of these 405 participants, 134 were randomized to the metformin group, 136 to the empagliflozin group, and 135 to the 5:2 MR group, all included in the intention-to-treat analysis ( Figure 1 ). Finally, 332 patients completed the 16-week treatment, for a completion rate of 82.0%.

At weeks 8 and 12, no patients in the 5:2 MR group required additional metformin for FPG level of 180.2 mg/dL or more and 2-hour plasma glucose of 250.5 mg/dL or more (to convert glucose to millimoles per liter, multiply by 0.0555). Only 1 patient in the metformin group had a FPG level of 218.0 mg/dL and consequently received additional empagliflozin.

At week 16, patients in the 5:2 MR group showed the greatest reduction in HbA 1c level (LSM, −1.9% [SE, 0.2%]), significantly greater than patients receiving metformin (−1.6% [SE, 0.2%]; adjusted LSM difference, −0.3% [95% CI, −0.5% to −0.1%]) and empagliflozin (−1.5% [SE, 0.2%]; adjusted LSM difference, −0.4% [95% CI, −0.6% to −0.2%]) ( Figure 2 A; eTable 4 in Supplement 2 ). However, there was no difference between the 2 drug groups (adjusted LSM difference, –0.2% [95% CI, –0.4% to 0.01%]; P  = .06). Post hoc subgroup analysis revealed that, apart from individuals aged 60 years or older, 5:2 MR mirrored the trend of HbA 1c reduction seen in the primary analysis ( Figure 3 ). The unadjusted baseline characteristics of patients supported these findings (eTable 5 in Supplement 2 ). Similarly, analyses of patients who completed the 16-week treatment also yielded consistent results (eTable 6 in Supplement 2 ). Among individuals with obesity, 5:2 MR significantly reduced HbA 1c compared with metformin (LSM difference, −0.4% [95% CI, –0.6% to –0.1%]) and empagliflozin (LSM difference, −0.4% [95% CI, –0.7% to –0.1%]) ( Figure 2 B). More patients in the 5:2 MR group (88.9% [120 of 135]) achieved an HbA 1c level less than 7% compared with the metformin (73.9% [99 of 34]; P  = .002) and empagliflozin (70.6% [96 of 136]; P  < .001) groups ( Figure 2 C). Similarly, in the 5:2 MR group, 80.0% of patients (108 of 135) achieved an HbA 1c level of less than 6.5%, surpassing metformin (60.4% [81 of 134]; P  < .001) and empagliflozin (55.1% [75 of 136]; P  < .001). Fasting plasma glucose levels in the 5:2 MR group decreased by −30.3 mg/dL (95% CI, −46.7 to −13.7 mg/dL) ( Figure 2 D). At the end of 8-week follow-up, 72 of 94 participants (76.6%) in the 5:2 MR group maintained an HbA 1c less than 6.5% (eTable 7 in Supplement 2 ).

At week 16, patients in the 5:2 MR group showed greater weight loss (LSM, −9.7 kg [SE, 2.2 kg]) than those in the metformin group (−5.5 kg [SE, 2.3 kg]; adjusted LSM difference, −4.2 kg [95% CI, −6.2 to −2.2 kg]) and empagliflozin group (−5.8 kg [SE, 2.3 kg]; adjusted LSM difference, −3.9 kg [95% CI, −5.9 to −1.9 kg]; eFigure 3A and eTable 4 in Supplement 2 ), with a greater proportion of those in the 5:2 MR group achieving weight loss (eFigure 3B in Supplement 2 ). Subgroup analyses confirmed this trend (eFigure 4 in Supplement 2 ). In addition, patients in the 5:2 MR group had significant reduction in waist and hip circumference and systolic and diastolic blood pressure, but showed no notable differences in most metabolic markers, except for triglyceride and HDL-C, compared with patients receiving antidiabetic drugs (eTable 4 in Supplement 2 ).

In the 5:2 MR group (n = 135), 1 patient experienced constipation, and 8 individuals (5.9%) had hypoglycemic symptoms, likely related to the low-energy diet ( Table 2 ). In the metformin group (n = 134), 26 individuals (19.4%) had mild gastrointestinal symptoms, and 8 individuals (6.0%) had hypoglycemia. In the empagliflozin group (n = 136), 3 patients (2.2%) experienced urinary symptoms, 5 patients (3.7%) experienced hypoglycemia, and 1 patient reported thirst. Two patients in the empagliflozin group experienced serious adverse events, including severe rash and hospitalization due to increased blood ketones, which resolved with treatment.

We found that among Chinese adults with overweight or obesity and newly diagnosed type 2 diabetes, the 5:2 MR approach achieved significant improvements in glycemic control and weight loss within a 16-week period, while also improving blood pressure and triglyceride and HDL-C levels. Therefore, 5:2 MR may potentially serve as an effective initial lifestyle intervention instead of antidiabetic drugs for early-stage type 2 diabetes.

Effective lifestyle interventions for patients with overweight and obesity and type 2 diabetes are crucial for achieving glycemic control and weight loss. Two single-center, small sample randomized clinical trials have confirmed that intermittent fasting can effectively reduce HbA 1c levels in these patients. 20 , 24 The 5:2 intermittent fasting diet for 12 months resulted in a reduction of 0.5% in HbA 1c level compared with a continuous energy restriction diet, with no difference in weight loss. 20 For patients with type 2 diabetes treated with insulin therapy, a 12-week 3:4 intermittent fasting intervention (3 days consuming 25% of recommended calories and 4 days without calorie restriction) led to a mean (SD) decrease of HbA 1c by 7.3 (12.0) mmol/mol (0.6% [1.1%]) and a mean (SD) weight loss of 4.8 (5.0) kg, with a daily total mean (SD) insulin dose reduction of 9 (10) IU. 24 A recent systematic review reported that the changes in HbA 1c after intermittent fasting intervention ranged from −1.5% to −0.3%. 25 Moreover, a meta-analysis of 2112 studies showed that partial or complete MR significantly reduced HbA 1c levels compared with conventional diabetes diets (−0.7% to −0.3%). 26 Our results found that after a 16-week intervention with the 5:2 MR, the mean HbA 1c reduction was 1.9%, greater than those achieved with metformin (0.3%) and empagliflozin (0.4%). According to American Diabetes Association recommendations, individuals with an HbA 1c of less than 6.5% for at least 6 months after the initiation of lifestyle interventions are considered to achieve diabetes remission. 27 In this study, 80.0% of patients reached this target with a 16-week 5:2 MR intervention. We acknowledge that the duration of our intervention was less than the recommended minimum of 6 months. Furthermore, at the end of the 8-week follow-up, 72 of 94 participants in the 5:2 MR group (76.6%) maintained an HbA 1c level of less than 6.5%, indicating that the 5:2 MR approach significantly and sustainably improves HbA 1c levels in patients with early type 2 diabetes.

In addition, our findings demonstrated that 5:2 MR reduced FPG levels, fasting insulin levels, C-peptide levels, and HOMA-IR. However, when compared with metformin and empagliflozin, the differences in fasting insulin levels, C-peptide levels, and HOMA-IR were not statistically significant. Animal studies have shown that fasting in diabetic mice can downregulate the expression of inflammatory factors, thereby alleviating inflammation. 28 A 5:2 MR plan may reshape the gut microbiota, promote white adipose tissue browning, and consequently reduce insulin resistance and the occurrence of obesity. 29 , 30 The MR used in this study contained omega-3 fatty acids and medium-chain fatty acids. Omega-3 fatty acids regulate leptin, inhibit fat synthesis, and promote fat breakdown. 31 Medium-chain fatty acids reduce heterotopic fat, enhance brown fat thermogenesis, and increase insulin sensitivity. 32

Compared with 2 antidiabetic drugs, 5:2 MR showed more significant and sustained benefits in weight loss and waist circumference reduction. Metformin exerts its effects by suppressing appetite, reducing insulin secretion, and improving gut microbiota. 33 Sodium-glucose cotransporter-2 inhibitors directly reduce body weight by increasing glucose excretion in the kidneys. 34 The DiRECT study confirmed that diabetes can be partially reversed through weight loss and proposed the “double cycle hypothesis,” suggesting that type 2 diabetes results from fat infiltration into the liver, pancreas, and muscle tissue, leading to the destruction of pancreatic β cells and tissue insulin resistance. Weight loss educes liver fat and significantly improves insulin resistance, and maintaining ideal body weight assists in β-cell function recovery, thus slowing down or even reversing the development of diabetes. 13 , 35 Our study cannot conclusively determine whether the glycemic improvement in patients with type 2 diabetes is due to weight loss or the 5:2 MR approach itself, requiring further investigation. The 5:2 MR reduced blood pressure and total cholesterol and increased HDL-C levels, consistent with previous studies indicating improved metabolic parameters with intermittent fasting and MR, 26 , 32 , 36 , 37 suggesting a potential cardiovascular protective effect.

The incidence of hypoglycemia was comparable across all 3 groups. When implementing a 5:2 MR intervention, it is essential to prevent hypoglycemia associated with low-energy diet. However, compared with medications, the 5:2 MR demonstrates favorable safety.

The 2020 China Guidelines emphasize lifestyle intervention as the foundational treatment for type 2 diabetes, with medication initiated only if lifestyle intervention fails to achieve glycemic control. 38 The EARLY study, for the first time to our knowledge, directly compared 5:2 MR with 2 widely used antidiabetic medications, providing evidence for the 5:2 MR approach as an effective initial lifestyle intervention for Chinese patients with early-stage type 2 diabetes.

This study has some limitations. First, it enrolled only patients not taking antidiabetic medication with a baseline HbA 1c level of less than 9%, so the efficacy of 5:2 MR for those taking medication or with a greater baseline HbA 1c needs further validation. Second, the 3-month washout period for eligibility regarding antidiabetic agents, including insulin, was short. A longer period without medication use (6 or 12 months) could offer more insights into prior medications’ effects. Third, the 5:2 MR intervention’s short duration means its long-term efficacy, especially for newly diagnosed patients with type 2 diabetes and overweight or obesity remains to be confirmed. Long-term follow-up studies are under way to assess the durability of 5:2 MR.

This randomized clinical study found that, for patients with newly diagnosed type 2 diabetes, a 16-week intervention with 5:2 MR could improve glycemic control and weight loss while also improving blood pressure, triglyceride levels, and HDL-C levels. Therefore, 5:2 MR may serve as an initial lifestyle intervention for patients with type 2 diabetes, providing an alternative to the use of metformin and empagliflozin medications.

Accepted for Publication: April 11, 2024.

Published: June 21, 2024. doi:10.1001/jamanetworkopen.2024.16786

Open Access: This is an open access article distributed under the terms of the CC-BY License . © 2024 Guo L et al. JAMA Network Open .

Corresponding Author: Lixin Guo, MD, Department of Endocrinology, Beijing Hospital, National Center of Gerontology, Institute of Geriatric Medicine, Chinese Academy of Medical Sciences, No. 1 Da Hua Rd, DongDan, Beijing 100730, China ( [email protected] ).

Author Contributions: Dr Guo had full access to all of the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis. Drs Guo, Xi, and Jin are co–first authors.

Concept and design: Guo, Yu.

Acquisition, analysis, or interpretation of data: Guo, Xi, Jin, Yuan, Qin, Chen, Zhang, Y. Liu, Cheng, W. Liu.

Drafting of the manuscript: Guo, Xi, Jin, Yuan, Qin, Chen, Zhang, Y. Liu, Cheng, W. Liu.

Critical review of the manuscript for important intellectual content: Guo, W. Liu, Yu.

Statistical analysis: Guo, W. Liu.

Obtained funding: Guo, Qin.

Administrative, technical, or material support: Guo, Xi, Jin, Yuan, Chen, Zhang, Y. Liu, Cheng, Yu.

Supervision: Guo.

Conflict of Interest Disclosures: None reported.

Funding/Support: This trial was funded by Beijing MetabolicControl Technology Co Ltd.

Role of the Funder/Sponsor: The funding source had no role in the design and conduct of the study; collection, management, analysis, and interpretation of the data; preparation, review, or approval of the manuscript; and decision to submit the manuscript for publication.

Data Sharing Statement: See Supplement 3 .

Additional Contributions: We would like to thank the participants for their contributions to this trial. We also thank Bin Qiao, PhD, for providing medical writing support for the manuscript; he was not compensated for his contribution.

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Linear algebra teaching and learning: themes from recent research and evolving research priorities

• Survey Paper
• Published: 01 November 2019
• Volume 51 , pages 1017–1030, ( 2019 )

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• Sepideh Stewart 1 ,
• Christine Andrews-Larson 2 &
• Michelle Zandieh 3  

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In this survey paper, we describe the state of the field on linear algebra research. We synthesize themes, questions, results, and perspectives emphasized in the papers that appear in this issue, as well as a selection of those published between 2008 and 2017. We highlight the extensive base of empirical research detailing how students reason about a variety of topic areas in linear algebra, as well as studies that provide evidence of promising directions for supporting students’ success through various teaching interventions and experiments in the classroom. From this survey paper, we identify areas for future research into the teaching and learning of this increasingly important area for modern applications.

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Acknowledgments

We are deeply appreciative of an incredible team of graduate students who worked with us to identify and organize the set of recent studies highlighted in this paper: B. Athey, I. Lee, M. Mauntel, J. Sipes, and J. Smith. We also thank the reviewers for their insightful feedback that helped us improve this paper.

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Stewart, S., Andrews-Larson, C. & Zandieh, M. Linear algebra teaching and learning: themes from recent research and evolving research priorities. ZDM Mathematics Education 51 , 1017–1030 (2019). https://doi.org/10.1007/s11858-019-01104-1

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Accepted : 15 October 2019

Published : 01 November 2019

Issue Date : December 2019

DOI : https://doi.org/10.1007/s11858-019-01104-1

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