Local officials in California failed to properly account for millions of dollars spent on homeland security efforts in the state, made dubious purchases that may not make communities safer, and could have overpaid millions by not seeking competitive bidding for equipment, according to an audit by the inspector general of the US Department of Homeland Security.
In one example cited, a California county bought a $96,600 generator to provide its public works department with emergency power during a catastrophe but didn't factor in a $130,000 overhaul of its electrical system needed to accommodate the generator. So nearly two years after the purchase, the new equipment wasn't ready for a disaster and might never be, county leaders admitted.
Another county entered into a multimillion-dollar contract for communications equipment without competitive bidding, which an independent assessment found could have cost taxpayers as much as 26 percent less if other companies had participated. A third county paid more than a half-million dollars to outfit a witness interview room, which was rejected as an illegitimate use of homeland security funds.
In a letter responding to the audit, California Office of Homeland Security Director Matthew Bettenhausen argued that the state has a system in place for approving local expenditures and that grant managers in his office receive extensive training throughout the year. A spokesman for Bettenhausen said "We're reviewing the report" but declined to address individual findings in the audit.
The March 13 report focused on $265 million given to the state from 2004-2006 through the State Homeland Security Program, only a fraction of the at least $1.7 billion awarded to California between 2003 and 2008 through the most common grant programs made available by the federal government. The state has received additional millions from Washington for projects that exclusively target port security, expanded interoperable radio systems, hiring new firefighters, border protection and more.
In overall dollars received, California is the leading beneficiary of homeland security grants compared to other states, according to Matt Mayer, a former top official in the federal Department of Homeland Security's Office of Grants and Training.
The audit also found that despite receiving hundreds of millions of dollars in homeland security grants since 2001, California has not adequately evaluated improvements made in emergency response efforts, so it's difficult to tell which areas of the state are better prepared to respond to a major catastrophe.
Bettenhausen, in his response letter, disputed that California had failed to measure changes in its ability to manage disasters. He wrote that the state has tracked improvements in its emergency response capabilities and can gauge today how it is better equipped for calamity than before California began receiving the grants. "California has gone above and beyond what is required at the federal level in the measurement of preparedness and capabilities," Bettenhausen wrote.
The audit does not name any of the counties singled out in examples of questionable grant expenditures, and Bettenhausen's office would not provide them, although it made assurances to the inspector general that the purchases were valid.
Two California counties CIR was able to independently identify from the report, Alameda and Contra Costa, committed $33.4 million to a new emergency communications system using grants funds, but they need at least $34 million more for additional equipment and are unsure where it will come from, according to auditors. So for now, the system is only partially operational. In addition, a contractor may have overcharged them, and auditors are questioning the entire investment.
For the most part, local California recipients of the homeland security grants Congress began handing out after Sept. 11 must purchase what they need first, such as equipment and terrorism response training, before being reimbursed with federal money controlled by Sacramento. The grants have allowed cities of all sizes across the country to buy new mobile command vehicles, gas masks, hazmat suits, decontamination shelters, surveillance equipment for identifying terrorists and more.
A number of states have struggled to comply with rules created to increase accountability over the grant spending such as seeing to it that cities and counties keep their pricey new gear in a "state of readiness" and maintain documents that validate the purchases they've made.
Suspect equipment purchases were made in cases identified by the auditors. One of the two aforementioned counties not named hired an engineering firm after determining that the $96,600 generator it had purchased with grant funds would be too difficult for its own employees to install. The consultants discovered that an electrical system would first need to be expensively renovated before the generator could be used. Because the county wouldn't put up the money needed to cover the previously unrealized costs, the generator has sat inactive for almost two years. A public works director from the county didn't believe the money would ever be made available, according to the report.
Another recipient spent nearly $600,000 on audio recorders and additional equipment to outfit a witness interview room for law enforcement purposes, but such items do not "enhance preparedness for terrorists' attacks or natural disasters," the report found. Auditors recommended that the state recover the costs of both projects.
Bettenhausen countered in his letter that the audio equipment "can be used to monitor and record suspects and witnesses involved in a terrorist event." Bettenhausen also promised to make sure that the emergency generator was eventually installed.
Many local authorities in California, according to the report's findings, also are not meeting federal procurement guidelines that require competitive bidding for major homeland security purchases, an obligation that's designed to make certain taxpayers receive the most value possible. "As a result, the grants' requirements for fair and open competition in procurement were not always practiced and subgrantees may have paid more than was necessary," the report states.
Local grant administrators bought large-ticket items but were not familiar with federal procurement rules and did not have records showing what processes they did use to purchase the gear. The report cited examples totaling $11.6 million worth of questionable transactions, including a $294,000 bomb disposal robot, $81,000 worth of night-vision goggles and a $5.1 million communications system.
Multiple purchases were made without competition, or local officials failed to prove that there weren't enough suppliers of specific goods to compete with one another. Nor did grantees conduct cost analyses to determine whether prices were "fair and reasonable" when only one vendor was used without competition.
The $5.1 million communications system was supposed to be part of a much larger regional network in the San Francisco Bay Area of northern California, designed to serve multiple dispatch centers and emergency radios so that first responders could more readily talk to one another. Two Bay Area counties, Alameda and Contra Costa, which encircle several California cities including Oakland and Berkeley, manage the interoperability project together, one of many similar but expensive efforts that exist today around the country. Because of the system's original design, according to the report, most of the equipment for an early phase of the undertaking was purchased from the same company that conceived it.
The contractor, which is not named in the report but CIR identified as Motorola by searching documents online, estimated that the larger system ultimately would carry a price tag of about $54.8 million, including the cost of transmission towers, repeaters and other equipment. A later independent assessment, however, found that Motorola had added a substantial premium to the price of as much as 26 percent compared to what the cost may have been if more than one company competed to offer a better deal.
A spokesman for Motorola declined to comment on the audit's findings. The East Bay Regional Communications System Authority has spent approximately $10 million so far against the contract with Motorola. Bill McCammon, executive director of the authority, told CIR that it has since learned of other less costly alternatives, such as piggy backing on a contract that was competitively bid out to Motorola by Riverside County in southern California. The authority will pursue those options rather than continuing with the existing contract, but McCammon admits some service fees charged by Motorola so far were overpriced.
He said that when Alameda County sought a contractor initially for a new radio system, Motorola was the only bidder. Officials chose to proceed with the company anyway because the Department of Homeland Security requires grants to be obligated quickly for projects each year after the federal awards are announced.
Regardless of the system's estimated cost now, it doesn't include the additional millions it will take to provide the police, sheriffs, firefighters, paramedics, public works personnel and others who would benefit from interoperability with individual radios, thousands of which may be needed. McCammon isn't sure how much more that could cost.
As of May 2008, the two counties were looking for a way to meet a funding gap in the project of $34 million, but according to the audit, they don't know how to fill it. McCammon responds that "we're going through a very deliberate process to evaluate our options" with the aid of a communications consultant. Nonetheless, if the system does not eventually work, auditors say that all of the money used from homeland security grants for the project – more than $33 million earmarked so far – should be relinquished.
Bettenhausen responded in his letter that the state would not be requiring an "artificial date" for completion of the communications project but would review its implementation plan to see if the system is technically feasible and compatible with other interoperable radio initiatives in the state.
The report, meanwhile, did single out another project in California for commendation. The California National Guard created a unique truck-bound system that can bridge disconnected communications systems in a disaster area. But it would still take a few hours for the system to be deployed during an emergency with the help of a military airlift. An early prototype was used during Hurricane Katrina and also California's 2007 wildfire season. Six of the units are being manufactured and will be set up in regions across the state. Other states are examining the system's state-of-the-art design.
No user commented in " Golden State lacks control over grant spending "


We are thrilled to announce a new partnership with Colombia Immersion, an immersive Spanish language school based in Medellín. We'll be working closely with their team to provide immersive expeditions here in Colombia, giving students the opportunity to improve their Spanish skills while exploring authentic pueblos and adventuring throughout the mountains of Colombia.
The first trip will take us to the very top of Cerro Tusa and through the gorgeous, colonial pueblo of Jardín. From horseback riding among lush green mountains to sampling coffee with Don Andres these tours highlight the very best of Antioquia and allow Spanish-language students from Colombia Immersion to become well acquainted with this incredible region of Colombia. All-in-all this is a partnership we are incredibly proud to be announcing.
Colombia Immersion has been working in Medellín for the past two years. The curriculum is speaking-based in order to effectively enhance the language skills of students and allow them to connect with native speakers confidently.
Are you ready for an adventure? 2018 Dates are open! View available spots here.
Filed under Discovery, News, Trekking.
Tagged cerro tusa, colombia, Colombia Immersion, Medellin, partnership.


text:
PowerOut Theatre Company
Hot Pepper Theater
Take the Car
Take the Car: A Los Angeles play
Time changes everything. Right?
Sex, drugs, and bus stops—it's all part of Jenny and Meg's half-assed attempt to make it through high school. But once Jenny's son Lucas comes into the scene, their haphazard life goes from amusing to dangerous. How do you raise a child if you never really grew up yourself?
Take the Car weaves together past and present to tell the story of who we are, who we were, and what the hell we think we're doing.
Part of the Williamstown Theatre Festival's 2015 Director's Lab.
Shown: Lindsay Rico, Amanda Shechtman, Doron JePaul Mitchell
Photos by Paul Fox
a new play by Nora Sørena Casey | directed by Ben Ferber
Featuring: Lindsay Rico (Meg), Amanda Shechtman (Jenny/Sofia), Doron JePaul Mitchell (Lucas/Randall)
Alex Petersen (scenic designer), Miriam Michaels (lighting designer), Zachary Heffner (master electrician), Jeff Sherwood (sound designer),
Elizabeth O'Donnell (sound board operator), Robin Piatt Stegman (costume designer), Chris Carcione (graphic design)
PowerOut Theatre Company, A WordPress.com Website.

redpajama_set_name: RedPajamaCommonCrawl


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The story of how I built my business around my passion for horses.
Making quick money is definitely possible but making big bucks quickly is something else.
I believe you have to have more than a bit of luck to have that happen.
Success in business takes passion.
And this is way I'm willing to share my story of how I choose to make money on the internet from my personal passion for horses.
Build a business around one of your true passions ( horses, beagles, sailing, quilting, whatever.
So...what I decided to do was to build a website dedicated to all things horses: horse gifts, pictures of horses, and many cool gift ideas for horse lovers all over the world.
During my investigation of how to make money on the internet with my passion, I bumped into one of the most amazing sites.
I found this cool web-building service that allowed me to take my passion of horses and mold it into a real money-making business.
I started out slow...and watched my business follow passion...and it start to grow.
Oh, And, Yes...there were inevitable glitches, goofs, and unplanned events.
But I was able to look at these as learning opportunities, not hurdles.
I overcame these hurdles and enjoy the immense satisfaction I get from seeing your efforts begin to pay off.
My passion for business is the driving force behind my effort.
Having that passion will dramatically increase your probability of success, plus you'll have a great time getting there.
Our business came from our passion for horses, and that passion has made putting our business together just plain fun!
And...we had fun assembling our gift site for horse-lovers by including much more than a site that housed pictures of horses, or general horse site filled will general horse stuff.
Oh, and here's an interesting thing we found to be true in watching our business grow: credibility is key.
Lack of this is probably the major cause of small business failure.
We established credibility by being truthful about our product and service, and by demonstrating expertise about what we offer.
My sincere advice to you: take your business very seriously or be prepared to fail.
You don't need to know rocket science but you'll need to know a lot of things, and you'll need to do a lot of things.
By doing the right things you reduce the possibility of failure.
By reducing, and eliminating, the possibility of failure, you insure success.
So...if you ask me my "secret" to success, I'd have to tell you that it is nothing more than - PERSEVERANCE.
As the saying goes, "When you get a lemon, make lemonade"
Most new businesses that do become profitable take some time getting there.
You won't begin making big bucks overnight.
And no...I'm not suggesting you lower your goals or expectations.
Just realize that good things take time.


text:
Follow the things
'Follow the things' is a phrase coined by anthropologist Arjun Appadurai in 1988 and expanded upon by anthropologist George Marcus in 1995. Both encouraged academic researchers to undertake multi-site research with people whose lives were (often unknowingly) intertwined through, among other factors, the making, trading, purchase, use and disposal of things. The 'Follow the things' website is run by Ian Cook, an Associate Professor of Geography at the University of Exeter in the UK, with considerable help from Keith Brown, Associate Research Professor at the Watson Institute for International Studies, at Brown University in the USA. Most of the pages have been produced by students taking Ian's 'Geographies of Material Culture' module at the Universities of Birmingham and Exeter and Keith's 'Anthropologies of Global Connection' module at Brown.
One of these pages, by Eeva Kemppainen, addresses the commodification of human organs:
http://www.followthethings.com/kidneytrade.shtml . Kemppainen generated an interesting map to examine the flows and sites specific to international trade in kidneys: https://maps.google.co.uk/maps/ms?msa=0&msid=215401690512249291683.0004c39af5a05b539fb94&ie=UTF8&t=m&ll=23.563987,13.007813&spn=110.040698,246.09375&z=2&source=embed
Medical tourism in mainstream TV and film
Friday Night Lights - 'Backfire' episode (2007) - Stem cell therapy in Mexico
Love and Other Drugs (Zwick, 2010) - Pharmaceutical tourism to Canada
The Best, Exotic Marigold Hotel (Madden, 2011) - Retirement migration to India


asked 1 year ago viewed 3902 times active 1 year ago Visit Chat Related 10How often should I run CHKDSK?3chkdsk "An unspecified error occurred (696e647863686b2e e19)"19Chkdsk Questions0Canceled chkdsk, chkdsk still runs0Windows Else block in if-then-else statement missing in SPD2013 Why I cannot modify the "Created" .... "Modified" columns after updating SP library file? You are going to: I understand, and I'm going in Cancel You are leaving LeagueofLegends.com Facechecking the brush is dangerous, and so is facechecking the web. CHKDSK is verifying files (stage 1 of 5)... 768 file records processed.
Riot Games, League of Legends and PvP.net are trademarks, services marks, or registered trademarks of Riot Games, Inc. I saw all the files when boot using USB. it is new and of good configuration and working with all other things very well. Instead, head on over to our [Report a Bug](http://boards.na.leagueoflegends.com/en/c/bug-report) board.
more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed Not the answer you're looking for? So success is not very likely at this point. I ran chkdsk both from command box and from the drive C tools window.
I tried again today, and chkdsk got to the 35% mark and slowly continued to uptick the number of files it had read and stayed at 35% for a long time. They are there in my computer list but not opening; even not responding to right click. Which action helped you to get access of the drive?Soumimessage edited by Soumi See More: chkdsk not working after step 2 Report • #1 Johnw November 23, 2014 at 22:15:37 "chkdsk local drive G is listed in my computer but not accessible.Im using win7.I have run chkdsk for it using chkdsk g:/fafter stage two it is showing - Index verification completed.
share|improve this answer answered Jan 29 '15 at 17:06 magicandre1981 54k1375110 When chkdsk was finished and I clicked on the device in Windows Explorer it showed it was not Need a crash course in basic phone... He then plugged the drive back in and restarted chkdsk. windows-7 hard-drive chkdsk share|improve this question edited Jan 30 '15 at 1:04 asked Jan 29 '15 at 13:52 steve 613 1 nice to hear that a new cable fixed it.
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switch... 100% Disk space slowing down... Riot Games, League of Legends and PvP.net are trademarks, services marks, or registered trademarks of Riot Games, Inc. Accept my answer to "close" the question. –magicandre1981 Jan 31 '15 at 15:24 add a comment| 2 Answers 2 active oldest votes up vote 2 down vote 0xc000009d = STATUS_DEVICE_NOT_CONNECTED Looks See More ↓ #4 Derek November 24, 2014 at 14:53:16 Thanks Johnw.SoumiIt is most unlikely that Sobrien will come back after all this time and spot this new post.
At this point, you're running the risk of encountering bigger problems than what Chkdsk is typically good at resolving. Now Diskeeper is still complaining that it can't run because the chkdsk is schedules to run and hasn't--but it has. fish tank problem Why are some programming languages turing complete but lack some abilities of other languages?


For more information on this issue contact Nancy Cline at (212) 376-3041 or by email at ncline@vera.org.
The Vera Institute of Justice's Speaking of Justice Series explores issues of justice and fairness through ongoing discussions with some of the most thoughtful and knowledgeable government officials, scholars, and nonprofit leaders from around the country. The series is funded through a gift by the Lucille Lortel Foundation.


text:
Richard Henry "R. H." Tawney (Calcutá, Índia,  — Londres, ) foi um historiador econômico,  crítico social,  e socialista cristão inglês. e um proponente importante da educação de adultos. O The Oxford Companion to English Literature (1997) cita que Tawney teve um "impacto significativo" em todos esses papéis inter-relacionados".

O historiador acadêmico
Primeiro importante trabalho como historiador foi The Agrarian Problem in the Sixteenth Century (1912).  De 1917 a 1931, foi professor na London School of Economics (LSE). Em 1926, ele ajudou a fundar a Sociedade de História Econômica com Sir William Ashley, entre outros, e se tornou o editor conjunto de sua revista, The Economic History Review.

De 1931 até sua aposentadoria em 1949, ele foi professor de história econômica na LSE e Professor Emérito depois de 1949. Ele era Doutor Honoris Causa das universidades de Oxford, Manchester, Birmingham, Sheffield, Londres, Chicago, Melbourne e Paris.

As obras históricas de Tawney refletem suas preocupações éticas e com a história econômica. Ele era profundamente interessado na questão agraria na zona rural inglesa durante os séculos XVI e XVII e, na tese de Max Weber sobre a relação entre o surgimento do protestantismo e a ascensão do capitalismo.

Religion and the Rise of Capitalism (1926) foi a sua obra clássica e fez sua reputação como um historiador. Ele explorou a relação entre o protestantismo e o desenvolvimento econômico nos séculos XVI e XVII. Tawney "lamentou a divisão entre o comércio e a moralidade social provocada pela Reforma Protestante, levando a subordinação da doutrina cristã para a busca da riqueza material".

Crítica social
Dois dos livros de Tawney destacam-se pela crítica social:  The Acquisitive Society (1920), e Equality em  (1931). No primeiro, um de seus mais lidos livros, ele criticou o individualismo egoísta da sociedade moderna. O capitalismo, ele insistiu, incentiva a ganância e, assim, corrompe todos. No último livro, Tawney defende uma sociedade igualitária.

Em 1906, Tawney ingressou na Sociedade Fabiana que defendia uma mudança gradual para o socialismo em vez de uma mudança revolucionária, sendo eleito como membro executivo para o biênio 1921-1933.

Obras
 The Agrarian Problem in the Sixteenth Century (1912), London: Longman, Green and Co.
 The Acquisitive Society (1920), New York, Harcourt Brace and Howe (1920); Mineola, NY, Dover (2004) ISBN 0-486-43629-2
 Secondary Education for All (1922)
 Education: the Socialist Policy (1924)
 Religion and the Rise of Capitalism (1926), Mentor (1953) ISBN 0-7658-0455-7, Peter Smith (1962)
 Equality (1931) ISBN 0-04-323014-8
 Land and Labour in China (1932).
 The Radical Tradition: Twelve Essays on Politics, Education and Literature, (1964), Harmondsworth, Penguin, ISBN 0-14-020834-8

Ligações externas

Biografia de Tawney
Tawney's Essays introducing the 1923 edition of A Discourse Upon Usurye by Thomas Wilson
of the Tawney papers na Division da London School of Economics

Socialistas do Reino Unido
Historiadores da Inglaterra
Socialistas cristãos

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}
\label{sec:intro}
The chemical compositions of stars are valuable fossil records that preserve --- in many cases --- the chemistry of the interstellar medium (ISM) from which the stars were born. Observations of large samples of stars formed at different epochs therefore allow one to unfold the chemical enrichment and star formation history (SFH) of the host system with unprecedented accuracy. The Milky Way (MW) is an ideal laboratory to conduct such analysis on a galaxy scale where chemistry observations of large numbers of individual stars are achievable.
These observations have relatively recently extended to the Galactic bulge (compared to, e.g., the solar neighborhood and halo), which is the only example of a massive galaxy center that can be studied in this way. Our bulge is therefore of great importance to understand the stellar population and formation history of galactic bulges in general.

Early studies of MW bulge chemistry revealed the existence of multiple stellar populations with a wide range of metallicity, spanning $\sim-1$ to $\sim+0.5$ dex \citep{whitford1983,rich1988,mcwilliam1994}.
Large effort has since been devoted to mapping the detailed metallicity distribution function (MDF) in the Galactic bulge using tracers such as Red Giant Branch stars \citep{zoccali2008,rich2012,uttenthaler2012,johnson2013,schultheis2017,rojas2017,garcia2018},
red clump stars \citep{hill2011,ness2013,zoccali2017,rojas2014}, and even subgiant and dwarf stars \citep{bensby2013}.
The MDF in the bulge as derived in different works shows noticeable difference in peak structure, described as either two strong [Fe/H] peaks \citep[with a supersolar component at $>+0.3$ and a subsolar one at $<-0.4$; e.g.,][]{hill2011,schultheis2017} or multiple, less strong peaks within the same metallicity range \citep[e.g.,][]{bensby2013,ness2013}.

In addition to the MDF, $\alpha$-element abundances serve as important constraints on the SFH of stellar populations, in both our Galaxy and the general galaxy population. In particular, the relative abundance ratio of $\alpha$-elements to iron ([$\alpha$/Fe]) is a powerful diagnostic of star formation timescale, as rapid star formation can be recognized from an enhanced [$\alpha$/Fe] in stellar composition \citep{matteucci1994,thomas2010}.
It has become increasingly clear that the bulge is composed of at least two distinct populations: a dominant one comprising old, metal-poor, and $\alpha$-enhanced stars, and a less numerous one comprising intermediate-age, metal-rich stars with solar-like [$\alpha$/Fe] \citep[e.g.,][]{babusiaux2010,hill2011,schultheis2017,rojas2019,queiroz2020}.

A long-running debate persists over whether the $\alpha$-element enhancement in the old bulge stellar populations is identical with that of thick disk stars, such as those in the solar vicinity.
The important physical question underlying this discussion is whether the bulge and thick disk share the same early SFH or not. Slightly different, perhaps systematically higher, [$\alpha$/Fe] abundances in the old bulge stars were suggested in some works \citep{cunha2006,zoccali2006,lecureur2007,fulbright2007,zasowski2019}.
On the other hand, other studies found no significant difference \citep{melendez2008,alves2010}, evidence that favors a picture with no distinct chemical ``bulge'' structure besides the inner thick disk in the Galactic center \citep{fragkoudi2018,dimatteo2019}.
In this sense, the Milky Way would be a pure disk galaxy with a buckled bar \citep[e.g.,][]{nataf2017}.

On the theoretical side, impressive progress has been made
to interpret these observed chemical distributions in the bulge.
\citet{matteucci1990} performed one of the pioneering chemical evolution modelling of the Galactic bulge, which traces not only the enrichment history of global metal content but also Fe, Si, Mg, and O individually.
By comparing the prediction of various models with the observed MDF from \citet{rich1988}, the authors reached the conclusion that the bulge formed on a much shorter timescale ($<0.5\;$ Gyr) than the disk, with more efficient star formation and a flatter IMF.
Following the observations of two distinct stellar populations in the bulge, many chemical evolution models have been developed to account for possible multiple formation channels or phases in the bulge \citep[e.g.,][]{bekki2011,grieco2012,tsujimoto2012,haywood2018,matteucci2019}.
\citet{haywood2018} proposed a heuristic two-phase SFH connected by a rapid quenching episode to explain the density dip in the MDF and [$\alpha$/Fe] distribution functions ($\alpha$-DF) observed in the APOGEE survey \citep{majewski2017}. By exploring a broad radial range of the inner Galaxy ($R_{\rm GC} < 7\;$kpc), \citet{haywood2018} suggested that the bulge and inner disk are indistinguishable in terms of chemical properties and that their chemical evolution has followed a similar path.

Although enormous progress has been made in modelling the chemical evolution in the bulge, most works have been conducted in a qualitative way, in which viable models are commonly
assessed based on visual inspection on the comparison with observations.
In this work we aim to perform a quantitative analysis using recent chemical data from the APOGEE survey in order to reveal the detailed star formation and enrichment history of the Galactic bulge.
The paper is organized as follows: We introduce the sample selection and the observed MDF and $\alpha$--DF in \S\ref{sec:data}. In \S\ref{sec:model} we describe the chemical evolution model used in this work and explain the systematic search for the best-fit model and the result of this search.
More discussions about interpretation of the best-fit model and room for improvement of the model are presented in \S\ref{sec:discussion}. We briefly summarize our results in \S\ref{sec:summary}.

\section{Data}
\label{sec:data}

\subsection{Sample selection}
\label{sec:sample_selection}

Our sample is selected from the Apache Point Observatory Galactic Evolution Experiment survey \citep[APOGEE;][]{majewski2017}, an on-going core project of the Sloan Digital Sky Survey IV \citep[SDSS-IV;][]{blanton2017}. APOGEE provides high-resolution, near-infrared spectra for $\sim$5$\times$10$^5$ stars throughout the Milky Way's bulge, disc, and halo \citep{zasowski2013, zasowski2017}, using custom spectrographs \citep{wilson2019} at 2.5~m Sloan Telescope at the Apache Point Observatory \citep{gunn2006}, and the 2.5~m Ir\'en\'ee du~Pont telescope \citep{bowen1973} at Las Campanas Observatory. Radial velocities, stellar parameters, and chemical abundances are determined from the spectra
, using custom pipelines described in \citet{nidever2015}, \citet{garcia2016} and \citet{jonsson2020}. In this work we use the parameters and abundances of a recent APOGEE internal release that includes reduced observations until November 2019 \citep[using the DR16 pipeline;][]{ahumada2020},
and spectro-photometric distances based on the procedure described in \citet{rojas2017}.

To remove stars with problematic spectra or unreliable parameter determinations, we adopt a signal-to-noise (SNR) cut of 60 and use the APOGEE spectroscopic flags\footnote{http://www.sdss.org/dr16/algorithms/bitmasks/} to select stars with EXTRATARG$==0$; the 1st, 4th, 9th, 16th and 17th bit of STARFLAG$==0$, {which correspond respectively to COMMISSIONING, LOW\_SNR, PERSIST\_HIGH, SUSPECT\_RV\_COMBINATION, and SUSPECT\_BROAD\_LINES}; and the 19th and 23th bit of ASCAPFLAG$==0$, {which correspond to METALS\_BAD and STAR\_BAD}.
Since elemental abundance determinations in APOGEE tend to be less reliable at low effective temperature, {$T_{\rm eff}$}, we further exclude stars with $T_{\rm eff}<3200$~K. We have tested that our results do not change significantly when adopting an effective temperature cut at 3500~K for the sample selection.

In this work, we analyze the [Mg/Fe]-[Fe/H] abundance plane. We adopt magnesium ([Mg/Fe]) to trace the $\alpha$ abundance, as it is shown to be the $\alpha$ element most reliably measured by the pipeline, with least dependence on effective temperature, \citep[][J{\"o}nsson et al.~in prep]{jonsson2018,rojas2019}.
{Note that the [Fe/H] and other elements to iron ratio, [X/Fe], of APOGEE stars are not populated when a \texttt{PARAM\_MISMATCH\_WARN} flag is set. This flag is activated when the difference between the [Fe/H] abundance determined from Fe spectral features and the [M/H] (the abundance of total metal content determined from the entire spectrum) exceeds 0.1$\;$dex. The reason for this discrepancy is not yet completely clear to the APOGEE team, but the stars for which this flag is set are generally metal rich ([Fe/H] $>$ 0.0) and cool ($T_{\rm eff}$ $<$ 4000 K). We obtain $15\%$ more stars when using [M/H] instead of [Fe/H] for the sample selection. Because stars with this mismatch are mostly metal rich, the sample selected in this work based on [Fe/H] is likely biased with underestimated surface density at the high metallicity end.
We will discuss the potential effect on our results in \S\ref{sec:model_bestfit}. }

We select stars in the Galactic bulge region on our side with Galactocentric distances within $3\;$kpc ($R_{\rm GC}<3\;$kpc).
Since the APOGEE sampling is irregular at different heights above the mid-plane, we focus on the mid-plane ($|Z|<0.5\;$kpc), where the observations tend to be distributed more homogeneously. Our stellar sample on the mid-plane contains 4454 stars. To obtain the integrated abundance distribution for the whole bulge region, we adopt the metallicity- and $\alpha$-dependent scale height derived by \citet[][see more details in \textsection2.3]{bovy2012b}.

Figure~\ref{xy-rz} shows the spatial distribution of APOGEE stars (grey dots) and our sample (orange dots) in the $R_{\rm GC}-Z$ plane (left-hand panel) and the $X-Y$ plane (right-hand panel). The positions of the Sun and the Galactic center are marked for reference.

\begin{figure*}
	\centering
	\includegraphics[width=0.95\textwidth]{xy-rz.png}
	\caption{Spatial distribution of stars in the $R_{\rm GC}-Z$ plane (left-hand panel) and $X-Y$ plane (right-hand panel). APOGEE stars meeting the flag and SNR requirements as described in the text (\S\ref{sec:sample_selection}) are indicated by gray dots; stars in our final bulge sample with further spatial and effective temperature limits are denoted as orange dots.}
	\label{xy-rz}
\end{figure*}

\subsection{Sample abundance distribution in the plane}
\label{sec:sample_abundances}

The left panel of Figure~\ref{mgfe-feh} shows the observed density distribution of [Mg/Fe]--[Fe/H] for the sample on the mid-plane of the bulge region.
A bimodal distribution is present, with a density gap at solar metallicity and $\rm [Mg/Fe] \sim 0.15$, suggesting at least two phases of star formation in the Galactic bulge. This gap in the [Mg/Fe]--[Fe/H] plane was also reported in \citet{rojas2019} based on APOGEE DR14 data. The metal-rich and metal-poor populations follow two [Mg/Fe]--[Fe/H] sequences with a small vertical offset in [Mg/Fe], consistent with the finding in \citet{hill2011}. The top and right sub-panels of Figure~\ref{mgfe-feh} show the projected distribution functions in [Fe/H] and [Mg/Fe] (the MDF and $\alpha$-DF), respectively, in black solid lines. The grey dashed lines indicate the MDF and $\alpha$-DF of the inner disk ($4<R_{\rm GC}<8\;$kpc) taken from \citet{lian2020b}.
The bulge MDF shows a strong, wide peak at [Fe/H]$\sim$0.37$\;$dex, while the $\alpha$-DF exhibits a clear bimodal distribution, with two peaks at [Mg/Fe]$\sim0.32$ and $\sim0.05\;$dex, respectively and an interesting bump at [Mg/Fe]$\sim0.2$.

A similar bimodal distribution has been found across a large portion of the disk from the inner Galaxy to the solar vicinity \citep{nidever2014,hayden2015}.
The metal-poor, high-$\alpha$ branch is nearly invariant in chemical abundance space throughout the Galaxy except that the density distribution is more extended in both [Fe/H] and [Mg/Fe] and peaks at a slightly lower [Fe/H] in the bulge than in the solar vicinity.
In contrast, the metal-rich, low-$\alpha$ branch shifts systematically toward lower metallicity with increasing Galactic radius. This radial variation pattern suggests that the Galactic bulge and disk share a common early star formation history but follow different evolution paths in the more recent past, when the chemical thin disk (i.e., the low-$\alpha$ branch) formed.

\begin{figure*}
	\centering
	\includegraphics[width=0.95\textwidth]{mgfe-feh-data.png}
	\caption{Density distribution of [Mg/Fe]--[Fe/H] for stars on the mid-plane (left-hand panel) and for the whole bulge region (right-hand panel). The projected distribution functions in [Fe/H] and [Mg/Fe] are normalized to the total number of stars and shown at the top and right sub-panels, respectively. Grey dashed lines in the left-hand sub-panels indicate the MDF and $\alpha$--DF in the inner disk taken from \citet{lian2020b}.
	In the right-hand panel, the dash-dotted and solid contours outline the integrated density distribution with and without consideration of disk flare, respectively. The red dash-dotted and blue dashed curves in sub-panels denote the corresponding MDF and $\alpha$-DF, respectively. Black solid curves are the original MDF and $\alpha$-DF on the mid-plane for reference.

	}
	\label{mgfe-feh}
\end{figure*}

\subsection{Integrated abundance distribution}

It has been shown that the vertical structure of stellar populations is dependent on their metallicity and [$\alpha$/Fe] \citep[e.g.,][]{bovy2012b,bovy2016}. More metal-poor and $\alpha$-enhanced stellar populations tend to have larger scale height and therefore dominate the stellar density distribution
at larger vertical distance from the mid-plane.
In the APOGEE survey, pointings targeting the inner Galaxy do not provide complete vertical or even angular coverage, but are localized to certain sky regions by design.
The effect is that the observational sampling of the APOGEE survey at different heights in the bulge region is complex and irregular.
To derive abundance distributions of the global bulge region, simply summing up APOGEE observations at different heights, given the significant vertical abundance gradient \citep{garcia-perez2018},
will introduce considerable bias for which it is difficult to account.

An effective alternative approach is to use the observed distribution in the mid-plane and account for the vertical distribution based on the known vertical structure of mono-abundance sub-populations. So far such mono-abundance analysis has only been conducted for the disk outside the bulge region \citep{bovy2012b,mackereth2017}.
We therefore extrapolate the abundance-dependent disk scale height from \citet{bovy2012b}, taking the effect of the flare into account, to obtain the vertical structure of mono-abundance sub-populations in the bulge region.
When no disk flare is considered, the scale heights of the high- and low-$\alpha$ populations ([Mg/Fe]$>0.15$ and [Mg/Fe]$<0.15$) are 562 and 233~pc on average, respectively.
For the flare, we adopt an average value of $R^{-1}_{\rm flare}=-0.1\;{\rm kpc^{-1}}$ for the low-$\alpha$ population \citep[see Fig.~12 in][]{bovy2016} and assume no flare for the high-$\alpha$ population as in that same paper.
This choice of flare strength for the low-$\alpha$ population implies a 1.52 times lower scale height at $R_{\rm GC}=3\;$kpc compared to at the solar radius.

For each [Fe/H]--[$\alpha$/Fe] bin, assuming an exponential disk in the vertical direction, the integrated surface density is derived following
\begin{equation}
\label{eqn:integrated_density}
 d_{\rm tot}=\frac{d_{0.5}}{1-e^{-\frac{0.5}{h_{\rm Z}}}},
\end{equation}
where $d_{0.5}$ is the observed surface density within $|Z|<0.5\;$kpc and $h_{\rm Z}$ is the scale height of the given mono-abundance sub-population.

It is worth pointing out that this approach implicitly assumes a homogeneous vertical structure within the bulge region. It is known that the MW's bulge contains a buckled bar which complicates the situation. The Galactic bar barely affects the vertical distribution of metal-poor, high-$\alpha$ stars as evidenced by the consistent scale height ($\sim500\;$pc, \citealt{portail2017}) with their counterparts in the disk \citep{bovy2012b}. Therefore we assume that the structure of these stars near the solar neighborhood is applicable to the bulge. In contrast, the metal-rich, low-$\alpha$ stars in the bar are generally thicker than their counterparts in the disk due to bar buckling \citep{wegg2013}. Thus extrapolation of the structure of these stars from the disk could possibly underestimate their scale height in the bulge and therefore underestimate their integrated surface density.

Note that the original abundance distribution on the mid-plane is equivalent to the integrated abundance distribution when assuming the same scale heights for all stars.
Although the scale height of metal-rich, low-$\alpha$ stars in the bar is higher than that in the disk, it is still lower than the scale height of the metal-poor, high-$\alpha$ stars \citep{portail2017}. Hence taking the original observations on the mid-plane to represent the whole bulge would overestimate the scale height and thus the relative surface density of metal-rich, low-$\alpha$ stars. Therefore the original and integrated MDF and $\alpha$-DF could be considered as two extreme cases with overestimated and underestimated metal-rich, low-$\alpha$ stars, respectively.
Since we lack information on the scale height of the mono-abundance population in the bulge, we use both the original and the integrated observations to constrain the bulge SFH as described in the next section.

The right panel of Figure~\ref{mgfe-feh} shows the integrated density distribution of [Mg/Fe]--[Fe/H] and the projected MDF and $\alpha$-DF for the whole bulge region. The density map and solid contours show the integrated density distribution without consideration of the disk flare, while the dash-dotted contours outline the distribution with the disk flare taken into account. Comparing to the density distribution on the mid-plane (left panels), the integrated one exhibits a more prominent metal-poor, high-$\alpha$ branch.
This is because the denominator in Eq.~\ref{eqn:integrated_density} is larger for the metal-poor population, given their larger scale height compared to the metal-rich population.

In Figure~\ref{mgfe-feh}, the integrated MDFs and $\alpha$-DFs with and without consideration of the disk flare are indicated as red dash-dotted and blue dashed curves, respectively.
They are significantly different from their equivalent distributions in the mid-plane (black solid) with relatively more metal-poor, high-$\alpha$ stars due to those populations larger scale height.
However, the integrated MDF and $\alpha$-DF barely change when taking the disk flare into account.
There is an interesting bump in the $\alpha$-DF at [Mg/Fe]$\sim$+0.2, which extends the metal-poor, high-$\alpha$ branch in the MDF. The existence of this population reduces the significance of the bimodality in both the MDF and $\alpha$-DF.

\section{Chemical evolution model}
\label{sec:model}

\subsection{Key ingredients}
\label{sec:model_ingredients}

To interpret observed chemical abundances in gaseous and stellar components of local galaxies, we developed a numerical chemical evolution model that accounts for three basic astrophysical processes involved in chemical enrichment: star formation, gas accretion, and galactic winds \citep{lian2018a,lian2018b,lian2018c,lian2019}.
The model was further enhanced to be able to predict multiple elemental abundances of individual stars, in order to compare chemical evolution tracks with the enormous sets of stellar observations in the Milky Way (\citealt{lian2020a}, submitted).
Here, we briefly introduce basic assumptions and inputs of the model and recommend
\citet{lian2018a} and \citet{lian2020a} to the reader for a more detailed description of the model development.

We adopt the Kennicutt-Schmidt (KS) star formation law \citep{kennicutt1998} that connects the gas mass with star formation rate (SFR), but we allow the coefficient that regulates star formation efficiency (SFE: SFR/M$_{\rm gas}$) to vary (Table~\ref{tab:model_parameters}). This free coefficient is normalized to the original coefficient of the KS law (i.e., $2.5\times10^{-4}$).
A \citet{kroupa2001} stellar initial mass function is adopted in the model.
Considering the relatively high mass of our Galaxy, and the fact that metal outflow strength scales inversely with galaxy mass \citep{chisholm2018,lian2018a}, we assume no outflow in the model for the inner Milky Way.

Metal production from asymptotic giant branch (AGB) stars, Type-Ia supernovae (SN-Ia), and core-collapse supernovae (CCSNe) are included in the model.
Instead of assuming instantaneous mixing, we take the lifespan of stars fully into account, with
metal-enriched material from evolved stars released to mix with ISM at the end of their evolution.
We use metallicity-dependent AGB yields from \citet{ventura2013} and SN-Ia yields from \citet{iwamoto1999}. To study the effect of uncertainty in yields on our results, we use two sets of metallicity-dependent CCSNe yields from \citet[][hereafter CL04]{chieffi2004} and \citet[][hereafter K06]{kobayashi2006} and explore the best-fit model for each of them.
To account for the time delay of SN-Ia since the birth of the SN-Ia-producing binary system,
we adopt an empirical power-law SN-Ia delay-time distribution (DTD) with slope $-1.1$, which is calibrated to observed SN-Ia rates \citep{maoz2012}.
We left the parameter $f_{\rm SN-Ia}$ free (Table~\ref{tab:model_parameters}), which regulates the global SN-Ia rate -- i.e.,
the fraction of stars within $3<M_*<16\;M_{\odot}$ that are SN-Ia progenitors -- to test whether it could be constrained by the abundance observations. {We adopt a minimum SN-Ia delay time of 35~Myr, which is widely used in chemical evolution models in the literature \citep[e.g.,][]{matteucci2009}. However, we will discuss the effect of adopting a different minimum SN-Ia delay time in \S\ref{sec:discussion_improvement} . }

\subsection{Underestimate of magnesium production}
\label{sec:model_mg}

When comparing observed $\alpha$ element-iron abundance ratios, [$\alpha$/Fe], with our chemical evolution models, we noticed evidence that the magnesium production in the K06 and CL04 CCSNe yields may be underestimated.
To illustrate this, Figure~\ref{mg-test} shows the comparison between models with these yields and observed abundances in the [O/Fe]--[Fe/H] and [Mg/Fe]--[Fe/H] planes.
Observations within $6<R_{\rm GC}<7\;$kpc (also with APOGEE) are used in this comparison for two reasons. On one hand, the observed sample at smaller $R_{\rm GC}$ in the inner Galaxy is dominated by luminous stars for which APOGEE's oxygen abundances are in some cases less reliable \citep{jonsson2018,zasowski2019}. On the other hand, stellar populations in the outer Galaxy are intrinsically different from the bulge region, with far fewer high-$\alpha$ stars. As a compromise, stars within $6<R_{\rm GC}<7\;$kpc are used for the comparison.

We calculate a test model for both the K06 (magenta lines) and CL04 (orange lines) yields, with adopted parameter values listed in Table~\ref{para} as the ``Test model''.
In Figure~\ref{mg-test}, it can be seen that at a given [Fe/H], the offsets between the models and data are clearly larger in [Mg/Fe] (right panel) than in [O/Fe] (left panel). For the model with K06 yields, when the model track in [O/Fe]--[Fe/H] is above the high-$\alpha$ branch, the track of the same model in [Mg/Fe]--[Fe/H] lies at lower [Mg/Fe].
We have confirmed that the offset in [Mg/Fe] with respect to [O/Fe] is not dependent on the choice of model parameters.

{This systematic shift between different $\alpha$ elements is not likely to be due to systematics in the observed abundances. Based on the comparison with optical studies, oxygen and especially magnesium are both precisely and accurately measured in APOGEE
(see the discussion in \citealt{jonsson2020}).
According to \citet{jonsson2020}, a small number of stars show signs of systematics in the abundance measurements; however, because these stars are concentrated within a limited range of chemical abundance, this behavior is unlikely to explain the discrepancy seen across a wide range of metallicity in Fig.~\ref{mg-test}.
}

The most likely explanation for this discrepancy is that the magnesium production with respect to oxygen in both the K06 and CL04 CCSNe yields are underestimated. The underestimation of Mg in CCSNe yields has been highlighted in \citet{thomas1998}. A similar underestimated Mg/O ratio is also reported in other chemical evolution models in the literature \citep{sukhbold2016,rybizki2017,limongi2018}.

{As a result, the chemical evolution model struggles to match the observed [Mg/Fe]--[Fe/H] distribution with the default CCSNe magnesium yields given the adopted minimum SN-Ia delay time of 35$\;$Myr.
If no correction is applied to the Mg yields, then in order to reproduce the enhanced [Mg/Fe] of the high-$\alpha$ population, one needs to adopt a much higher minimum SN-Ia delay time. As we will discuss later on in \S\ref{sec:discussion_improvement}, with this modification and the original yields, the models cannot attain as good a match to the data.}

In this work, we take the oxygen production as a benchmark to estimate the relative deficiency in Mg production.
We calculate the net offset between the model tracks and the data in [Mg/Fe], with respect to [O/Fe], at a given [Fe/H]. The average offsets in K06 and CL04 yields are $-0.13$ and $-0.15\;$dex, respectively. We enhance the Mg production by the same offsets in these two CCSNe yields. The dashed lines in the right panel of Fig.~\ref{mg-test} show the model tracks with enhanced Mg production. With this modification, the models are able to match the observed oxygen and magnesium abundances simultaneously with physically plausible parameters.
While we adopt a constant enhancement, the modification of magnesium production required to match the data is possibly metallicity dependent \citep{matteucci2020}.
It is worth pointing out that the difference in magnesium production between the CL04 and K06 yields is much higher than the modification adopted in this work. Therefore we expect the effect of using different yields on our results would be much higher than the modification of magnesium production.

\begin{figure*}
	\centering
	\includegraphics[width=16cm]{mg-yield-test-koba.png}
	\caption{Prediction of models with original or Mg-enhanced CCSNe yields in [O/Fe]--[Fe/H] (left-hand panel) and [Mg/Fe]--[Fe/H] (right-hand panel). Solid lines indicate models with original CCSNe yields while dashed lines are models with Mg-enhanced yields. Observed stellar abundances in the disk within $6<R_{\rm GC}<7\;$ kpc and $|Z|<4\;$ kpc are shown in the blue density map and black contours.}
	\label{mg-test}
\end{figure*}

\subsection{Star formation framework}
\label{sec:model_framework}

To conduct a systematic search for the best-fit model, we adopt a simple and flexible star formation framework modified from the one in \citet{lian2020b}.
, in which we propose a novel multi-phase star formation framework to explain the joint distribution function of [Fe/H] and [Mg/Fe] in the inner disk ($4<R_{\rm GC}<8$~kpc). This framework consists of an early and a recent starburst that are associated with two relatively brief gas accretion events.
The early starburst is responsible for the formation of the chemical thick disk on a short timescale, while during the recent starburst, a metal-poor, moderately $\alpha$-enhanced population formed in the thin disk.
Our model based on this multi-phase star formation history successfully reproduces the observed abundance distribution functions in the disk, especially the bimodal distribution in the [$\alpha$/Fe]--[Fe/H] plane. In this scenario, the well-known $\alpha$-bimodality arises due to a dramatically rapid decrease of star formation (i.e., star formation quenching) following the early starburst.

We adopt a similar multi-phase star formation framework to interpret the abundance distribution functions observed in the Galactic bulge.
Note that the metal-poor, moderately $\alpha$-enhanced population observed in the disk is absent in the bulge, suggesting that the inner Galaxy is not affected by the recent gas accretion event manifested in the disc. We therefore adopt a simpler version of the star formation framework presented in \citet{lian2020b}.
without the second gas infall and starburst event.
There are four free parameters that characterize this star formation framework:
\begin{itemize}
	\item Initial coefficient of the KS law during the early starburst, $Coe_{\rm burst}$;
	\item Time when gas accretion is switched off, $t_{\rm acc,tran}$;
	\item Time when coefficient of the KS law changes, $t_{\rm SFE, tran}$;
	\item The coefficient of the KS law after the early starburst, $Coe_{\rm post-burst}$.
\end{itemize}
The initial gas accretion rate is fixed to be 1.5$\;M_{\odot}{\rm yr^{-1}kpc^{-2}}$, and no gas accretion is assumed to occur after $t_{\rm acc,tran}$.
Thus in total, there are five free parameters in our model: four parameters describing the SFH plus $f_{\rm SN-Ia}$, the parameter regulating the global SN-Ia rate (\S\ref{sec:model_ingredients}).

\subsection{Best-fit model}
\label{sec:model_bestfit}

\subsubsection{Systematic search for the best-fit model}
Although abundance distribution functions have frequently been used to infer the bulge SFH \citep[e.g.,][]{haywood2018,matteucci2019}, the result is usually evaluated via a qualitative comparison to the data without a systematic, quantitative search across a range of possible solutions.
One goal of this work is to determine the SFH of the Galactic bulge region in a quantitative way by exploring an extensive parameter space and assessing the robustness of the result.

To this end, we create a 5-D parameter grid and run our chemical evolution model for each combination of the parameters. The range and step size of each parameter grid dimension are listed in Table~\ref{grid}. These values are chosen based on our experience to achieve a compromise between a wide coverage of parameter space and acceptable computation time.
In total, 216,000 models are calculated for each of the CL04 and K06 CCSNe yields.
Then, given the derived SFH and chemical evolution history for each model,
we generate a mock stellar catalog of surviving stars, considering observational abundance uncertainties.
A conservative uncertainty of 0.02~dex and 0.03~dex is adopted for [Fe/H] and [Mg/Fe], respectively, which is generally higher than the uncertainties in the data release catalog by a factor of $\sim$2.
Some of the parameters in the best-fit models change very slightly if different [Fe/H] and [Mg/Fe] uncertainties are adopted, but the main conclusions of this paper remain unchanged.

The abundance distribution functions derived from these mock catalogs are then compared to the observed APOGEE catalogs to assess the goodness of match to the data.
This assessment is performed with an unweighted quasi-$\chi^2$ diagnostic defined as:
\begin{equation}
\label{eqn:chi2}
\chi^2_v = \sum_{\rm i}\frac{(X_{\rm mod,i}-X_{\rm obs,i})^2}{N\bar{\sigma}^2},
\end{equation}
where the sum is calculated over all [Fe/H] and [Mg/Fe] bins ($i$), $X_{\rm mod}$ and $X_{\rm obs}$ are the values of the model and observed DFs in each bin, and $\bar{\sigma}$ is the average Poisson error over all these bins. The quantity $N$ stands for the degrees of freedom in the fitting, which equals the number of [Fe/H] and [Mg/Fe] bins (here, 50) minus the number of free model parameters (5), so for the results described below, $N=45$.
We adopt a constant error $\bar{\sigma}$ in order to use the constraining power of both density peaks and troughs in MDF and $\alpha$-DF.
The model that minimizes this reduced quasi-$\chi^2$ is considered the best-fit model. Since a constant error is used in the $\chi^2$ calculation, it does not affect the choice of the best-fit model.

\begin{table}
	\caption{Parameter ranges of the SFH model grid. Quantities in parentheses are the grid step size of each parameter.  The first four parameters are described in \S\ref{sec:model_framework} and the fifth in \S\ref{sec:model_ingredients}.
	}
	\label{grid}
	\centering
	\begin{tabular}{l c}
	\hline\hline
	 Parameter & Range \\
	 \hline
     $Coe_{\rm burst}$ & 0.1-1 (0.1) \\
     $Coe_{\rm post-burst}$ & 0.01-0.28 (0.01-0.02) \\
     $t_{\rm acc,tran}$ &  0.1-1.5 (0.1) \\
     $t_{\rm SFE, tran}$ & 0.6-2.2 (0.2) \\
     $f_{\rm SN-Ia}$ & 0.016-0.030 (0.2) \\

     \hline
	\end{tabular}\\
	\label{tab:model_parameters}
\end{table}

\subsubsection{The best-fit models}
Figure~\ref{chi2-k06} shows the $\chi_\nu^2$ as a function of each free parameter, with the other four parameters fixed to their best-fit values, for models with K06 yields. Figure~\ref{chi2-cl04} shows the equivalent results for the models with CL04 yields.
These best-fit parameters and minimized $\chi_\nu^2$ values are listed in Table~\ref{para}.

To illustrate the robustness of the fit, we include $\chi_\nu^2$ calculated for the abundance distribution functions on the mid-plane (black dashed line) and for the whole bulge region with and without considering the disk flare (magenta dash-dotted and blue solid lines, respectively).
It can be seen that considering the disk flare barely affects the fitting results. This constancy is expected given the very small change in abundance distribution functions when taking the disk flare into consideration (see Fig.~\ref{mgfe-feh}). It also implies that the fitting results are robust against small variations in the data.

A slightly different model, regardless of CCSNe yield choice, is preferred by the data on the mid-plane.
As discussed in \textsection2, compared to the region high above the disk, the mid-plane in the bulge contains relatively more metal-rich, low-$\alpha$ stars and fewer metal-poor, high-$\alpha$ stars.
As a result, models with less star formation in the initial starburst phase (i.e., lower $Coe_{\rm burst}$) or more star formation in the following prolonged secular phase (i.e., higher $Coe_{\rm post-burst}$) are favored by the unscaled mid-plane observations.
However, the shape of the $\chi_\nu^2$ distribution of these fits are rather similar, which suggests the model parameters are consistently constrained by the observed abundance distribution functions without strong degeneracy.
{Note that the sample selected in this work likely underestimates the surface density of the metal-richest stars ($\rm [Fe/H] > +0.2$; \S\ref{sec:sample_selection}), and thus the fraction of the metal-rich, low-$\alpha$ sequence. This implies more star formation in the secular evolution phase than the prediction of the best-fit models presented here. To account for that, a higher post-burst SFE would be required, and some residual gas accretion may also be needed to fuel more star formation. We expect the scale of change in these parameters would be comparable to the difference between the best-fitted models for the observations on the mid-plane and for the whole bulge region which also show considerably different fraction of metal-rich, low-$\alpha$ stars.}

Compared to K06 yields, the best-fit model based on CL04 yields has a relatively shorter initial accretion phase and therefore short starburst episode. Another difference is that the SFE is generally higher in the models with CL04 yields. These differences are mainly because [Mg/Fe] in the CL04 yields is $\sim0.3\;$dex lower than in the K06 yields.
With this lower [Mg/Fe] in CCSNe yields, a higher star formation rate is needed to reproduce the observed [Mg/Fe] at a given [Fe/H]. More star formation results in more rapid enrichment and therefore reaching the density gap in [Fe/H] earlier, when the transition of star formation occurs. {Although the best-fit values of the parameters are different, the qualitative pictures of the two best-fit star formation scenarios are the same, and the fit quality to the data is comparable. More discussion on this is presented in \S\ref{sec:discussion}.}

Given the consistent model parameters obtained for the original data on the mid-plane and the integrated ones for the whole bulge, hereafter we focus on the two best-fit models, one with K06 yield tables and one with CL04 yields, for the integrated abundance distribution in the bulge that accounts for the disk flare.

\begin{figure*}
	\centering
	\includegraphics[width=16cm]{chi2-function-koba.png}
	\caption{$\chi_\nu^2$ distribution as functions of each model parameter, with other parameters fixed to their best-fit values. The CCSNe yields from K06 are used in the models. Fitting results for observations in the mid-plane are shown as black dashed lines, and results for the integrated bulge region, with or without consideration of disk flare, are indicated as with magenta dash-dotted lines and blue solid lines, respectively.
	}
	\label{chi2-k06}
\end{figure*}

\begin{figure*}
	\centering
	\includegraphics[width=16cm]{chi2-function-cl04.png}
	\caption{Same structure as Fig.~\ref{chi2-k06} but using CL04 CCSNe yields in the model.}
	\label{chi2-cl04}
\end{figure*}

\subsubsection{Evolutionary history of the best-fit models}

Figure~\ref{sfh} shows the SFR (upper-left panel), gas accretion (upper-right panel), [Fe/H] (bottom-left panel), and [Mg/Fe] (bottom-right panel) histories of the two best-fit models based on CL04 (orange) and K06 (green) CCSNe yields. The initial gas accretion episode in both models lasts 1$\;$Gyr or less, suggesting the inner Galaxy has evolved largely as a closed-box system.
There are two major phases of star formation: a short initial starburst associated with the initial gas accretion, followed by a prolonged secular evolution phase with low-level star formation activity.
Similar to the disk model history in \citet{lian2020b}.
the early starburst is responsible for generating the metal-poor, high-$\alpha$ branch (i.e., the chemical/geometric inner thick disc), while the metal-rich, low-$\alpha$ branch is slowly built up during the later secular evolution. Therefore, the bulk of the old, metal-poor bulge stars is formed during the short-lived initial starburst.
This rapid gas accretion and starburst in the early Universe that forms the bulge is consistent with the picture of coalescing giant clumps in a turbulent disk that has been proposed as a means of bulge formation in the literature \citep[e.g.,][]{elmegreen2008,inoue2012,clarke2019}.
Other channels can also trigger a short early starburst, including an initial dissipative collapse and/or early hierarchical mergers, which could possibly be involved in the bulge formation.

An important episode shared by both modeled SFHs is the dramatic decrease in SFR right after the peak, which drops by one order of magnitude within 1~Gyr. This rapid SFR cessation (the so-called ``quenching'') process is critical to reproduce the observed  $\alpha$-bimodality \citep{haywood2018,lian2020b}. Interestingly, during this quenching episode, both models favor a two-step decrease of SFR with a relatively slower decline followed by a nearly straight drop.
The first step is caused by the shutdown of the external gas supply (transition of gas accretion is earlier than the SFE).
The second step, instead, is caused by switching the SFE to a lower values.
This two-step cessation of SFR is required to explain the existence of the density bump in the $\alpha$-DF.
The higher [Mg/Fe] of the metal-poor bulge stars, compared to stars in the thick disk in the solar vicinity, implies that this quenching process likely takes place earlier in the inner Galaxy and then propagates outwards, consistent with the inside-out quenching picture proposed to explain external galaxy observations \citep{ellison2018,wang2018,lin2019}.

It is worth pointing out that the well-known bimodality of the general galaxy population in the color--magnitude (or color--mass) diagram---the blue cloud versus the red sequence---is also largely shaped by a rapid SFR cessation process that quiescent galaxies have experienced \citep[e.g.,][]{schawinski2014,lian2016}. This similarity to the quenching-induced $\alpha$-bimodality in the Milky Way suggests an intriguing link between the evolutionary path of our Galaxy and the general galaxy population, and that the SFR quenching process has been playing a critical role in shaping galaxies observed today, including our Galaxy. We offer a heuristic speculation that, if the recent gas accretion event that supplied metal-poor gas and boosted star formation in the disk (mostly in the outer disc; \citealt{lian2020a}) did not happen, our Galaxy would not appear to be star-forming today but instead be quiescent and red, like a S0 galaxy.

\begin{table*}
	\caption{Best-fitting bulge SFH parameters (Table~\ref{tab:model_parameters}) for models based on CCSNe yields from \citet[][K06]{kobayashi2006} and \citet[][CL04]{chieffi2004}, fitted to observed abundances distributions on the mid-plane and in the integrated bulge region with and without considering the disk flare.

	The last column lists the reduced-$\chi^2$ value for each model. The parameters of the ``Test model'' discussed in \S\ref{sec:model_mg} and shown in Fig.\ref{mg-test} are also included. }
	\label{para}
	\centering
	\begin{tabular}{l c c c c c c}
		\hline\hline
		Model & $Coe_{\rm burst}$ & $Coe_{\rm post-burst}$ & $t_{\rm acc,tran}$ &  $t_{\rm SFE, tran}$ & $f_{\rm SN-Ia}$ & $\chi^2_v$ \\
		\hline
		& - & - & Gyr & Gyr & - & - \\
		\hline

	   K06 (mid-plane) & 0.20 & 0.06 & 0.9 & 1.9 & 0.030 & 21.5\\
	   K06 (no flare) & 0.20 & 0.04 & 1.1 & 2.1 & 0.028 & 21.3 \\
	   K06 (flare) & 0.20 & 0.04 & 1.1 & 2.1 & 0.028 & 20.9 \\
	   CL04 (mid-plane) & 0.50 & 0.24 & 0.3 & 0.9 & 0.024 & 20.7 \\
	   CL04 (no flare) & 0.60 & 0.24 & 0.5 & 1.1 & 0.022 & 14.3 \\
	   CL04 (flare) & 0.60 & 0.24 & 0.5 & 1.1 & 0.022 & 14.6 \\
	   \hdashline
	   Test model & 1.00 & 0.2 & 4.0 & 4.0 & 0.028 & - \\
		\hline
	\end{tabular}\\
\end{table*}

\begin{figure*}
	\centering
	\includegraphics[width=16cm]{sfh-models.png}
	\caption{SFR (upper-left panel), Gas accretion (upper-right panel), [Fe/H] (bottom-left panel), and [Mg/Fe] (bottom-right panel)  histories of best-fit models based on CCSNe yields from K06 (solid green) and CL04 (dashed orange).

    }
	\label{sfh}
\end{figure*}

\section{Discussion}
\label{sec:discussion}

\subsection{A three-phase SFH for the bulge}
\label{sec:discussion_sfh}

The SFHs of our best-fit models comprise three phases: an initial starburst, followed by a rapid star formation quenching episode, and a long-term secular evolution phase with a low SFR. Figure~\ref{track} shows the integrated [Mg/Fe]--[Fe/H] distribution of the whole bulge region and the predicted evolution tracks of our best-fit models. Green solid and orange dashed lines indicate the K06- and CL04-based models, respectively, as in Figure~\ref{sfh}. The large black circles highlight important transition times in the models, when the gas accretion switches off and the SFE ramps down. Other circles along the model tracks mark constant time intervals of 0.2$\;$Gyr for the evolution in the first 2$\;$Gyr.

Given this three-phase SFH, the [Mg/Fe]--[Fe/H] diagram can be separated into three general regimes (vertical dashed lines in Figure~\ref{track}). The metal-poor, high-$\alpha$ branch is mostly formed during the initial starburst. The following density valley between the high- and low-$\alpha$ branches is due to a rapid star formation quenching process. The metal-rich, low-$\alpha$ branch is gradually built through low SFR in the long-term secular evolution phase. This interpretation suggests that the density gap between the two main branches represents a transition between two modes of star formation in the bulge, from violent, bursty star formation at early times to long-term, low-state star formation more recently. This transition is qualitatively similar to the transition of disk formation from the chemical thick to the chemical thin disk \citep{haywood2018,lian2020b}
except for an earlier onset in the inner Galaxy.

\begin{figure*}
	\centering
	\includegraphics[width=16cm]{mgfe-feh-track.png}
	\caption{Integrated [Mg/Fe]--[Fe/H] distribution of the bulge region, overplotted with the evolution tracks of  best-fit models based on K06 (green solid) and CL04 (orange dashed) yields (\S\ref{sec:model_bestfit}). Orange and green circles on the model tracks mark  time intervals of 0.2$\;$Gyr for evolution in the first 2$\;$Gyr. Larger black circles highlight the important transitions of gas accretion and SFE in the model.
	Vertical dashed lines separate the three primary phases of star formation, as discussed in the text.

	}
	\label{track}
\end{figure*}

\subsection{Comparison with earlier bulge SFH measurements}
\label{sec:discussion_comparisons}

To account for the complex MDF in the bulge, a multi-phase SFH framework is generally adopted in the literature \citep{tsujimoto2012,grieco2012,haywood2018,matteucci2019}.
The metal-poor bulge stars are generally believed to build up quickly at early times, while the metal-rich stars formed with a relatively longer time scale \citep{tsujimoto2012,grieco2012}.
\citet{haywood2018} argued that a multi-phase SFH involving a rapid quenching stage adopted from \citet{snaith2015} could both explain well the global [Mg/Fe]--[Fe/H] relation and broadly reproduce the $\alpha$-bimodality in the inner Galaxy. By comparing the model-predicted MDF and $\alpha$-DF with observations, they concluded that the star formation quenching process is critical to reproduce the density dip between the high- and low-$\alpha$ branches. This empirical result is qualitatively consistent with our quantitative result.

However, the SFH proposed in \citet{haywood2018} is quantitatively different from the SFH in our best-fit model. The early starburst of \citet{haywood2018} lasts $\sim1\;$Gyr longer, and the quenching episode starts much later, than in our bulge SFH ($\sim4\;$Gyr versus $\sim1\;$Gyr after initial star formation). The model in \citet{haywood2018} also faces challenges reproducing the observed MDF and $\alpha$-DF in detail, such as the relative number of high- and low-$\alpha$ stars.

An alternative scenario to explain the MDF in the bulge is a bursty SFH, as proposed by \citet{matteucci2019}. In this scenario, the bulge SFH consists of multiple, short-lived starbursts that are separated by non-star forming gaps. This SFH is shown to be able to reproduce multiple peaks in the bulge MDF \citep{bensby2017}. A strong prediction of this scenario is a complex $\alpha$-DF, possibly with numerous peaks, that is inconsistent with the clear double-peaked bulge $\alpha$-DF as shown in this and other works.

In addition to chemical evolution modelling, a large number of bulge SFHs have been derived using different methodologies \citep{nataf2015}. Most previous work relying on chemical compositions or photometric age estimates suggest that the bulge is a generally old component \citep{zoccali2003,grieco2012}. However, the inferred age distribution of bulge planetary nebulae peaks around $\sim3\;$Gyr, suggesting the presence of a considerable fraction of young populations in the bulge \citep{buell2013,gesicki2014},
a finding consistent with other photometric and spectoscopic studies \citep[e.g.,][]{bensby2013,bernard_2018_bulgeSFH}, and also present in barred galaxies in cosmological simulations of \citep{fragkoudi2020}.
A recent study of spectroscopic ages of bulge stars (Hasselquist et al., in prep) also confirms a non-negligible fraction of young stars (age $\sim$ 2--5~Gyr) that are preferentially found in the plane.

The bulk bulge population predicted by our best-fit SFH is generally old, with a mass-weighted integrated stellar age of 10.5 and 9.7$\;$Gyr for models with CL04 and K06 yields, respectively.
However, we note that a significant fraction of young stars is also predicted. Figure~\ref{age-dtr} shows the predicted age distribution of our best-fit models assuming an age uncertainty of 0.25$\;$dex.
The left-hand panel indicates the global age distribution while the right-hand sub-panels show age distribution in four metallicity bins that correspond to the metallicity range of each episode in the best-fit SFH. The two intermediate metallicity bins correspond to the two stages of the star formation quenching phase.
The ages are derived based on a simple assumption that the star formation in the protogalaxy that would become the Milky Way started at 13.7$\;$Gyr ago.

Given an age uncertainty of 0.25$\;$dex, the number fraction of living young stars (age$<5\;$Gyr) predicted by the best-fit models are 11.3\% and 15.8\% for CL04 and K06 CCSNe yields, respectively. Note that this fraction is subject to change when different age uncertainty is assumed. For example, a considerably higher fraction of 25-29\% is expected when assuming an age uncertainty of 0.4$\;$dex. Comparing to the age distribution predicted by four distinct bulge SFHs as shown in Fig.~1 in \citet{nataf2015}, our result is more consistent with the age distribution derived by \citet{bensby2013}.

\begin{figure*}
	\centering
	\includegraphics[width=16cm]{age-dtr-feh.png}
	\caption{Global age distribution (left-hand panel) and that in four metallicity bins (right-hand panels) predicted by the best-fit models based on CL04 and K06 CCSNe yields. A notable fraction of young stars (age$<5\;$Gyr), which are preferentially very metal-rich, is present.
	}
	\label{age-dtr}
\end{figure*}

\subsection{Room for improvement in the model}
\label{sec:discussion_improvement}

Figure~\ref{mgfe-feh-model} shows the density distribution of [Mg/Fe]--[Fe/H] as predicted by our best-fit models. The observed stellar distribution is shown as orange contours for comparison.
Both models reproduce the global [Mg/Fe]--[Fe/H] trend and the $\alpha$-bimodality remarkably well. A noticeable difference is the narrowness of the tracks predicted by the models, compared to the data, in both low- and high-$\alpha$ branches. Figure~\ref{mdf-comp} compares the observed and predicted MDF and $\alpha$-DF for the best-fit models. The matches to the observed abundance distribution functions are equally good for these two models. The bimodal $\alpha$-DF is reproduced remarkably well, including the bump at [Mg/Fe]$\sim$0.2. The bimodality in [Fe/H] predicted by the models is also less significant than that in [Mg/Fe], with an extended tail at the metal-poor end, which is in good consistency with observations.

Nevertheless, there remains room for improvement in the model fits to the abundance distribution functions. The low metallicity peaks in the MDF predicted by the models are slightly offset towards higher metallicity ($\sim-0.6$ versus observational $\sim-0.7$). This is a result of trade-offs to match the [Mg/Fe] distribution at the same time.
Another under-performance in the models is that the predicted width of the low-$\alpha$ population in the [Mg/Fe] distribution is narrower than the data. This is also visible in Fig.~\ref{mgfe-feh-model}. One possible reason for this discrepancy could be that the observational uncertainties adopted in the model simulations are still underestimated. Another possibility is that the residual scatter is intrinsic, which suggests that the evolution path may not be smooth and unique but rather (at least somewhat) stochastic and inhomogeneous. The SFH implied by the best-fit models in this work could be considered as an average solution. To account for stochasticity in SFH requires a much more complex star formation framework, which is beyond the scope of this paper aiming at understanding the global bulge SFH.

{Although our sample does not cover the low metallicity range ($\rm [Fe/H] < -1.2$), our best-fit models, especially the one with the K06 yields, keep increasing with decreasing [Fe/H] at $\rm [Fe/H] < -1$. This is not consistent with the observed flat [Mg/Fe] plateau at $\rm [Fe/H] < -1$ in many other previous works \citep[e.g.][]{mcwilliam1995}.
The models also seem to not reproduce well the ``knee'' (change of slope) at $\rm [Fe/H]\sim-0.6$.
We hypothesize this mismatch is possibly due to the original [Mg/Fe] ratio in the CCSNe yields at low metallicity not matching the [Mg/Fe] value of the plateau, and/or the adopted minimum SN-Ia delay time being too low.
}

{To test this latter idea, we calculated another set of test models, named `late-Ia' models here, for each of the CCSNe yields, adopting a higher minimum SN-Ia delay time (150$\;$Myr for CL04 and 70$\;$Myr for the K06 yields), the default yield tables (i.e., no Mg enhancement), and lower fractions of SN-Ia progenitors (0.012 for CL04 and 0.018 for K06 yields), which are required for the default Mg production to match the [Mg/Fe] of the low-$\alpha$ sequence. The value of minimum SN-Ia delay time is adopted for an example to demonstrate the impact of changing SN-Ia DTD, and all the other changes have to happen at the same time to maintain the [Mg/Fe]--[Fe/H] trend.
The SFH of these late-Ia models are adopted as the same as the corresponding best-fit models.
Figure~\ref{a1} shows the evolutionary track of the late-Ia models in the [Mg/Fe]--[Fe/H] plane. The predicted MDF and $\alpha$-DF, compared with the observed data, are shown in Figure~\ref{a2}. It can be seen that these late-Ia models reproduce well the global [Mg/Fe]--[Fe/H] trend, and indeed show a more pronounced ``knee'' at $\rm [Fe/H]\sim-0.6$ and tend to flatten at lower metallicity.}

{However, the predicted abundance distribution functions do not match the observed ones as well as the best-fit models with a lower minimum SN-Ia delay time. The predicted MDF shifts systematically towards lower [Fe/H]. We have tested that the under-performance of late-Ia models remains when allowing SFH to vary.
Since we focus on APOGEE observations in this work, we present the best-fit models to those data as our fiducial models. However, these late-Ia models illustrate one direction of future improvements when more observations of metal-poor stars are used to constrain the chemical evolution history.}

\begin{figure*}
	\centering
	\includegraphics[width=18cm]{mgfe-feh-model.png}
	\caption{Density distribution of [Mg/Fe]--[Fe/H] as predicted by the best-fit models for CL04 yields (left-hand panel) and K06 yields (right-hand panel). The observed  distribution for the whole bulge region is overplotted as orange dashed contours for comparison.}
	\label{mgfe-feh-model}
\end{figure*}

\begin{figure*}
	\centering
	\includegraphics[width=16cm]{mdf-comp.png}
	\caption{Comparison between the observed and modeled [Fe/H] (left-hand panel) and [Mg/Fe] (right-hand panel) distribution functions. The gray shaded region indicates the 3$\sigma$ scatter of the observed MDF and $\alpha$-DF, representing the average Poison uncertainty used to calculated the $\chi_\nu^2$ in Eq.~\ref{eqn:chi2}.

    }
	\label{mdf-comp}
\end{figure*}

\begin{figure}
	\centering
	\includegraphics[width=10cm,viewport=15 0 550 430,clip]{mgfe-feh-track-sn1atest.png}
	\caption{The same as Fig.\ref{track} except that a higher minimum SN-Ia delay time and default Mg production are assumed for the models.}
	\label{a1}
\end{figure}

\begin{figure*}
	\centering
	\includegraphics[width=16cm]{mdf-comp-sn1atest.png}
	\caption{The same as Fig.\ref{mdf-comp} except that a higher minimum SN-Ia delay time and default Mg production are assumed for the models.}
	\label{a2}
\end{figure*}

\iffalse
\subsection{Uncertainties in the best-fit parameters}
We estimate the uncertainties of parameters in the best-fit models by analyzing the distribution function of $\chi_\nu^2$ ($\chi^2$ derived from Eq. (2) divided by degree of from) around the minimum point.
Figure~\ref{chi2-k06} and \ref{chi2-cl04} show the $\chi_\nu^2$ value as a function of each free parameter
with other four parameters fixed to the best-fit values for models with K06 and CL04 yields, respectively.
To illustrate the robustness of these best-fit parameters, we include $\chi_\nu^2$ calculated for abundance distribution functions without correction for vertical distribution (orange dashed line), with correction but not considering disk flare (red dash-dotted line), and with correction considering disk flare (blue solid line).
It can be seen that considering disk flare or not barely affect the fitting results. This is expected given the small change in abundance distribution functions when taking disk flare into consideration (see Fig.~\ref{mgfe-feh}). It also implies that the fitting results are robust against small variations in data.

In contrast, in the case of data without correction, a considerably different model is preferred.
As discussed in \textsection2, compared to the bulge region high above the disk, the mid-plane have relatively more metal-rich, low-$\alpha$ stars and less metal-poor, high-$\alpha$ stars.
As a result, models with less star formation in the initial starburst phase with lower initial SFE and more star formation in the following prolonged suppression phase with higher post-starburst SFE are favored by the mid-plane observations.
However, the shape of difference distribution among fittings to different data are similar which suggests model parameters are well constrained by the observed abundance distribution functions without strong degeneracy.

The degree of freedom in the our fitting to the observed abundance distributions equals to the number of [Fe/H] and [Mg/Fe] bins (40) minus the number of free model parameters (5) which is 35. Given this degree of freedom and the corresponding $\chi_\nu^2$ probability density function, we estimate the 99.5\% probability of $\chi_\nu^2$ which are indicated as black dashed line in each panel in Fig.~\ref{chi2-k06} and Fig.~\ref{chi2-cl04}.
The parameter value with such reduced-$chi^2$ are considered as the 3$\sigma$ uncertainty which are also included in Table~\ref{para}.
\fi

\section{Summary}
\label{sec:summary}

We investigate the star formation history of the Galactic bulge by quantitatively modelling the [Fe/H] and [Mg/Fe] distribution functions simultaneously and matching to observations from the APOGEE survey.
To avoid potential bias introduced by the varying sampling of the APOGEE survey at different vertical distances, we select our sample close to the mid-plane ($|z|<0.5\;$kpc) and account for the vertical structure of mono-abundance sub-populations as in \citet{bovy2012b,bovy2016} to infer integrated abundance distribution functions. A major effect of accounting for the vertical structure is an increase in the number of metal-poor, high-$\alpha$ stars with respect to the metal-rich, low-$\alpha$ stars because of the former's larger scale height.

In both the raw and integrated [Mg/Fe] distribution functions, a clear bimodal distribution is present, with peaks at $+0.03$ and $+0.33\;$~dex and a gap at $+$0.15~dex. An interesting weak bump is also present at $+$0.2~dex.
The integrated [Fe/H] distribution function exhibits three peaks at $\sim-0.67$, $\sim-0.27$ and $\sim+0.37\;$~dex with a dip at solar metallicity.

To extract SFH information from abundance distribution functions, we apply our numerical chemical evolution model based on a star formation framework that allows single or multiple phases of star formation.
To fully exploit the data, we explore a wide parameter space and systematically search for the best-fit model based on a quantitative assessment of fit quality.
We obtain the best-fit model for the observations on the mid-plane as well as for the integrated ones across the global bulge region.
To test the effect of stellar yields on the results, we run two sets of models with different CCSNe yields (from K06 and CL04) and describe both sets of best-fit results.

The best-fit models, regardless of the CCSNe yields used,
suggest a three-phase SFH that consists of an early and intense starburst, followed by a rapid star formation quenching episode (one order of magnitude decrease in SFR within 1$\;$Gyr), and a prolonged stage of low star formation. The early starburst is responsible for the formation of the metal-poor, high-$\alpha$ branch; during the prolonged third phase, the metal-rich, low-$\alpha$ branch is gradually built up. These two star formation phases are connected by a rapid star formation quenching episode (an order of magnitude decrease in SFR within $\sim1\;$Gyr) that occurs in the early Universe. Because of the rapid decrease in SFR, the [Mg/Fe] ratio in the ISM also drops significantly and rapidly, leaving few stars formed with intermediate [Mg/Fe].
This quenching process is therefore critical to reproduce the $\alpha$-bimodality observed in the Milky Way, not only in the bulge region but also in the disk \citep{lian2020b}.
Future work will further address whether this quenching process took place simultaneously throughout the Galaxy or propagated inside-out or outside-in by studying radial variation of abundance distribution functions in more detail.
Given the best-fit SFH in this work, a non-negligible fraction of young stars (age$<5\;$Gyr) is expected to exist in the bulge, a prediction that is broadly consistent with several recent age determinations of bulge stars \citep[e.g.,][Hasselquist et al., in prep]{bernard_2018_bulgeSFH}.

There are mild differences between the best-fit models when using K06's or CL04's CCSNe yields. In general, the model with CL04 yields requires a higher SFE in both star formation phases to balance the lower intrinsic [Mg/Fe] in the yields table. However, the main characteristics of the three-phase SFH are not affected by the choice of CCSNe yields. We conclude that the uncertainty in CCSNe yields mildly affects the best-fit parameter values but does not significantly weaken the constraining power of observed abundance distributions on the bulge's SFH.

\section*{Acknowledgements}
ARL acknowledge partial financial support to the APOGEE2-S survey through the QUIMAL project 130001 and FONDECYT project 1170476. DMN acknowledges support from NASA under award Number 80NSSC19K0589.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges
support and resources from the Center for High-Performance Computing at
the University of Utah. The SDSS web site is www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science,
Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics,
Instituto de Astrof\'isica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) /
University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory,
Leibniz Institut f\"ur Astrophysik Potsdam (AIP),
Max-Planck-Institut f\"ur Astronomie (MPIA Heidelberg),
Max-Planck-Institut f\"ur Astrophysik (MPA Garching),
Max-Planck-Institut f\"ur Extraterrestrische Physik (MPE),
National Astronomical Observatories of China, New Mexico State University,
New York University, University of Notre Dame,
Observat\'ario Nacional / MCTI, The Ohio State University,
Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group,
Universidad Nacional Aut\'onoma de M\'exico, University of Arizona,
University of Colorado Boulder, University of Oxford, University of Portsmouth,
University of Utah, University of Virginia, University of Washington, University of Wisconsin,
Vanderbilt University, and Yale University.

\section*{Data Availability Statements}
The data underlying this article is from an internal incremental release of SDSS-IV/APOGEE survey, following the SDSS-IV Data Release 16. This incremental release is not publicly available now but will be included in the final public data release of SDSS-IV in 2021.
The chemical evolution model results are available upon request.

\bibliographystyle{mnras}

redpajama_set_name: RedPajamaArXiv


1. Fernando Alonso - Renault Fernando Alonso wins his first race of the season ahead of Jarno Trulli and Nick Heidfeld.
Juan Pablo Montoya, Ralf Schumacher, David Coulthard, Michael Schumacher and Christian Klien make the top 8.
The 2 BAR Hondas retire with engine failures early in the race.
Fisichella struggling with his tyres tangles with Webber and both retire.
Kimi Raikkonen suffers a puncture but manages 9th.
Ferrari struggle on the Bridgestones with Michael finishing 7th and Barrichello retiring.
Toyota, McLaren, Red Bull and Jordan managed to finish both cars.
13 cars were classified as finishers.
1. Michael Schumacher - Ferrari Michael Schumacher wins the race ahead of Juan Pablo Montoya and Jenson Button. Williams and BAR were closer to Ferrari here than they were in Melbourne.
Michael led from start to finish. Mark Webber starting from second has a bad start, struggles with Ralf Schumacher during the race and eventually spins and retires. Montoya takes second at the start but can't catch Michael. Button starting from 6th finishes on the podium. The first for him.
Fernando Alonso starting from the back after spinning in qualifying moves into 8th in just a few laps but can't get ahead of Coulthard. He finishes in 7th.
Ferrari, BAR, Renault, Sauber, Toyota and Minardi managed to finish both cars.
16 cars were classified as finishers.


text:
Периодаты меди(II) — неорганические соединения, соли металла меди и иодной кислоты.

Метапериодаты
 Метапериодат меди(II) Cu(IO) — синие кристаллы, образуются при кипячении раствора ацетата меди с ортопериодной кислотой:

Ортопериодаты
 Кислый ортопериодат меди(II) CuHIO·nHO (где  = 1, 2, 3) — зеленовато-жёлтые кристаллы. Образуется при действии ортопериодной кислоты на дигидроксодикарбонат меди(II):

 или действие ортопериодата натрия на избыток сульфата меди:

 Ортопериодат меди(II) Cu(IO)·nHO (где  = 3, 5, 7) — зелёные кристаллы.

Дипериодаты
 Дипериодат меди(II) CuIO·6HO — зеленые кристаллы, получают при упаривании суспензии гидроксида меди(II) в ортопериодной кислоте:

Литература

медь
Соединения меди
Списки химических соединений