text:
#ifndef _LOG_H_
#define	_LOG_H_

/*******************************************************
 * DBREG:
 *	The DB file register code keeps track of open files.  It's stored
 *	in the log subsystem's shared region, and so appears in the log.h
 *	header file, but is logically separate.
 *******************************************************/
/*
 * The per-process table that maps log file-id's to DB structures.
 */
typedef	struct __db_entry {
	DB	*dbp;			/* Open dbp for this file id. */
	int	deleted;		/* File was not found during open. */
} DB_ENTRY;

/*
 * FNAME --
 *	File name and id.
 */
struct __fname {
	SH_TAILQ_ENTRY q;		/* File name queue. */

	int32_t   id;			/* Logging file id. */
	DBTYPE	  s_type;		/* Saved DB type. */

	roff_t	  name_off;		/* Name offset. */
	db_pgno_t meta_pgno;		/* Page number of the meta page. */
	u_int8_t  ufid[DB_FILE_ID_LEN];	/* Unique file id. */

	u_int32_t create_txnid;		/*
					 * Txn ID of the DB create, stored so
					 * we can log it at register time.
					 */
	int	  is_durable;		/* Is this file durable or not. */
};

/* File open/close register log record opcodes. */
#define	DBREG_CHKPNT	1		/* Checkpoint: file name/id dump. */
#define	DBREG_CLOSE	2		/* File close. */
#define	DBREG_OPEN	3		/* File open. */
#define	DBREG_RCLOSE	4		/* File close after recovery. */

/*******************************************************
 * LOG:
 *	The log subsystem information.
 *******************************************************/
struct __db_log;	typedef struct __db_log DB_LOG;
struct __hdr;		typedef struct __hdr HDR;
struct __log;		typedef struct __log LOG;
struct __log_persist;	typedef struct __log_persist LOGP;

#define	LFPREFIX	"log."		/* Log file name prefix. */
#define	LFNAME		"log.%010d"	/* Log file name template. */
#define	LFNAME_V1	"log.%05d"	/* Log file name template, rev 1. */

#define	LG_MAX_DEFAULT		(10 * MEGABYTE)	/* 10 MB. */
#define	LG_MAX_INMEM		(256 * 1024)	/* 256 KB. */
#define	LG_BSIZE_DEFAULT	(32 * 1024)	/* 32 KB. */
#define	LG_BSIZE_INMEM		(1 * MEGABYTE)	/* 1 MB. */
#define	LG_BASE_REGION_SIZE	(60 * 1024)	/* 60 KB. */

/*
 * DB_LOG
 *	Per-process log structure.
 */
struct __db_log {
/*
 * These fields need to be protected for multi-threaded support.
 *
 * !!!
 * As this structure is allocated in per-process memory, the mutex may need
 * to be stored elsewhere on architectures unable to support mutexes in heap
 * memory, e.g., HP/UX 9.
 */
	DB_MUTEX  *mutexp;		/* Mutex for thread protection. */

	DB_ENTRY *dbentry;		/* Recovery file-id mapping. */
#define	DB_GROW_SIZE	64
	int32_t dbentry_cnt;		/* Entries.  Grows by DB_GROW_SIZE. */

/*
 * These fields are always accessed while the region lock is held, so they do
 * not have to be protected by the thread lock as well.
 */
	u_int32_t lfname;		/* Log file "name". */
	DB_FH	 *lfhp;			/* Log file handle. */

	u_int8_t *bufp;			/* Region buffer. */

/* These fields are not protected. */
	DB_ENV	 *dbenv;		/* Reference to error information. */
	REGINFO	  reginfo;		/* Region information. */

#define	DBLOG_RECOVER		0x01	/* We are in recovery. */
#define	DBLOG_FORCE_OPEN	0x02	/* Force the DB open even if it appears
					 * to be deleted. */
	u_int32_t flags;
};

/*
 * HDR --
 *	Log record header.
 */
struct __hdr {
	u_int32_t prev;			/* Previous offset. */
	u_int32_t len;			/* Current length. */
	u_int8_t  chksum[DB_MAC_KEY];	/* Current checksum. */
	u_int8_t  iv[DB_IV_BYTES];	/* IV */
	u_int32_t orig_size;		/* Original size of log record */
	/* !!! - 'size' is not written to log, must be last in hdr */
	size_t	  size;			/* Size of header to use */
};

/*
 * We use HDR internally, and then when we write out, we write out
 * prev, len, and then a 4-byte checksum if normal operation or
 * a crypto-checksum and IV and original size if running in crypto
 * mode.  We must store the original size in case we pad.  Set the
 * size when we set up the header.  We compute a DB_MAC_KEY size
 * checksum regardless, but we can safely just use the first 4 bytes.
 */
#define	HDR_NORMAL_SZ	12
#define	HDR_CRYPTO_SZ	12 + DB_MAC_KEY + DB_IV_BYTES

struct __log_persist {
	u_int32_t magic;		/* DB_LOGMAGIC */
	u_int32_t version;		/* DB_LOGVERSION */

	u_int32_t log_size;		/* Log file size. */
	u_int32_t mode;			/* Log file mode. */
};

/*
 * LOG --
 *	Shared log region.  One of these is allocated in shared memory,
 *	and describes the log.
 */
struct __log {
	/*
	 * Due to alignment constraints on some architectures (e.g. HP-UX),
	 * DB_MUTEXes must be the first element of shalloced structures,
	 * and as a corollary there can be only one per structure.  Thus,
	 * flush_mutex_off points to a mutex in a separately-allocated chunk.
	 */
	DB_MUTEX fq_mutex;		/* Mutex guarding file name list. */

	LOGP	 persist;		/* Persistent information. */

	SH_TAILQ_HEAD(__fq1) fq;	/* List of file names. */
	int32_t	fid_max;		/* Max fid allocated. */
	roff_t	free_fid_stack;		/* Stack of free file ids. */
	u_int	free_fids;		/* Height of free fid stack. */
	u_int	free_fids_alloced;	/* N free fid slots allocated. */

	/*
	 * The lsn LSN is the file offset that we're about to write and which
	 * we will return to the user.
	 */
	DB_LSN	  lsn;			/* LSN at current file offset. */

	/*
	 * The f_lsn LSN is the LSN (returned to the user) that "owns" the
	 * first byte of the buffer.  If the record associated with the LSN
	 * spans buffers, it may not reflect the physical file location of
	 * the first byte of the buffer.
	 */
	DB_LSN	  f_lsn;		/* LSN of first byte in the buffer. */
	size_t	  b_off;		/* Current offset in the buffer. */
	u_int32_t w_off;		/* Current write offset in the file. */
	u_int32_t len;			/* Length of the last record. */

	DB_LSN	  active_lsn;		/* Oldest active LSN in the buffer. */
	size_t	  a_off;		/* Offset in the buffer of first active
					   file. */

	/*
	 * Due to alignment constraints on some architectures (e.g. HP-UX),
	 * DB_MUTEXes must be the first element of shalloced structures,
	 * and as a corollary there can be only one per structure.  Thus,
	 * flush_mutex_off points to a mutex in a separately-allocated chunk.
	 *
	 * The s_lsn LSN is the last LSN that we know is on disk, not just
	 * written, but synced.  This field is protected by the flush mutex
	 * rather than by the region mutex.
	 */
	int	  in_flush;		/* Log flush in progress. */
	roff_t	  flush_mutex_off;	/* Mutex guarding flushing. */
	DB_LSN	  s_lsn;		/* LSN of the last sync. */

	DB_LOG_STAT stat;		/* Log statistics. */

	/*
	 * !!! - NOTE that the next 7 fields, waiting_lsn, verify_lsn,
	 * max_wait_lsn, maxperm_lsn, wait_recs, rcvd_recs,
	 * and ready_lsn are NOT protected
	 * by the log region lock.  They are protected by db_rep->db_mutexp.
	 * If you need access to both, you must acquire db_rep->db_mutexp
	 * before acquiring the log region lock.
	 *
	 * The waiting_lsn is used by the replication system.  It is the
	 * first LSN that we are holding without putting in the log, because
	 * we received one or more log records out of order.  Associated with
	 * the waiting_lsn is the number of log records that we still have to
	 * receive before we decide that we should request it again.
	 *
	 * The max_wait_lsn is used to control retransmission in the face
	 * of dropped messages.  If we are requesting all records from the
	 * current gap (i.e., chunk of the log that we are missing), then
	 * the max_wait_lsn contains the first LSN that we are known to have
	 * in the __db.rep.db.  If we requested only a single record, then
	 * the max_wait_lsn has the LSN of that record we requested.
	 */
	DB_LSN	  waiting_lsn;		/* First log record after a gap. */
	DB_LSN	  verify_lsn;		/* LSN we are waiting to verify. */
	DB_LSN	  max_wait_lsn;		/* Maximum LSN requested. */
	DB_LSN	  max_perm_lsn;		/* Maximum PERMANENT LSN processed. */
	u_int32_t wait_recs;		/* Records to wait before requesting. */
	u_int32_t rcvd_recs;		/* Records received while waiting. */
	/*
	 * The ready_lsn is also used by the replication system.  It is the
	 * next LSN we expect to receive.  It's normally equal to "lsn",
	 * except at the beginning of a log file, at which point it's set
	 * to the LSN of the first record of the new file (after the
	 * header), rather than to 0.
	 */
	DB_LSN	  ready_lsn;

	/*
	 * During initialization, the log system walks forward through the
	 * last log file to find its end.  If it runs into a checkpoint
	 * while it's doing so, it caches it here so that the transaction
	 * system doesn't need to walk through the file again on its
	 * initialization.
	 */
	DB_LSN	cached_ckp_lsn;

	u_int32_t regionmax;		/* Configured size of the region. */

	roff_t	  buffer_off;		/* Log buffer offset in the region. */
	u_int32_t buffer_size;		/* Log buffer size. */

	u_int32_t log_size;		/* Log file's size. */
	u_int32_t log_nsize;		/* Next log file's size. */

	/*
	 * DB_LOG_AUTOREMOVE and DB_LOG_INMEMORY: not protected by a mutex,
	 * all we care about is if they're zero or non-zero.
	 */
	int	  db_log_autoremove;
	int	  db_log_inmemory;

	u_int32_t ncommit;		/* Number of txns waiting to commit. */
	DB_LSN	  t_lsn;		/* LSN of first commit */
	SH_TAILQ_HEAD(__commit) commits;/* list of txns waiting to commit. */
	SH_TAILQ_HEAD(__free) free_commits;/* free list of commit structs. */

	/*
	 * In-memory logs maintain a list of the start positions of all log
	 * files currently active in the in-memory buffer.  This is to make the
	 * lookup from LSN to log buffer offset efficient.
	 */
	SH_TAILQ_HEAD(__logfile) logfiles;
	SH_TAILQ_HEAD(__free_logfile) free_logfiles;

#ifdef HAVE_MUTEX_SYSTEM_RESOURCES
#define	LG_MAINT_SIZE	(sizeof(roff_t) * DB_MAX_HANDLES)

	roff_t	  maint_off;		/* offset of region maintenance info */
#endif
};

/*
 * __db_commit structure --
 *	One of these is allocated for each transaction waiting to commit.
 */
struct __db_commit {
	DB_MUTEX	mutex;		/* Mutex for txn to wait on. */
	DB_LSN		lsn;		/* LSN of commit record. */
	SH_TAILQ_ENTRY	links;		/* Either on free or waiting list. */

#define	DB_COMMIT_FLUSH		0x0001	/* Flush the log when you wake up. */
	u_int32_t	flags;
};

/*
 * Check for the proper progression of Log Sequence Numbers.
 * If we are rolling forward the LSN on the page must be greater
 * than or equal to the previous LSN in log record.
 * We ignore NOT LOGGED LSNs.  The user did an unlogged update.
 * We should eventually see a log record that matches and continue
 * forward.
 * If truncate is supported then a ZERO LSN implies a page that was
 * allocated prior to the recovery start pont and then truncated
 * later in the log.  An allocation of a page after this
 * page will extend the file, leaving a hole.  We want to
 * ignore this page until it is truncated again.
 *
 */

#ifdef HAVE_FTRUNCATE
#define	CHECK_LSN(redo, cmp, lsn, prev)					\
	if (DB_REDO(redo) && (cmp) < 0 &&				\
	    !IS_NOT_LOGGED_LSN(*(lsn)) && !IS_ZERO_LSN(*(lsn))) {	\
		ret = __db_check_lsn(dbenv, lsn, prev);			\
		goto out;						\
	}
#else
#define	CHECK_LSN(redo, cmp, lsn, prev)					\
	if (DB_REDO(redo) && (cmp) < 0 && !IS_NOT_LOGGED_LSN(*(lsn))) {	\
		ret = __db_check_lsn(dbenv, lsn, prev);			\
		goto out;						\
	}
#endif

/*
 * Helper for in-memory logs -- check whether an offset is in range
 * in a ring buffer (inclusive of start, exclusive of end).
 */
struct __db_filestart {
	u_int32_t	file;
	size_t		b_off;

	SH_TAILQ_ENTRY	links;		/* Either on free or waiting list. */
};

#define	RINGBUF_LEN(lp, start, end)					\
	((start) < (end) ?						\
	    (end) - (start) : (lp)->buffer_size - ((start) - (end)))

/*
 * Internal macro to set pointer to the begin_lsn for generated
 * logging routines.  If begin_lsn is already set then do nothing.
 */
#undef DB_SET_BEGIN_LSNP
#define	DB_SET_BEGIN_LSNP(txn, rlsnp) do {				\
	DB_LSN *__lsnp;							\
	TXN_DETAIL *__td;						\
	__td = R_ADDR(&(txn)->mgrp->reginfo, (txn)->off);		\
	while (__td->parent != INVALID_ROFF)				\
		__td = R_ADDR(&(txn)->mgrp->reginfo, __td->parent);	\
	__lsnp = &__td->begin_lsn;					\
	if (IS_ZERO_LSN(*__lsnp))					\
		*(rlsnp) = __lsnp;					\
} while (0)

/*
 * These are used in __log_backup to determine which LSN in the
 * checkpoint record to compare and return.
 */
#define	CKPLSN_CMP	0
#define	LASTCKP_CMP	1

/*
 * Status codes indicating the validity of a log file examined by
 * __log_valid().
 */
typedef enum {
	DB_LV_INCOMPLETE,
	DB_LV_NONEXISTENT,
	DB_LV_NORMAL,
	DB_LV_OLD_READABLE,
	DB_LV_OLD_UNREADABLE
} logfile_validity;

#include "dbinc_auto/dbreg_auto.h"
#include "dbinc_auto/dbreg_ext.h"
#include "dbinc_auto/log_ext.h"
#endif /* !_LOG_H_ */


Integrating mobile phone augmented reality technology with traditional Chinese art and storytelling, the AR installation of 'Shadow Play: Tales of Urbanization of China' portrays a society undergoing an unprecedented process of urbanization. The narrative of the 'Family of the Village Head' is based upon a series of real-life incidents that have taken place during this time of tumult; clashes over land eviction, kidnapping of children, suicides of migrant workers, and man-made environmental catastrophes are among them. Utilizing the capabilities of augmented reality technology, the audience may then experience the installation in real time and space through their mobile phones. The animated, classically-inspired shadow-puppet characters are superimposed upon photographs taken by the artists in the rural and urbanizing regions of China. Through phone cameras, two layers of imagery are merged; the foreground comes to life in a digital application atop the background of reality. It is this dynamic between the new and the traditional, between modern realities and the age-old human condition that embodies the implications of the work. A collection of screenshots from the piece is presented here.
How to Experience the AR installations in Real-Time: First, download and launch the AR browser application Layar on your iPhone/iPad or Android phone. Simply aim the device's camera at the photographs, press phone screen to scan the images and obtain AR experience. To take a step farther, you may also capture the artworks with your mobile device's camera.
The Shadow Play AR is currently under development for a public presentation during Jamaica Flux 2016. The project receives grants from New Radio & Performing Arts INC, Queens Council on the Arts, Jamaica Cennter for Arts & learning, New York Foundation for the Art, and Creatival Capital.
"I saw four demons hold him down..."
That winter, the village head died..
for its Turbulence.org website. It was made possible with funding from the Jerome Foundation.


Justin Verlander: "system is broken"
Justin Verlander is not one of those players who are having a tough go of it on the free agent market. He's all set. But he's looking out for his fellow players who are not finding the free agent market as hospitable as he and others who might've a few years back. On a few occasions this offseason he has voiced displeasure with the state of things in Major League Baseball.
Some may quibble with 26-36 being a "great performance window" as players tend to peak at the early end of that span and decline as it goes on, but that's not the end of the story when it comes to long-term contracts. One thing that is rarely mentioned is that, while a lot of big long-term deals may look bad toward the end, teams often get more than their money's worth on the front end. In the case of Harper and Machado, cited by Verlander, it's not at all difficult to imagine each of those guys being worth far more than what they'll be paid on the front half of a deal and pretty fairly compensated as they begin to decline. While the last couple of years may look bad, it's likely that they'll have given their teams far more aggregate value over time than that for which they were paid. Such is life when you sign an MVP-caliber 26 year-old.
That aside, it's hard to take issue with Verlander in the grand scheme here. Teams may say they can't afford a big contract to Harper and Machado, but they're never willing to give you the denominator in their financial analysis. I suspect there is no team that can't afford them in an absolute sense and that most passing on them simply are unwilling to make that kind of a commitment. Why they are not is a question for which they rarely give you a straight answer and even more rarely give you an honest one.
As for "rebuilding," well, we covered that this morning.
Verlander is throwing heat today, y'all.
That last time Major League Baseball held the All-Star Game in Philadelphia, the year was 1996.


ANN ARBOR, Mich. – The Ohio State track and field teams competed this weekend at the Phil Diamond Invitational in Ann Arbor, Mich. There was no team scoring. Leading the Buckeyes were veterans Erika Goines (Sr., Zanesville, Ohio) winning the shot put with a heave of 46'8″ and Sekou Smith (Sr., Chicago, Ill.) winning the 400 hurdles (50.97).
After two seasons with a nagging hamstring injury, Joe Crallie (Jr., East Sparta, Ohio) is back on track with a first-place finish in the long jump (24'2-1/4″). The women's and men's 4 x 400 relay teams reigned supreme over the competition with times of 3:48.50 and 3:11.71 respectively.
The Buckeyes will travel to Bloomington, Ind. to compete in the Billy Hayes Invitational this week for a final tune-up before the Big Ten Championships May 22-24.


In November, my Rt Hon Friend the Prime Minister convened a cross-party Working Group to establish a new independent complaints and grievance procedure, in response to reports of sexual harassment and bullying in parliament.
As Chairman of the Working Group, I am pleased to confirm that all members of the Working Group and all Party Leaders have agreed a report which is being published today.
I attach a copy of the report of the Working Group to this statement for the convenience of members.


Unwanted hair growth may occur on many areas of the body such as upper lip, sideburns, chin, ears, chest, back, armpits (axillae), legs, fingers, feet, or toes and is embarrassment for both men and women. Therefore for solving this problem there are various method such as shaving, plucking, electronic tweezers, radio frequency tweezers, waxing, depilatories (creams), and electrolysis of hair remoaval which gives temporary effect. But laser hair removal on other hand is completely different and powerful in comparision with those techniques and it gives permanent results. Laser is the advanced technique formulated for the removal of hair and it help eliminate unwanted hair. It is useful to remove hair from any part of the body whether it's on your face, arms, legs or other areas.
Advantages & Disadvantages of Laser Hair Removal - Laser hair removal is costly and needs some sittings in the cosmetic clinic. It acts as the permanant solution to the excess hair problem.
Traditional hair removal techniques, such as shaving, plucking and waxing, provide only temporary relief. Laser hair removal is non-invasive, and safely removes unwanted body hair without damaging the delicate pores and structures of the skin. Facial and bikini areas are typically completed in below 45 minutes, legs and larger areas can take longer.
First approved by the FDA in 1995, laser hair removal or "photo-epilation" is a highly popular treatment used by both women and men to remove unwanted hair and to give the skin a smooth and youthful look. Laser hair removal contains beams of highly concentrated light designed to selectively penetrate into the hair follicles, to be absorbed by the pigment in the hair follicles and to destroy the hair within that hair follicle. The procedure was originally described to be performed on dark hairs on light skin; newer technologies have made the procedure safer for patients with darker skin and those of color.
Excessive hair growth is classified as either hypertrichosis or hirsuitism and this laser hair removal helps in removing any kind of hair growth in any kind of skin. Laser hair removal is today considered well tolerated, safe and effective as it is developed with various of pocedure using light and laser surgery. Laser hair removal has become one of the most common cosmetic procedures. A variety of lasers and light sources have been developed that may be employed in laser hair removal. Various types of lasers are utilized with laser hair removal which contain alexandrite, diode, Neodymium YAG and intense pulsed light sources.
At present, repeated laser hair removal treatments are between 70 and 90% effective in the long term, lasting six to 12 months. Laser hair removal may be an option if you seek long-term results and at present days only laser is one of the safe and effective method which may gives permanent results.


It has been an absolute pleasure to be a part of Cohort 8 of the Pathways to Law scheme at the University of Exeter for the past two years. It has given me a multitude of opportunities – from seminars with some of the best lecturers in the country, to workshops with legal professionals, and many trips to various universities and locations across the country. For me personally, as an aspiring Barrister, the placement I attended at Magdalen Chambers in Exeter was by far the best part of my Pathways experience. It allowed me to shadow Barristers in a variety of areas of the law, in different courts around Devon and Cornwall, and to witness first-hand the inner workings of a modern chambers. The professionals I had the privilege to work alongside made me feel immensely welcome and made sure they answered any questions and challenged me so that I could achieve my full potential whilst I was with them. The advice, support and future opportunities that I have received since then have also been truly invaluable, and have helped me to start making legal contacts at a young age, standing me in good stead for a future career in the law. This simply wouldn't have been a possibility without the support of Pathways to Law and our Pathways co-ordinator.
Pathways to Law has definitely taught me the importance of taking any opportunity offered to me, even if it is completely out of my comfort zone. I have taken part in many debate-style sessions whilst at university and also argued a Plea-in-Mitigation case in a mock hearing at the Inner Temple, London, both of which have taught me the importance of confidence and self-belief which will help me going forward. I have also represented the student cohort on the Pathways to Law Management Board (Exeter). This has enabled me to put across suggestions and feedback from other pathways students on the programme whilst experiencing a professional working environment within the university. Working with a variety of senior people as a seemingly inexperienced teenager can be very daunting, but being on this programme gave me the confidence to be able to hold my own and thrive in these situations. The E-Mentoring has also been really useful; I had a fantastic mentor who was a huge source of advice and support throughout the programme. I now feel much more prepared to study law at University and have been offered a place at both the University of Exeter and Cardiff University this September.
Finally, it has also taught me that nothing should be a barrier to my future – be it financially, physically, mentally or geographically. I can achieve what I aspire to achieve just by working hard, being committed, and grasping every opportunity I have to better myself and succeed. I highly recommend the Pathways to Law programme to any aspiring young person.
Ellie completed her GCSE's at Tiverton High School before studying Law, Sociology and Psychology A level's at Exeter College.


text:
\section{Introduction}\label{sec-intro}
In the non-relativistic limit, Einstein's general relativity reduces to Newtonian gravity.
In analogy to Einstein-Cartan gravity, Newtonian gravity has a coordinate-independent formulation in terms of frame fields, namely Newton-Cartan  gravity \cite{Cartan:1923zea,Cartan:1924yea,Duval:1984cj,Bekaert:2014bwa, Bekaert:2015xua}.
Since Einstein-Cartan gravity can be obtained by gauging the relativistic Poincar\'e algebra,
one may attempt to derive Newton-Cartan gravity by gauging a non-relativistic symmetry algebra.
The first candidate algebra would be the Galilei algebra ${\mathfrak g}_0$, which is the non-relativistic limit of the Poinca\'re algebra.
However, it turns out that one instead needs to consider the Bargmann algebra ${\mathfrak g}_{0}^+$,
which is an extension of the Galilei algebra with a central generator $M$ called ``mass''.
This mass generator is related to an additional $U(1)$ gauge field in Newton-Cartan gravity.
If we introduce a non-zero cosmological constant,
then the counterparts of the Galilei algebra ${\mathfrak g}_0$ and the Bargmann algebra ${\mathfrak g}_0^+$
are the Newton-Hooke algebra ${\mathfrak g}_\L$ and the extended Newton-Hooke algebra ${\mathfrak g}_\L^+$.

Since the equations of motion of Newton-Cartan gravity
can be obtained by gauging the Bargmann algebra ${\mathfrak g}_{0}^+$ \cite{Andringa:2010it},
Newton-Cartan gravity can be considered as a gauge theory of the Bargmann algebra.
Another way to derive Newton-Cartan gravity is to consider a limit of Einstein-Cartan gravity
mimicing the \.In\"on\"u-Wigner contraction of the relativistic algebra \cite{Bergshoeff:2015uaa}.
However, in general spacetime dimensions, these equations of motion do not have a simple action principle.

In three dimensions, the Bargmann algebra ${\mathfrak g}_0$ admits a second central extension by the ``spin'' generator $S$ \cite{Grigore:1993fz, Jackiw:2000tz}.
The resulting algebra is referred to as the extended Bargmann algebra ${\mathfrak g}_0^{++}$\,.
In parallel, in the cosmological case, the three-dimensional Newton-Hooke algebra admits double extensions.
We refer to this as the doubly extended Newton-Hooke (NH) algebra ${\mathfrak g}_{\L}^{++}$\,.
We summarize the zoo of three-dimensional non-relativistic algebras in Table \ref{3d-nonrel-alg}.
Note that the subscript $\L$ denotes the cosmological constant and the superscript $+$ or $++$ means the central extension
by $M$ or $M, S$\,.
Due to the second central extension,
both ${\mathfrak g}_0^{++}$ and ${\mathfrak g}_{\L}^{++}$ have non-degenerate invariant bilinear forms \cite{Papageorgiou:2009zc, Papageorgiou:2010ud}.
Consequently, the corresponding theories admit action principles through  Chern-Simons formulation.
These  non-relativistic gravity theories are sometimes called extended Bargmann gravity and extended Newton-Hooke (NH) gravity, respectively.
Along the same lines, the non-relativistic Chern-Simons theories admit various extensions according to the symmetry algebras
\cite{Hartong:2016yrf, Bergshoeff:2016lwr,Bergshoeff:2016soe}.

\begin{table}[h!]
\centering
 \begin{tabular}{|c || c | c |}
 \hline
  & Flat & (A)dS \\
 \hline
 No extension & Galilei  ${\mathfrak g}_0$ & Newton-Hooke (NH) ${\mathfrak g}_\L$ \\
 Extended by $M$  & Bargmann ${\mathfrak g}_{0}^+$ & Extended NH ${\mathfrak g}_\L^+$ \\
 Extended by $M$, $S$ &  Extended Bargmann  ${\mathfrak g}_{0}^{++} $ &
  Doubly extended NH ${\mathfrak g}_\L^{++}$ \\
 \hline
\end{tabular}
\caption{Three-dimensional non-relativistic isometry algebras.}
\label{3d-nonrel-alg}
\end{table}

The presence of two central generators $M$ and $S$ are hence crucial to
the action principle of three-dimensional non-relativistic gravities in the frame formulation.
In fact, as we shall show later, these doubly extended non-relativistic algebras
can be obtained from the relativistic ones with two $u(1)$ generators after suitable contractions (see also \cite{Bergshoeff:2016lwr, Hartong:2017bwq}).
For instance, in AdS, the relativistic isometry algebra with two $u(1)$'s,
\be
	so(2,2)\oplus u(1) \oplus u(1)
	= \Big(sl(2,\mathbb R)\oplus u(1)\Big)\oplus \Big(sl(2,\mathbb R) \oplus u(1)\Big)\,,
	\label{assoc}
\ee
contracts to the doubly extended NH ${\mathfrak g}_\L^{++}$ under a suitable
scaling limit. In fact, both the relativistic and non-relativistic algebras
can be split into left- and right-moving sectors, so the contraction can be performed in each sector.
After the contraction, the $u(1)$ generators become central elements,
and the resulting algebra cannot be arranged as a direct sum anymore.
The is the key mechanism for a non-degenerate bilinear form --- hence for the action principle.

We notice that another peculiar utility of $u(1)$ can be found in a different extension of
three-dimensional gravity, namely the colored gravity
which has been studied recently in \cite{Gwak:2015vfb,Gwak:2015jdo, Joung:2017hsi}.
Several no-go results are known for theories of multiple gravitons,\footnote{See e.g. \cite{Boulanger:2000rq} for the no-go result and \cite{Boulanger:2000ni} for an exception.}
but in three dimensions
they can be evaded by extending the isometry algebra $so(2,2)$ with
two $u(1)$ generators.
It is remarkable that the resolution of the problems in non-relativistic limit and color-decoration
are the same: replace $sl(2,\mathbb R)$ by $sl(2,\mathbb R)\oplus u(1)$.
This observation leads us to the following question:
does the three-dimensional colored gravity have a natural non-relativistic limit?

Let us explain a few more details about the colored gravity.
It can be constructed in the Chern-Simons formulation
with an appropriate choice of gauge algebra.
For the multi-graviton interpretation, this algebra should contain
multiple copies of isometry generators,
hence a direct product of the isometry $so(2,2)$ and
the internal symmetry.
Usually, the direct product of two Lie algebras does not define a Lie algebra,
but the case of two associative algebras does so.
The internal symmetry can be taken as $U(N)$ which obviously has an associative structure.
About the isometry algebra, we can extend the Lie algebra
$sl(2,\mathbb R)\oplus sl(2,\mathbb R)$ to
an associative algebra
  $gl(2,\mathbb R)\oplus gl(2,\mathbb R)$
  by adding  two copies of $u(1)$.
 We refer to the internal symmetry as ``color symmetry'' for the reason which will become
clear later,
and the resulting gravity theory will be correspondingly referred to as ``colored gravity''.
The colored gravity has several interesting features:
{\it i)} It has a non-trivial potential with many saddles points
corresponding to the (A)dS vacua of different cosmological constants.
These vacua can be classified according to how much the internal symmetry is preserved.
{\it ii)} Around the color singlet vacuum, the theory has one graviton, $(N^2-1)$ massless spin-two fields charged under the color symmetry
and $U(N)\times U(N)$ Chern-Simons gauge field.
The color non-singlet spin-two fields are strongly interacting for large $N$.
{\it iii)} Around a color-symmetry breaking background,
a Higgs-like mechanism takes place and one finds partially-massless spin-two fields
\cite{Deser:1983tm,Deser:2001us}
appearing in the spectrum.

The organization of the paper is as follows.
In Section \ref{sec-non-relativistic},
we present a concise overview of several aspects of the 3d non-relativistic algebras.
In Section \ref{sec-non-relativistic-colored-alg},
we derive the non-relativistic colored-gravity algebras
from the \.In\"on\"u-Wigner contractions of the relativistic ones.
In Section \ref{sec-non-relativistic-colored-action},
we construct the actions of non-relativistic colored gravities using the Chern-Simons formulation.
In Section \ref{sec: discussion}
we conclude the work with a brief summary.
Appendix includes technical details and additional materials.

\section{Non-relativistic Algebras and their Bilinear Forms}\label{sec-non-relativistic}

\subsection{Various non-relativistic algebras in three dimensions}\label{subsec-3d-nonrel}

We will give a concise review of various three-dimensional non-relativistic algebras:
the Galilei, Bargmann, extended Bargmann, Newton-Hooke (NH), extended NH, and doubly extended NH algebras,
which have been briefly discussed in Introduction.

\subsubsection*{Galilei algebra ${\mathfrak g}_0$}

The three-dimensional Galilei algebra  is an eight-dimensional Lie algebra with the non-trivial Lie brackets:
\be
[J,\,G_i]=2\,\epsilon_{ij}\, G_j,\qquad
[J,\,P_i]=2\,\epsilon_{ij}\,  P_j, \qquad
[H,\,G_i]=2\,\epsilon_{ij}\, P_j.\label{Gal}
\ee
Here, $H, P_i, J$ and $G_i$ are the generators of
time translations, spatial translations, spatial rotations and Galilean boosts, respectively.
The Levi-civita symbol $\e_{ij}$ is defined as
\be
\epsilon_{12}=-\epsilon_{21}=1,
\quad\epsilon_{11}=\epsilon_{22}=0.
\ee
Remark that, in higher dimensions, there exist non-zero brackets between two spatial rotation generators.

\subsubsection*{Newton-Hooke algebra ${\mathfrak g}_\L$}

When the cosmological constant $\L$ is turned on,
the Galilei algebra is deformed to the so-called Newton-Hooke algebra.
As a result, we have an additional set of Lie brackets,
\be
[H,\,P_i]=-2\,\Lambda\,\epsilon_{ij}\, G_j, \label{NH}
\ee
besides those in \eqref{Gal}.

\subsubsection*{Bargmann algebra ${\mathfrak g}_0^+$ and extended NH algebra ${\mathfrak g}_\L^+$}

In general dimensions, both the Galilean and Newton-Hooke (NH) algebras can be centrally extended by the mass generator $M$.\footnote{In a relativistic theory,
the energy of a particle in the rest frame
is proportional to the rest mass.
However, in a non-relativistic theory,
spacetime symmetries do not require energy-mass equivalence,
so we can consider an independent generator for mass
in the form of a central extension.
Intuitively speaking, the energy-mass relation is lost in the large speed of light limit.
}
The resulting algebras are known as the Bargmann algebra and the extended NH algebra, respectively.
In three dimensions, the additional Lie brackets are
\be
[G_i, P_j]=2\,\epsilon_{ij}\,M.\label{Bar}
\ee

\subsubsection*{Extended Bargmann algebra ${\mathfrak g}_0^{++}$
and doubly extended NH algebra ${\mathfrak g}_\L^{++}$}

In three dimensions, the Bargmann and extended NH algebras admit a second central extension by
the spin generators $S$.\footnote{To interpret $S$ as a spin generator is subtle.
See \cite{Duval:2002cw, Hagen:2002pg} for more details.}
The extended Bargmann algebra, or equivalently the doubly extended Galilei algebra, has one more set of
Lie brackets,
\be
	[G_i,G_j]=2\,\epsilon_{ij}\, S.\label{ext}
\ee
In the doubly extended Newton-Hooke algebra,
the additional Lie brackets are \eqref{ext} and
\be
[P_i,P_j]=-2\,\Lambda\,\epsilon_{ij}\,S.\label{extNH}
\ee

\medskip

All the non-relativistic algebras presented above can be
obtained from the most general one, the doubly extended NH algebra, by various \.In\"on\"u-Wigner contractions.
The cosmological constant dependent terms can be removed by the usual \.In\"on\"u-Wigner contraction.
The mass and spin generators only appear in the right hand sides of the Lie brackets,
hence can be eliminated by a suitable contraction:  rescale them first by a contraction parameter,
 then send the parameter to zero.

\subsection{Non-relativistic limits as contractions}

It is useful to realize these non-relativistic limits as contractions.
In fact, the three-dimensional relativistic isometry algebra with a non-zero cosmological constant and two additional $u(1)$ generators
can contract to the doubly extended NH algebra (hence all non-relativistic algebras discussed in section \ref{subsec-3d-nonrel}).
Let us provide more details on this point.
The relativistic isometry algebra in three dimensions has the Lie brackets,
\be
	[\,\mathcal J_{a}\,,\,\mathcal J_{b}\,]=2\,\epsilon_{ab}{}^{c}\,\mathcal J_{c}\,,
	\qquad
	[\,\mathcal J_a\,,\,\mathcal P_{b}\,]=2\,\epsilon_{ab}{}^{c}\,\mathcal P_{c}\,,
	\qquad
	[\,\mathcal P_{a}\,,\,\mathcal P_{b}\,]=-2\,\Lambda\,\epsilon_{ab}{}^{c}\,\mathcal J_{c}\,,
	\label{com rel}
\ee
where $\cP_a$ and $\cJ_a$ are the translation and Lorentz generators,
and
 $\L$ is the cosmological constant.

\subsubsection*{Contraction to Newton-Hooke algebra}

For an illustration, let us see how
the relativistic isometry algebra \eqref{com rel}
contracts to the Newton-Hooke algebra ${\mathfrak g}_\L$.
We first redefine the generators using the parameter $c$ as
\be
G_i=\frac 1 {c}\,\mathcal J_i\,,
\qquad
P_i=\frac1{c}\,\mathcal P_i\,,
\label{PG}
\ee
and
\be
J=\cJ_0\,,\qquad
H=\cP_0\,,
\ee
then derive their Lie brackets.
Notice that the temporal components scale differently in $c$ from the spatial ones.
The \.In\"on\"u-Wigner contraction is performed in the $c\to \infty $ limit which
effectively removes certain terms in the structure constants.
One can easily check that the resulting algebra is the NH algebra ${\mathfrak g}_\L$.
Hence, one can view this contraction  as  the non-relativistic limit
and interpret the parameter $c$ as the speed of light.

\subsubsection*{Contraction to doubly extended Newton-Hooke algebra}

Let us now show how the
doubly extended NH algebra $\mathfrak g^{++}_{\L}$
can be derived by a contraction.
As the algebra $\mathfrak g^{++}_{\L}$ has dimension eight due to the central generators $M$ and $S$,
it cannot be obtained as a contraction of the relativistic isometry algebra which has dimension six.
To fix the mismatch between the numbers of generators, we need to include two additional $u(1)$ generators.
We refer to these generators as $\cZ_\cP$ and $\cZ_\cJ$.
In terms of them, we change the definitions of $H$ and $J$ to
\be
J=\cZ_\cJ+\cJ_0, \qquad
H=\cZ_\cP+\cP_0,
\label{HJ mix}
\ee
but keep those of $P_i$ and $G_i$ the same as \eqref{PG}.
The mass and spin generators are given by the other combinations with different rescalings,
\be
	S=\frac {\cZ_\cJ-\cJ_0}{2\,c^2},
	\qquad
	M=\frac {\cZ_\cP-\cP_0}{2\,c^2}.
	\label{MS mix}
\ee
With these definitions, one can succesfully reproduce
the doubly extended NH algebra $\mathfrak g_\L^{++}$
by taking the $c\to \infty$ limit.
Note that in \eqref{HJ mix} and \eqref{MS mix} the mixings with $\cZ_\cP$ and $\cZ_\cJ$ are independent,
hence the corresponding central extensions by $M$ and $S$ are also independent.

\subsubsection*{Chiral decomposition}

When $\L\neq0$\,, the relativistic isometry algebra can be decomposed into two parts:
\be
	\cL_a=\frac 1 2\left(\cJ_a+\frac1{\sqrt{-\L}}\,\cP_a\right)\,,
	\qquad
	\tilde\cL_a=\frac 1 2\left(\cJ_a-\frac1{\sqrt{-\L}}\,\cP_a\right)\,,
	\label{LL JP}
\ee
and each of them forms an $sl(2)$\,:
\be
	[\cL_a,\cL_b]=2\,\e_{ab}{}^c\,\cL_c\,,
	\qquad
	[\tilde\cL_a,\tilde\cL_b]=2\,\e_{ab}{}^c\,\tilde\cL_c\,.
\ee
Accordingly, it is natural to decompose the $u(1)\oplus u(1)$ part as
\be
	\cZ=\frac 1 2\left(\cZ_\cJ+\frac 1 {\sqrt{-\L}}\,\cZ_\cP\right),\qquad
	\tilde\cZ=\frac 1 2\left(\cZ_\cJ-\frac 1 {\sqrt{-\L}}\,\cZ_\cP\right).
	\label{chiral-Z}
\ee

For the non-relativistic limit $c\to \infty$,
one can check that the redefinitions \eqref{PG}, \eqref{HJ mix} and \eqref{MS mix}
are compatible with the chiral decomposition.
More precisely, with the redefinitions,
\be
	K=\cZ+\cL_0,\qquad
	L_i=\frac1c\,\cL_i,
	\qquad
	Z=\frac{\cZ-\cL_0}{2\,c^2},
	\label{chiral def}
\ee
\be
	\tilde K=\tilde \cZ+\tilde \cL_0,\qquad
	\tilde L_i=\frac1c\,\tilde \cL_i,
	\qquad
	\tilde Z=\frac{\tilde \cZ-\tilde \cL_0}{2\,c^2},
	\label{anti-chiral def}
\ee
the two copies of the relativistic chiral algebra $sl(2)\oplus u(1)$ contract to
the following non-relativistic chiral algebras,
\be
	[K,\,L_i]=2\,\e_{ij}\,L_j, \qquad
	[L_i,\, L_j]=2\,\e_{ij}\,Z,
\ee
\be
	[\tilde K,\,\tilde L_i]=2\,\e_{ij}\,\tilde L_j,\qquad
	[\tilde L_i,\, \tilde L_j]=2\,\e_{ij}\,\tilde Z,
\ee
where $Z$ and $\tilde Z$ are the centers of the chiral algebras.
The chiral subalgebra $\mathfrak{h}$, spanned by $K, L_i$ and $Z$, can be viewed as
a central extension of
the two-dimensional Euclidean algebra
where $K$ and $L_i$ can be interpreted as two-dimensional translation and rotation generators, respectively.
One can equally interpret  $\mathfrak{h}$ as a harmonic oscillator,
where $L_1\pm i\,L_2$ are creation/annihilation or position/momentum operators,
$K$ is the Hamiltonian, and $Z$ is the Plank contant $\hbar$.\footnote{The same
algebra can be obtained from the subalgebra of Virasoro algebra spanned by
$L_m/m^{3/2}$, $L_{-m}/m^{3/2}$ and $L_0/m$ (here $L_m$ are the standard generators of Virasoro algebra)
in the $m\to\infty$ limit.  The Virasoro central charge $c$ becomes the central element $Z$.}

The doubly extended NH  algebra $\mathfrak g_{\L}^{++}$ can be directly decomposed into
two copies of  the chiral algebra $\mathfrak h$ and $ \tilde {\mathfrak h}$\,:
\be
	\mathfrak g_{\L}^{++}=\mathfrak h \oplus \tilde {\mathfrak h},
	\label{chiral decomp}
\ee
where the explicit decomposition reads
\ba
	G_i=L_i+\tilde L_i,\qquad
	&J=K+\tilde K,\qquad
	&S=Z+\tilde Z,\nn
	P_i={\sqrt{-\L}}\,  (L_i-\tilde L_i),\qquad
	&H={\sqrt{-\L}}\, (K-\tilde K),\qquad
	&M={\sqrt{-\L}}\, (Z-\tilde Z).
	\label{DENH decomp}
\ea

\subsection{Invariant bilinear form}
\label{sec: inv bf}

One may attempt to use the non-relativistic algebras in section \ref{subsec-3d-nonrel} to construct
non-relativistic gravity theories.
In three dimensions, as in the relativistic case, one can rely on the Chern-Simons formulation
where non-degeneracy of the bilinear form is important for
the dynamics of the resulting theory.
It turns out that, among the algebras that we have considered,
only those with double extensions by $M$ and $J$ admit a non-degenerate bilinear form.
In below, we focus on the doubly extended NH algebra and
discuss its bilinear form.

Since the doubly extended NH algebra $\mathfrak g_\L^{++}$ admits
the decomposition \eqref{chiral decomp},
we can consider
the invariant bilinear form $\la \cdot, \cdot \ra$ for each chiral sector.
The non-diagonal part are the $2\times 2$ symmetric matrices,
\be
	B=\begin{pmatrix}
	\la K, K \ra & \la Z, K\ra \\
	\la K, Z \ra & \la Z, Z \ra
	\end{pmatrix},
	\qquad
	\tilde B=\begin{pmatrix}
	\la \tilde K, \tilde K \ra & \la \tilde Z, \tilde K\ra \\
	\la \tilde K, \tilde Z \ra & \la \tilde Z, \tilde Z \ra
	\end{pmatrix}.
\ee
Due to the rotational symmetry, the diagonal part is fixed as
\be
	\la L_i, L_j \ra=\ell\,\delta_{ij},
	\qquad \la \tilde L_i, \tilde L_j \ra=\tilde \ell\,\delta_{ij}\,,
\ee
with undetermined constants $\ell$ and $\tilde \ell$,
while the other bilinear forms $\la L_i, K\ra$, $\la L_i, Z\ra$ and  their tilde counterparts vanish.
Symmetry also determines the matrices $B$ and $\tilde B$ to be
\be
	B=\begin{pmatrix}
	\b & \ell\, \\
	\ell & 0
	\end{pmatrix},
	\qquad
	\tilde B=\begin{pmatrix}
	\tilde \b & \tilde \ell \\
	\tilde \ell & 0
	\end{pmatrix},
	\label{bl L}
\ee
where $\b$ and $\tilde \b$ are additional allowed constants.
The  non-degeneracy of the bilinear form requires  $\ell\neq0$ and $\tilde \ell\neq0$,
but $\b$ and $\tilde \b$ are left unconstrained.

The bilinear form of the doubly extended NH algebra
\eqref{DENH decomp} is given by those of the left and right chiral algebras $\mathfrak h$ and $\tilde{\mathfrak h}$,
together with possible cross terms.
However, one can show that the only cross term
compatible with symmetry is
\be
	\la K,\tilde K\ra=\g.
	\label{K tilde K}
\ee
In the end, the invariant bilinear form of $\mathfrak{g}_\L^{++}$
has  five undetermined constants:
\be
	\ell,\quad  \tilde \ell,\quad  \b, \quad \tilde \b, \quad \g.
\ee
The number of free parameters is consistent with the analysis of \cite{Hartong:2016yrf}.

As in the usual relativistic Chern-Simons gravity, we set $\ell$ and $\tilde\ell$ as
\be
	\ell=-\tilde \ell=\frac{1}{2\sqrt{-\L}}\,,
\ee
 so the spatial part takes the standard form.
The non-vanishing components of the bilinear form are
\be
	\la J, M\ra=\la S, H\ra=1\,,\qquad
	\la P_i, G_j\ra=\delta_{ij}\,,
\ee
and
\be
	\begin{pmatrix}
	\la H,H \ra & \la H,J\ra \\
	\la J,H \ra & \la J,J \ra
	\end{pmatrix}
	=
	\begin{pmatrix}
	-\L\,(\b+\tilde \b -2\,\g) &{\sqrt{-\L}}\,( \b-\tilde \b)\\
	{\sqrt{-\L}}\,( \b-\tilde \b) &  \b+\tilde \b+2\,\g
	\end{pmatrix}.
\ee
The above symmetric matrix remains completely arbitrary.

The bilinear form that we have derived using symmetry can also be obtained from the relativistic one by a contraction.
To begin with, let us focus on the chiral sector. The bilinear form of $\mathfrak{h}=sl(2)\oplus u(1)$
is unique for each of the $sl(2)$ and $u(1)$ part, up to overall factors:
\be
	\la \cZ,\cZ\ra=\z,\qquad
	\la \tilde\cZ,\tilde\cZ\ra=\tilde\z,
	\label{rel bil}
\ee
\be
	\la \cL_a,\cL_b\ra =\l\, \eta_{ab},\qquad
	\la \tilde \cL_a,\tilde \cL_b\ra =\tilde \l\, \eta_{ab}.
	\label{LL-tr}
\ee
Using \eqref{chiral def} and \eqref{anti-chiral def},
these give the bilinear form for the rescaled generators as
\be
	\la L_i,L_j\ra =\frac{\l}{c^2}\,\delta_{ij}\,,
	\qquad
	\la \tilde L_i,\tilde L_j\ra =\frac{\tilde \l}{c^2}\,\delta_{ij}\,,
\ee
\be
	B=\begin{pmatrix}
	\z-\l & ({\z+\l})/(2c^2) \\
	({\z+\l})/(2c^2) & ({\z-\l})/(4c^4)
	\end{pmatrix},
	\quad
	\tilde B=\begin{pmatrix}
	\tilde \z-\tilde \l & (\tilde \z+\tilde \l)/(2c^2) \\
	(\tilde \z+\tilde \l)/(2c^2) & (\tilde \z-\tilde \l)/(4c^4)
	\end{pmatrix}.
\ee
Hence by choosing
\be
	\z=c^2\,\ell+ \b/2 ,\quad
	\tilde \z=c^2\,\tilde\ell+\tilde \b/2,\quad
	\l=c^2\,\ell- \b/2,\quad
	\tilde\l=c^2\,\tilde \ell-{\tilde \b}/2,
	\label{l z}
\ee
we recover the bilinear form \eqref{bl L} in the contraction limit.
Then we consider the brackets of the left and right sectors.
We have the freedom to introduce the cross term,
\be
	\la \cZ, \tilde \cZ\ra=\g,
\ee
which leads to \eqref{K tilde K} after the contraction.

We have seen that all the non-relativistic algebras and their bilinear forms
can be obtained from the relativistic one by contractions.
In addition, it is also sufficient to consider the chiral sector $sl(2)\oplus u(1)$.
At the level of Lie algebra, there is no relation between $sl(2)$ and $u(1)$,
hence their bilinear forms  \eqref{rel bil} are also independent.

On the other hand, one simple way to introduce an invariant bilinear form is through
an invariant linear form, i.e. trace:
\be
	\la X, Y\ra =\tr(X\, Y).
\ee
A well-defined trace requires an associative product between  $X$ and $Y$.
This can be done by representing $X$ and $Y$ as matrices. But the price to pay is that
we have to work with a larger space because the associative products of the original elements will generate
additional elements. The enlargement can be minimized by choosing the smallest representation.
For $sl(2)$, the smallest representation is the fundamental representation,
which is two dimensional and requires only one more element, i.e. the identity, to have the associative structure.
The associative product of the relativistic chiral algebra reads
\be
	\cL_a\,\cL_b=\eta_{ab}\,\cI+\e_{ab}{}^c\,\cL_c,
	\qquad
	\tilde \cL_a\,\tilde \cL_b=\eta_{ab}\,\tilde \cI+\e_{ab}{}^c\,\tilde \cL_c,
	\label{associative prod}
\ee
and
we recover the bilinear form \eqref{LL-tr} using
\be
	\tr(\cI)=\l,\quad
	\tr(\tilde \cI)=\tilde \l,
	\quad \tr(\cL_a)=\tr(\tilde\cL_a)=0.
	\label{h trace}
\ee
There is no a priori reason to identify the identities $\cI$ and $\tilde \cI$ with the $u(1)$ generators $\cZ$ and $\tilde \cZ$.
However, if they coincide up to some factors $z(c)$ and $\tilde z(c)$, the coefficients $\l, \tilde \l$ and $\z, \tilde\z$ in \eqref{rel bil}, \eqref{LL-tr} will be related as
\be
	\cZ=z(c)\,\cI \quad \Rightarrow \quad  \z=z^2(c)\,\l,
	\label{Z-id}
\ee
\be
	\tilde\cZ=\tilde z(c)\,\tilde\cI \quad \Rightarrow \quad \tilde \z=\tilde z^2(c)\,\tilde\l.
	\label{Z-id-tilde}
\ee
These relations together with \eqref{l z} fix the form of the functions $z(c)$ and $\tilde z(c)$ as
\be
	z^2(c)=\frac{c^2\ell+\b/2}{c^2\ell-\b/2},\qquad
	\tilde z^2(c)=\frac{c^2\tilde \ell+\tilde \b/2}{c^2\tilde \ell-\tilde \b/2}\,.
	\label{z-k-l}
\ee
Therefore, we fully recover the non-relativistic algebra $\mathfrak h$ and $\tilde{\mathfrak h}$, and
their bilinear forms using the associative structure of the relativistic counterpart, $gl(2)$.
As we mentioned in the introduction,
the associative structure of the isometry algebras plays a central role in the color decoration of
three-dimensional gravity.
From this observation, we are driven to look more closely the possible interplay
between non-relativistic and color deformations of gravity
in three dimensions.

\section{Colored Gravity and its Non-relativistic Limits}\label{sec-non-relativistic-colored-alg}

If a theory contains more than one graviton and they are related by internal symmetry,
the global symmetry should be extended
from the usual isometry $\mathfrak{g}_i$ to a
color-decorated one $\mathfrak{g}_i \otimes \mathfrak g_c$,
where $\mathfrak g_c$ is a finite vector space generated by the color label.
This leads to severe algebraic constraints:
there is only trivial possibility if we assume the color-decorated algebra
does not include additional generators,
which means extra fields are necessary \cite{Gwak:2015vfb,Gwak:2015jdo}.
Even if we find a good candidate of colored gravity algebras by including a reasonable amount of additional fields,
the construction of equations of motion or/and actions is not guaranteed to be straightforward.
In three dimensions, the situation is much simpler.
First, thanks to the Chern-Simons formulation, we obtain the action directly from a given algebra.
Second, a color-decorated algebra can be constructed by
extending the isometry algebra with only two additional $u(1)$ generators.

Technically speaking, we want to find a Lie algebra structure on
the tensor-product vector space $\mathfrak{g}_i \otimes \mathfrak g_c$\,.
The Lie brackets between two elements of the above space may be written as
\be
	[\,M_{X}\otimes \bm \cT^{I},M_{Y}\otimes \bm \cT^{J}\,]
	=\frac12\,[\,M_{X},M_{Y}\,]\otimes \{\,\bm \cT^{I} ,\bm \cT^{J}\,\}
	+\frac12\,\{\,M_{X}, M_{Y}\,\} \otimes [\,\bm \cT^{I},\bm \cT^{J}\, ] \,,
	\label{color bracket}
\ee
where $M_X$ and $M_Y$ belong to $\mathfrak{g}_i$ whereas $\bm \cT^I$ and $\bm \cT^J$ to $\mathfrak{g}_c$\,.
Note that we have used both the commutator and anti-commutator of the generators of each space.
Therefore, in order to have well-defined Lie brackets \eqref{color bracket},
we should begin with two associative algebras $\mathfrak{g}_i$ and $\mathfrak{g}_c$\,.
For the color algebra $\mathfrak{g}_{c}$\,, we use the matrix algebra $u(N)$.\footnote{In principle, any finite-dimensional associative algebra can
be used as the color algebra.}
For the isometry algebra $\mathfrak{g}_{i}$\,,
the minimal choice is   $gl(2)\oplus gl(2)$.
This is precisely the same as the setting \eqref{assoc} for the doubly extended
NH algebra.

\subsection{Chiral sector}

Let us start from the non-relativistic limits of one chiral sector of the colored gravity algebra.
It is generated by
the following generators,
\be
	\cZ,\qquad \cL_a\,,\qquad \bm \cT^I,\qquad \bm\cL_a^I,
\ee
where $\cL_a$ and $\cZ$ form the original relativistic algebra $sl(2)\oplus u(1)=gl(2)$
and  $\bm \cT^I$ are the $su(N)$ generators:
\be
	[\bm \cT^I,\bm \cT^J]=2\,i\,f^{IJ}{}_K\,\bm \cT^K.
\ee
The generators $\bm\cL_a^I$ are the color decoration of the spin-two generators and
they transform covariantly under $sl(2)$ and $su(N)$:
\be
	[\cL_a,\bm\cL^I_b]=2\,\e_{ab}{}^c\,\bm\cL^I_c,
	\qquad
	[\bm \cT^I,\bm\cL^J_a]=2\,i\,f^{IJ}{}_K\,\bm\cL^K_a,
\ee
but the Lie brackets between them are non-trivial:
\be\label{Lie-bra-colored-isometry}
	[\bm\cL_a^I,\bm\cL^J_b]=
	2\,i\,\eta_{ab}\,f^{IJ}{}_K\,\bm \cT^K+
	2\,\e_{ab}{}^c\left(\frac 1 N\d^{IJ}\,\cL_c+g^{IJ}{}_K\,\bm\cL^K_c\right).
\ee
In \eqref{Lie-bra-colored-isometry}, we used the associative algebra of $u(N)$
\be
\bm \cT^I\,\bm \cT^J=\frac 1 N\,\d^{IJ}I
+g^{IJ}{}_K\,\bm \cT^K
+i\,f^{IJ}{}_K\,\bm \cT^K,
\ee
where $g_{IJK}$ is totally symmetric and $f_{IJK}$ totally antisymmetric.
We assign its bilinear form as
\be
	\la \bm\cT^I,\bm\cT^J\ra
	=\tr(\bm\cT^I\,\bm\cT^J)=\delta^{IJ},
	\label{TT bf}
\ee
where we used $\tr(I)=N$ while $\tr(\bm \cT^I)=0$.

We want to construct the non-relativistic analogs of the above colored gravity algebra,
which we assume to have the same number of generators as the relativistic one.
The non-relativistic algebra is spanned by
\be
	Z, \quad L_i, \quad K,\quad \bm \cT^I, \quad \bm L_i^I,\quad \bm K^I.
\ee
Their Lie brackets depend on the non-relativistic limits.

\subsubsection*{Color decoration after contraction}

One of the possible ways to obtain the non-relativistic colored gravity algebra
is by colored-decorating the doubly extended NH algebra.
For that, the latter algebra should admit an associative algebra structure.
In the previous section, we have shown that it can be obtained
from an associative relativistic algebra by a suitable contraction.
However, we have not examined the
full consistency of the associative structure of the resulting non-relativistic algebra.
An associative product can be split into commutators, which reproduce the Lie brackets,
and anti-commutators.
Therefore, we have to check whether the contraction limit \eqref{chiral def} is equally well-defined for the anti-commutators.
From \eqref{associative prod}, \eqref{Z-id} and \eqref{z-k-l},
most of the anti-commutators are regular in the contraction limit:
\be
	\{K,\,L_i\}=2\,L_i,
	\quad
	\{ K, \,Z\}=2\,Z,
	\quad
	\{L_i,\,L_j\}=2\,\delta_{ij}\,Z,
	\quad
	\{L_i, \,Z\}=0,
	\quad
	\{Z,\,Z\}=0.
	\quad
\ee
However, before taking the contraction limit,
the $\{K,K\}$ anti-commutator reads
\be\label{anti-com-KK}
	\{ K, K \}=2\,(K-2\,c^2\,Z).
\ee
In the $c\to \infty$ limit, the anti-commutator \eqref{anti-com-KK} becomes ill-defined,
so the corresponding associative product is problematic.
The contraction of the $gl(2)$ gives a good Lie algebra structure and bilinear form or trace,
but not the fully associative algebra structure.

This issue does not affect the Chern-Simons action of the doubly extended NH gravity.
The reason we consider the anti-commutator $\{K,K\}$ here
is that from \eqref{color bracket} they appear in the Lie bracket of its color-decorated counterparts,
$[\bm K^I,\bm K^J]$.
We can avoid the divergence by defining the colored generators as
\be
	\bm K^I=\frac1c\,K\otimes \bm \cT^I,
	\qquad
	\bm Z^I=Z\otimes \bm \cT^I,
\ee
which lead to a regular commutator
\be
	[\bm K^I,\bm K^J]
	=-4\,i\,f^{IJ}{}_K\,\bm Z^K.
\ee
Concerning the color decoration of $L_i$ generators,
we consider two options:
\begin{itemize}
\item
In the first option, by defining them as
\be
	\bm L^I_i=\frac1{c}\,L_i\otimes \bm \cT^I,
\ee
we have
\be
	[\bm K^I,\,L_i]=[K,\,\bm L_i^I]=2\,\e_{ij}\,\bm L_j^I.
\ee
\item
In the second option, with the definition
\be
	\bm L^I_i=L_i\otimes \bm \cT^I,
\ee
we obtain different Lie brackets,
\be
	[K,\,\bm L_i^I]=2\,\e_{ij}\,\bm L_j^I,
	\qquad
	[L_i, \bm L^I_j]=2\,\e_{ij}\,\bm Z^I,
\ee
\be
	[\bm L^I_i, \bm L^J_j]=
	2\,\epsilon_{ij}\left(\frac 1 N\,\e_{ij}\,\delta^{IJ}\,Z
	+2\,g^{IJ}{}_K \bm Z^K\right)
	+2\,i\,\delta_{ij}\,f^{IJ}{}_K \,\bm Z^K.
\ee
\end{itemize}
The rest of the Lie brackets vanish in both cases.

In the above paragraph, we obtained two non-relativistic colored gravity algebras with suitable contractions.
However, it turns out that they do not have non-degenerate invariant bilinear form
due to the additional contraction related to the color decoration.
This problem can be fixed by considering the mixing between the two chiral algebras,
which will be explained in section \ref{non-rel-alg}.
In the next subsection,
we will introduce the generalized \.In\"on\"u-Wigner contraction,
whose concept is crucial to the systematic construction of non-relativistic algebras from the relativistic ones.

\subsection{Generalized \.In\"on\"u-Wigner contraction}

\.In\"on\"u-Wigner contraction \cite{Inonu:1953sp}  is a procedure
for generating Lie algebras from a starting Lie algebra
by taking certain limits of the group-contraction parameters.
The contraction procedure involves singular changes of bases,
so the contracted algebra is not isomorphic to the original algebra.
If a Lie algebra $\mathfrak g$ has a subalgebra $\mathfrak h$, we can decompose $\mathfrak g$ into
\be
\mathfrak g=\mathfrak h\oplus\mathfrak i.
\ee
We then rescale the generators of $\mathfrak i$ by a contraction parameter $c$
and introduce the new generators as
\be
\mathfrak j= \frac 1 c\,\mathfrak i.
\ee
The Lie brackets become
\be
[\mathfrak h,\mathfrak h]\subset \mathfrak h,\qquad
[\mathfrak h,\mathfrak j]\subset \frac 1 c \, \mathfrak h \oplus \mathfrak j,\qquad
[\mathfrak  j,\mathfrak j]\subset \frac 1 {c^2} \,\mathfrak h \oplus \frac 1 c\, \mathfrak j.
\ee
In the singular limit where $c\to\infty$,
we obtain a new Lie algebra,
\be
	\cG(\mathfrak g)=\mathfrak h\oplus \mathfrak j,
\ee
with the Lie brackets,
\be
[\mathfrak h,\mathfrak h]\subset \mathfrak h,\qquad
[\mathfrak h,\mathfrak j]\subset \mathfrak j,\qquad
[\mathfrak j,\mathfrak j]\subset \emptyset ,
\ee
where $\mathfrak j$ corresponds to an Abelian ideal.
As an example, the Poincar\'e algebra can be obtained from the (A)dS algebra
by sending the radius of curvature to infinity,
and the translation generators in the Poincar\'e algebra generate an Abelian ideal.

\subsubsection*{Three-level  contraction}

The standard contraction procedure can be generalized.
Assume the original Lie algebra $\mathfrak g$ admits a decomposition into three parts:
\be
\mathfrak g=\mathfrak i_0\oplus \mathfrak i_1 \oplus \mathfrak i_2,
\ee
with the condition,
\be
	[\mathfrak i_0,\mathfrak i_0]\subset \mathfrak i_0,
	\qquad
	[\mathfrak i_0,\mathfrak i_1]\subset \mathfrak i_0 \oplus \mathfrak i_1.
\ee
If we rescale the generators $\mathfrak i_1$ and $\mathfrak i_2$ by $c^{-1}$ and $c^{-2}$, respectively:
\be
\mathfrak j_0=\mathfrak i_0,\qquad
\mathfrak j_1=\frac 1 c\,\mathfrak i_1,\qquad
\mathfrak j_2=\frac 1 {c^2}\,\mathfrak i_2,
\ee
the contraction limit $c\to\infty$ gives a new algebra,
\be
	\cG(\mathfrak g)=\mathfrak j_0\oplus \mathfrak j_1\oplus \mathfrak j_2,
\ee
with the Lie brackets,
\be
[\mathfrak j_0,\mathfrak j_0]\subset \mathfrak j_0,\quad
[\mathfrak j_0,\mathfrak j_1]\subset \mathfrak j_1,\quad
[\mathfrak j_0,\mathfrak j_2]\subset \mathfrak j_2,\quad
[\mathfrak j_1,\mathfrak j_1]\subset \mathfrak j_2,
\ee
and
\be
[\mathfrak j_1,\mathfrak j_2] \subset \emptyset,\qquad
[\mathfrak j_2,\mathfrak j_2] \subset \emptyset.
\ee
Note that $\mathfrak j_2$ and $\mathfrak j_1+\mathfrak j_2$ are two different ideals of the contracted Lie algebra.
This contraction
actually covers all the non-relativistic limits considered in this paper.

\subsubsection*{Multi-level contraction}

We can further generalize the contraction procedure by considering
arbitrary many ``levels'',
i.e. several integer rescaling dimensions.
The starting point is again a decomposition of the original algebra
\be
	\mathfrak g=\mathfrak i_0\oplus \mathfrak i_1 \oplus \cdots \oplus \mathfrak i_{L},
	\label{multi decomp}
\ee
with the condition
\be
	[\mathfrak i_m,\mathfrak i_n]\subset \mathfrak i_0\oplus \cdots \oplus \mathfrak i_{m+n},
	\qquad
	\mathfrak i_{n>L}=\emptyset.
	\label{i cond}
\ee
After the rescaling
\be
	\mathfrak j_k=\frac 1 {c^k}\,\mathfrak i_k,
\ee
and the contraction $c\rightarrow \infty$, we obtain a new algebra,
\be
	\cG(\mathfrak g)
	=\mathfrak j_0 \oplus \mathfrak j_1
	\oplus \cdots \oplus \mathfrak j_L,
\ee
with the Lie brackets
\be
	[\mathfrak j_m, \mathfrak j_n]\subset \mathfrak j_{m+n},
	\qquad \mathfrak j_{n>L}=\emptyset.
	\label{grading}
\ee
When the levels of the elements are higher than $L/2$,
their commutators will vanish
as the maximum is $L$.
At least half of the ideals $\mathfrak j_n\oplus \mathfrak j_{n+1}\oplus \cdots
\oplus \mathfrak j_L$ are Abelian.
We need $L>1$ to have a non-abelian ideal.

Mathematically, the condition \eqref{i cond} is nothing but a filtration structure,
for which it is more nature to work with
\be
	\mathfrak f_n=\mathfrak i_0\oplus \cdots \oplus \mathfrak i_n.
\ee
Then, the decomposition \eqref{multi decomp} and the condition \eqref{i cond} can be recast as
the definition of the filtered Lie algebra:
\be
	\mathfrak f_0 \subset \cdots\subset \mathfrak f_{L-1}  \subset \mathfrak f_L=\mathfrak g,
	\quad
	{\rm with}\quad
	[\mathfrak f_m, \mathfrak f_n]\subset \mathfrak f_{m+n}.
\ee
For a given filtered (Lie) algebra $\mathfrak g$,
there exists a graded (Lie) algebra $\cG(\mathfrak g)$
with \eqref{grading}.
The latter is referred to as \emph{associated graded (Lie) algebra}.
In the mathematical physics literature, this generalized \.In\"on\"u-Wigner contraction has been studied in \cite{Weimar-Woods}.

An example of multi-level contractions can be found in higher spin algebras.
By definition, the vector space of a higher spin algebra consists of
the spaces $\mathfrak{K}_{s}$ of spin $s$ Killing tensors,
\be
	\mathfrak{g}_{HS}
	=\bigoplus_{s}\mathfrak{K}_{s}.
\ee
Each $\mathfrak{K}_{s}$ is generated by the traceless tensor $M_{a_1\cdots a_{s-1},b_1\cdots b_{s-1}}$
with the permutation symmetry of two-row Young diagram $(s-1,s-1)$.
The Lie brackets of the higher spin algebra satisfy
\be\label{higher-spin-bra}
	[\mathfrak{K}_{s_1}, \mathfrak{K}_{s_2}]
	\subset \mathfrak{K}_{s_1+s_2-2}\oplus \mathfrak{K}_{s_1+s_2-3} \oplus \cdots.
\ee
The spins on the right hand side of \eqref{higher-spin-bra} is bounded from above
because one cannot construct $(r,r)$ tensors with $r\ge s_1+s_2-2$
in terms of $M_{(s_1-1,s_2-1)}$ and $M_{(s_2-1,s_2-1)}$.\footnote{The case of $r=s_1+s_2-2$ is ruled out
by the antisymmetricity of Lie bracket.}
Hence, taking $\mathfrak{i}_{s-2}=\mathfrak{K}_s$, one can
see that the higher spin algebra enjoys the filtration structure \eqref{i cond}.
An unusual point is that a higher spin algebra is typically infinite dimensional,
so it involves infinitely many levels ($L=\infty$).
In three dimensions, one can consider finite dimensional higher spin algebras.
For instance, $sl(N,\mathbb{R})$ is isomorphic to the chiral part
of the higher spin algebras involving spin $2$ to $N$.
The contracted higher spin algebra still satisfies all the higher spin covariance conditions,
but does not admit an invariant bilinear form.
It would be interesting to check whether a central extension of this contracted higher spin algebra
may admit a non-degenerate bilinear form,
as in the case of three-dimensional non-relativistic spin-two algebras.

\subsection{Non-relativistic colored gravity algebra}\label{non-rel-alg}

In this subsection, we will study the non-relativistic limits of the three-dimensional colored gravity algebra
using the generalized \.In\"on\"u-Wigner contraction.
The guiding principle
is that the limiting algebra should have a non-degenerate bilinear form.
This non-degeneracy is related to the physical requirement
that the kinetic terms are present.\footnote{In
a more general setting, we may consider degenerate bilinear form.
Then some fields will not have kinetic terms and only appear in the cubic terms.}
As we have seen before,
this condition cannot be met in the chiral sector,
so we will work with the full algebra from now on.

We shall define the generators of the non-relativistic colored gravity algebras
in terms of the relativistic ones and the contraction parameter $c$.
For the color-singlet generators,
we will use the same contraction ansatz as the doubly extended NH algebra
$\mathfrak g^{++}_{\L}$, i.e. \eqref{PG}, \eqref{HJ mix} and \eqref{MS mix}.
This choice corresponds to the three-level  ansatz:
\ba
	&& \mathfrak{j}_0\quad : \quad
	\cZ+\cL_0\,,\quad \tilde \cZ+\tilde \cL_0\,,\nn
	&& \mathfrak{j}_1\quad : \quad
	\frac{\cL_i}c,\quad \frac{\tilde\cL_i}c\,, \nn
	&& \mathfrak{j}_2\quad : \quad
	\frac{\cZ-\cL_0}{c^2}\,, \quad \frac{\tilde\cZ-\tilde\cL_0}{c^2}\,.
\ea
The non-singlet generators of the colored gravity algebra
should be appended to $\mathfrak{j}_0$, $\mathfrak{j}_1$ and $\mathfrak{j}_2$
such that the contracted algebra admits a non-degenerate bilinear form.
From the symmetry consideration, the bilinear form of the non-singlet sector generators
are determined by that of the singlet sector as
\be
	\la a\otimes \bm\cT^I, b\otimes \bm\cT^J\ra
	=\la a,b\ra \,\delta^{IJ}\,,
\ee
where $a$ and $b$ are elements of $gl(2)\oplus gl(2)$ and their bilinear form is given in Section \ref{sec: inv bf}\,.
Let us summarize it here for the reader's convenience:
due to the associative structure,
the bilinear form should be diagonal, i.e. $\la \cZ, \tilde \cZ\ra=\g=0$, then we have
\ba
	&& \la \cZ,\cZ\ra=c^2\,\ell+\b/2\,,
	\qquad \la \cL_a,\cL_b\ra=\eta_{ab}\,(c^2\,\ell-\b/2)\,,\nn
	&& \la \tilde\cZ,\tilde\cZ\ra=c^2\,\tilde\ell+\tilde \b/2\,,
	\qquad \la \tilde \cL_a,\tilde \cL_b\ra=\eta_{ab}\,(c^2\,\tilde\ell-\tilde \b/2)\,.
\ea
As one can see, they all scale as $c^2$ hence diverge unless we
rescale them appropriately.
About the spatial part of non-singlet generators, we take the analogous rescaling as the singlet ones:
\be
\mathfrak{j}_1\quad : \quad
 \frac{\cL_i\otimes \bm \cT^I}{c},\quad \frac{\tilde \cL_i\otimes \bm \cT^I}{c},
\ee
hence these generators belong to $\mathfrak{j}_1$\,.
About the remaining generators, there are a few consistent possibilities.
In the classification, an important guideline is the fact that  $\mathfrak{j}_0$ must form a subalgebra
for a consistent contraction.
Moreover, due to the presence of $su(N)$ part, the $gl(2)\oplus gl(2)$ part of the non-singlet elements in $\mathfrak{j}_0$
should be closed under anti-commutator:
let us write the non-singlet part of $\mathfrak{j}_0$ as $\mathfrak{A}\otimes \bm \cT^I$,
then $\mathfrak{A}$ should satisfy
\be
	\{\mathfrak{A},\mathfrak{A}\}\subset \mathfrak{A}\,,
\ee
where $\mathfrak{A}$ is a subspace of $gl(2)\oplus gl(2)$.
With the above condition, let us examine what are the possible candidates
for the $\mathfrak{A}$ elements.
The first guess is the combinations analogous to the singlet sector:
\be
 (\cZ+\cL_0)\otimes \bm \cT^I,\qquad (\tilde \cZ+\tilde \cL_0)\otimes \bm \cT^I,
\ee
but one can easily see that neither $\cZ+\cL_0$ nor $\tilde \cZ+\tilde \cL_0$
is closed under anti-commutator.
In addition, we cannot consider $\cZ$ or $\tilde\cZ$ because their bilinear form will diverge.
In fact, the only possible way to have finite bilinear form in $\mathfrak{j}_0$ part --- that is, without rescaling
the generators --- is to cancel the $c^2$ terms: for instance, $\la \cZ+\cL_0,\cZ+\cL_0\ra
=\la \cZ,\cZ\ra+\la\cL_0,\cL_0\ra=(c^2\,\ell+\b/2)-(c^2\,\ell-\b/2)=\b$  in the singlet sector.
Noticing this, we can see
\be
	\mathfrak{j}_0\quad :\quad (\cZ+\tilde \cZ)\otimes \bm \cT^I
	\label{ZZ0}
\ee
has a finite bilinear form: $\la \cZ+\tilde \cZ,\cZ+\tilde \cZ\ra=\frac{\b+\tilde \b}2$, and it is closed
under anti-commutator: $(\cZ+\tilde \cZ)^2=\cZ+\tilde \cZ$.
The similar combination $\cZ-\tilde \cZ$ has a finite bilinear form but is not closed.
For the non-degeneracy of the bilinear form, the choice \eqref{ZZ0} should be
accompanied by
\be
	\mathfrak{j}_2\quad :\quad \frac{(\cZ-\tilde \cZ)\otimes \bm \cT^I}{c^2}.
	\label{ZZ2}
\ee
Now there remain only two generators $\cL_0$ and $\tilde \cL_0$\,,
and there are two non-trivial choices:
\be
	\mathfrak{j}_0\ :\  (\cL_0\pm \tilde \cL_0)\otimes \bm \cT^I,
	\qquad
	\mathfrak{j}_2\ :\  \frac{(\cL_0\mp \tilde \cL_0)\otimes \bm \cT^I}{c^2}.
	\label{L+-}
\ee
Note that both signs give consistent contractions.
There also exist rather trivial options:
{\it i)} the entire non-singlet generators are in $\mathfrak{i}_1$;
{\it ii)}
we have \eqref{ZZ0} and \eqref{ZZ2} but $\cL_0\otimes \bm\cT^I$ and $\tilde \cL_0\otimes \bm\cT^I$ are in $\mathfrak{i}_1$.
The latter two cases are mathematically consistent options, but
they treat the color-decorated isometry generators $\cL_a\otimes \bm\cT^I$
in the Lorentz invariant manner, hence they are unnatural for
the interpretation as non-relativistic physics.
In the following, we consider only the ``non-trivial'' cases. See
Appendix \ref{sec: case01} for the other two cases.
Another implicit assumption is that the $su(N)$ symmetry is not broken in the non-relativistic limits.
There will be more possibilities if we relax this assumption.

\section{Chern-Simons Formulation }\label{sec-non-relativistic-colored-action}

In this section, we derive the actions of
the non-relativistic colored gravity theories.
There are two ways to compute these non-relativistic actions.
The first way is to directly evaluate the Chern-Simons actions
whose gauge fields take values in the non-relativistic gauge algebras.
The second way is to take the non-relativistic limits of the relativistic Chern-Simons action
according to the contractions.
We will use the first method in the current section and
discuss the second one in Appendix \ref{contraction-action}.

\subsection{Gauge algebras}

Before starting our discussion, let us summarize
the algebraic structure of non-relativistic colored gravity symmetry.
For a more clear physical interpretation, we use ``$\cJ$, $\cP$'' bases
rather than the chiral and anti-chiral bases.
Since the singlet sector is identical to the doubly extended NH algebra,
we directly start with the non-singlet ones.
The contractions we have considered in the previous section are
\be
\bm C^I=\cZ_{\cJ}\otimes \bm \cT^I,\quad
\bm D^I=\frac 1 {c^2}\,\cZ_{\cP}\otimes \bm \cT^I,
\ee
\be
\bm G_i^I=\frac 1 {c} \cJ_i\otimes \bm \cT^I,\quad
\bm P_i^I=\frac 1 {c} \cP_i\otimes \bm \cT^I.
\ee
These generators transform covariantly under the NH symmetry as
\be
[J,\,\bm G_i^I]
=2\,\epsilon_{ij}\,\bm G_j^I,
\,\quad
[J,\,\bm P_i^I]
=2\,\epsilon_{ij}\,\bm P_j^I,
\ee
\be
[H,\,\bm G_i^I]
=2\,\epsilon_{ij}\, \bm P_j^I,
\,\quad
[H,\,\bm P_i^I]
=-2\,\L\,\epsilon_{ij}\,\bm G_j^I,
\ee
and they have non-trivial Lie brackets among themselves:
\be
[\bm C^I,\, \bm C^J]=2\,i\, f^{IJ}{}_K\, \bm  C^K,
\quad
[\bm C^I,\,\bm D^J]=2\,i\, f^{IJ}{}_K\, \bm D^K,
\ee
\be
[\bm C^I,\,\bm G_i^J]=2\,i\, f^{IJ}{}_K\, \bm G_i^K,
\quad
[\bm C^I,\, \bm P_i^J]=2\,i\, f^{IJ}{}_K\,  \bm P_i^K.
\ee
Besides $\bm C^I, \bm D^I, \bm G_i^I$ and $\bm P_i^I$,
we also have the temporal and central generators.
The two possibilities of the sign in \eqref{L+-}
are translated into the following two cases:
\be
 {\rm Type\ A}\quad :\quad \bm J^I=\cJ_0\otimes \bm \cT^I,\quad
\bm M^I=\frac 1 {c^2}\, \cP_0\otimes \bm \cT^I,
\ee
\be
 {\rm Type\ B}\quad :\quad
\bm H^I=\cP_0\otimes \bm \cT^I,\quad
\bm S^I=\frac 1 {c^2}\,\cJ_0\otimes \bm \cT^I.
\ee
Only one of the two central generators in the doubly extended NH algebra admits color decoration.
Let us present the Lie brackets of the two cases.

\subsubsection*{Type A}
The non-zero commutators involving one singlet and one non-singlet are
\be
[\bm J^I,\,G_i]
=2\,\epsilon_{ij}\, \bm G_j^I
,\quad
[\bm J^I,\,P_i]
=2\,\epsilon_{ij}\, \bm P_j^I,
\ee
\be
[ \bm G_i^I,\, P_j]
=[ G_i,\,\bm P_j^I]
=-2\,\epsilon_{ij}{}\, \bm  M^I.
\ee
The non-zero commutators involving only the non-singlet sectors are
\be
[\bm C^I,\,\bm J^J]=2\,i\, f^{IJ}{}_K\, \bm J^K,
\quad
[\bm C^I,\, \bm M^J]=[\bm D^I,\,\bm J^J]
=2\,i\, f^{IJ}{}_K\,  \bm M^K,
\ee
\be
[\bm J^I,\,\bm J^J]
=
-2\,i\, f^{IJ}{}_K\, \bm C^K,
\quad
[\bm J^I,\, \bm M^J]
=
-2\,i\,f^{IJ}{}_K\,\bm D^K,
\ee
\be
[\bm J^I,\,\bm G_i^J]
=
2\,\epsilon_{i j}\left(\frac 1 N\d^{IJ} G_j+g^{IJ}{}_K\,\bm G_j^K\right)
,
\ee
\be
[\bm J^I,\, \bm P_i^J]
=
2\,\epsilon_{i j}\left(\frac 1 N\d^{IJ} P_j+g^{IJ}{}_K\,\bm P_j^K\right),
\ee
\be
[\bm G_i^I,\,
\bm G_j^J]=
-\frac 1 \L \,[\bm P_i^I,\,
\bm P_j^J]
=
\frac 2{N}\,\epsilon_{ij}\,\d^{IJ}S,
\ee
\be
[\bm G_i^I,\, \bm P_j^J]
=
\frac 2 {N}\,\epsilon_{ij}\,\d^{IJ}\,M
-2\,\epsilon_{ij}\,g^{IJ}{}_K\, \bm M^K
+
2\,i\,\d_{ij}\,f^{IJ}{}_K\,\bm D^K.
\ee

\subsubsection*{Type B}

The non-zero commutators containing one singlet and one non-singlet sectors are
\be
[\bm H^I,\,G_i]
=2\,\epsilon_{ij}\, \bm P_j^I
,\quad
[\bm H^I,\, P_i]
=-2\,\L\,\epsilon_{ij}\,\bm G_j^I,
\ee
\be
[G_i,\,  \bm G_j^I]
=-\frac 1 \L\, [ P_i,\,\bm P_j^I]
=-2\,\epsilon_{ij}{}\, \bm  S^I.
\ee
The non-zero Lie brackets involving only the non-singlet sector are
\be
[\bm C^I,\, \bm H^J]=2\,i\, f^{IJ}{}_K\,  \bm H^K,
\quad
[\bm C^I,\,\bm S^J]=-\frac 1 \L [\bm D^I,\,
 \bm H^J]=
 2\,i\, f^{IJ}{}_K\, \bm S^K,
\ee
\be
[ \bm H^I,\,
 \bm H^J]
=
2\,i\,\L\, f^{IJ}{}_K\,\bm C^K
,
\quad
[\bm S^I,\, \bm H^J]
=
-2\,i\,f^{IJ}{}_K\,\bm D^K,
\ee
\be
[\bm H^I,\,\bm G_i^J]
=
2\,\epsilon_{i j}\left(\frac 1 N\d^{IJ} P_j+g^{IJ}{}_K\,\bm P_j^K\right)
,
\ee
\be
[\bm H^I,\, \bm P_i^J]
=
-2\,\L\,\epsilon_{i j}\left(\frac 1 N\,\d^{IJ} G_j+g^{IJ}{}_K\,\bm G_j^K\right),
\ee

\be
[\bm G_i^I,\,
\bm G_j^J]=
-\frac 1 \L\, [\bm P_i^I,\,
\bm P_j^J]
=
\frac 2 {N}\,\epsilon_{ij}\,\d^{IJ}S
-2\,\epsilon_{ij}\,g^{IJ}{}_K \bm S^J,
\ee
\be
[\bm G_i^I,\, \bm P_j^J]
=
\frac 2 {N}\,\epsilon_{ij}\,\d^{IJ}M
+
2\,i\,\d_{ij}\,f^{IJ}{}_K\,\bm D^K.
\ee

\subsubsection*{Bilinear form}

Let us also remind the reader that the invariant bilinear form reads
\be
	\la J, M\ra=\la S, H\ra=1\,,\qquad
	\la P_i,\,G_j\ra=\delta_{ij}\,,
\ee
\be
	\la \bm C^I ,\,\bm D^J \ra=\d^{IJ},\qquad
	\la \bm  P_i^I,\, \bm G_j^J \ra=\d_{ij}\,\d^{IJ},
\ee
and
\be
	\la \bm  J^I ,\,\bm M^J \ra=-\d^{IJ},\qquad\
		\la \bm  H^I ,\,\bm S^J \ra=-\d^{IJ},
\ee
for type A and B, respectively.
Here, we took the choice $\b=\tilde\b=\g=0$ for simplicity.

\subsection{Actions}\label{explicit-action}

Now we consider the Chern-Simons actions of the non-relativistic colored gravities.
The one-form gauge fields are
 \ba\label{one-form-A}
\text{Type A}\quad : \quad
 \cA&=&j J+s\, S+h\, H+m\, M+\O^i\, G_i+E^i\, P_i
 \nn&&
+\s_I\,\bm C^I+ \varsigma_I\, \bm D^I
+ \vartheta_I\, \bm J^I+ \varpi_I\, \bm M^I
+ \theta_I^i\, \bm G^I_i+ \pi_I^i\,\bm P^I_i,
\ea
 \ba\label{one-form-B}
 \text{Type B}\quad : \quad
 \cA&=&j J+s\, S+h\, H+m\, M+\O^i\, G_i+E^i\, P_i
 \nn&&
+\s_I\,\bm C^I+ \varsigma_I\, \bm D^I
+ \vartheta_I\, \bm S^I+ \varpi_I\, \bm H^I
+ \theta_I^i\, \bm G^I_i+ \pi_I^i\,\bm P^I_i,
\ea
which correspond to the non-relativistic colored gravity algebras constructed in Section \ref{non-rel-alg}.
It is now sufficient to plug the above into the Chern-Simons action,
\be
	S=\frac{\k}{4\pi}\,\int \la \cA \wedge d\cA\ra
	+\frac23\,\la \cA\wedge \cA\wedge \cA\ra.
\ee
In terms of the component fields, this action can be decomposed into several pieces:
\be
	S=S_{\text{gravity}}+S_{\text{gauge}}+S_{\text{matter}}.
	\label{Action 3}
\ee
Here, $S_{\text{gravity}}$ is the usual uncolored non-relativistic action,
\ba\label{gra-action}
S_{\text{gravity}}&=&
\frac {\kappa\,N} {2\pi}
\int E^i\wedge \nabla\,\O_i+m\wedge d\,j
+h\wedge [d\,s+\epsilon_{ij}\,(\O^i\wedge \O^j
-\L\,E^i\wedge E^j)],
\ea
where the covariant derivative $\nabla$ acts on ``spatial'' one-form $\o_i$ as
\be
	\nabla\,\o_i=d\,\o_i+\epsilon_{ij}\left(\o^j\wedge j-j\wedge \o^j\right).
\ee
About the other pieces of the action,
it is convenient to introduce the $su(N)$-valued fields as
\be
\bm\s=\s_I\,\bm \cT^I,\quad \bm \varsigma=\varsigma_I\, \bm \cT^I,\quad
\bm\vartheta=\vartheta_I\, \bm \cT^I,\quad \bm \varpi=\varpi_I\, \bm \cT^I,\quad
\bm \theta^i=\theta_I^i\, \bm \cT^I,\quad \bm\pi^i=\pi_I^i\,\bm \cT^I,
\ee
and write the action in terms of $\text{Tr}$, the trace in the adjoint representation.
Then, the gauge field part $S_\text{gauge}$ takes the simple form,
\be
S_{\text{gauge}}
=
\frac {\kappa } {2\pi}
\int
 \text{\tr }\left[
\bm  \varsigma\wedge (d\bm \s+\bm\s\wedge\bm\s)
\right].
\label{S gauge}
\ee
The matter field part $S_\text{matter}$
depends whether we consider the type A or B:
\be
	S_\text{matter}=S_\text{matter,0}+S_\text{type A/B}\,.
\ee
The common part $S_\text{matter,0}$ is
\be
S_{\text{matter,0}}=
\frac {\kappa } {2\pi}
\int
 \tr\left[
 -\bm \vartheta\wedge D\,\bm  \varpi
+ \bm \theta^i\wedge D\,\bm \pi_i
+ \e_{ij}\,h\wedge(\bm \theta^i\wedge\bm \theta^j-\L\, \bm\pi^i \wedge \bm\pi^j)\right],
\label{S matter 0}
\ee
where the covariant derivative $D$ acts on
the $su(N)$ valued  one-form $\bm\o$ as
\be
D\,\bm \o=\nabla\,\bm\o+\bm\s\wedge\bm \o+\bm\o\wedge\bm\s,
\ee
and $\bm\o$ can have spatial indices.

Finally, the last piece which depends on the type of the non-relativistic limit is
\ba
S_\text{type\ A}\eq\frac {\kappa } {2\pi}
\int
\tr\left[-\bm\varsigma\wedge\bm \vartheta\wedge\bm \vartheta
+2\,\e_{ij}\,\bm\vartheta\wedge\left(\bm\pi^i\wedge\O^j
 +\bm\theta^i\wedge E^j+ \bm  \theta^i \wedge \bm \pi^j
 \right)\right],\\
S_\text{type\ B}\eq \frac {\kappa } {2\pi}
\int\tr\Big[
\L\,\bm\varsigma\wedge\bm \varpi\wedge\bm \varpi
+
\epsilon_{ij}\,\bm\varpi\wedge\left(
2\,\bm\theta^i\wedge\O^j
 -2\,\L\,\bm\pi^i\wedge E^j+\bm  \theta^i \wedge \bm \theta^j
 -\L\,\bm  \pi^i \wedge \bm \pi^j\right)
 \Big].\nn
 \ea

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

In this work, we have studied the non-relativistic limits of three-dimensional colored gravity in terms of generalized \.In\"on\"u-Wigner contractions.
Let us
conclude our work with a brief summary of what have been done.
In Section \ref{sec-non-relativistic}, we revisited the three-dimensional non-relativistic isometry algebras and their bilinear forms.
We reviewed various central extensions of the Galilei and Newton-Hooke algebras.
Then we discussed their chiral decompositions, how they arise from the contractions of relativistic algebras,
and the issue of invariant bilinear forms.
In Section \ref{sec-non-relativistic-colored-alg},
we explained the obstruction in the chiral decompositions of non-relativistic colored gravity
and constructed the consistent non-relativistic colored gravity algebras using  generalized \.In\"on\"u-Wigner contractions.
In Section \ref{sec-non-relativistic-colored-action},
we derived the non-relativistic colored gravity actions
in the Chern-Simons formulation using the non-relativistic algebras
constructed in Section \ref{sec-non-relativistic-colored-alg}.

\acknowledgments

We thank Karapet Mkrtchyan for useful discussions in the early stages of the work.
WL wants to thank Yang Lei for useful discussions.
This work is supported by a grant from Kyung Hee University in 2017 (KHU-20171742).

redpajama_set_name: RedPajamaArXiv


text:
Drama Film Film Genre Home Page Romance
[FILM REVIEW] TRUE THINGS Review (2022)
December 31, 2022 December 31, 2022 Dave Alex First, Ann Firbank, Chris Brazier, David Griffiths, Elizabeth Rider, Frank McCusker, Harry Wootliff, Hayley Squires, Michael Moreland, Molly Davies, Nathan Ampofo, Peter Krausz, Ruth Wilson, Scarlett Leigh, Tom Burke, Tom Weston-Jones, True Things, UK
Alex First, David Griffiths and Peter Krausz deliver their review of True Things.
Most Popular Stars 16/1/2015-23/1/2015
January 24, 2015 Dave Anna Kendrick, Benedict Cumberbatch, Bradley Cooper, Chloe Grace Moretz, Chris Pratt, Dakota Johnson, Eddie Redmayne, Ellar Coltraine, Emma Stone, Evangeline Lilly, Hayley Atwell, Jamie Dornan, Jennifer Aniston, Jennifer Lawrence, Lee Pace, Maggie Grace, Michael Keaton, Rosamund Pike, Ruth Wilson, Scarlett Johansson, Shailene Woodley, Sienna Miller, Taylor Negron, Theo James, Tom Hardy
We take a look at the most popular stars online over the past week. Rosamund Pike Hayley Atwell Chloe Grace
Film Film and TV Articles Film News News
2015 Golden Globes Nominations Announced
December 12, 2014 Dave A Most Violent Year, Alan Cumming, Alejandro Gonzalez Inarritu, Alexandre Desplat, Allison Janney, Allison Tolman, Amy Adams, Annie, Antonio Sanchez, Ava DuVernay, Benedict Cumberbatch, Big Eyes, Big Hero 6, Bill Murray, Billy Bob Thornton, Birdman, Boyhood, Cake, Christoph Waltz, Claire Danes, Clive Own, Colin Hanks, Common, David Fincher, David Oyelowo, Dominic West, Don Cheadle, Downton Abbey, Eddie Redmayne, Edie Falco, Edward Norton, Emily Blunt, Emma Stone, Ethan Hawke, Fargo, Felicity Jones, Force Majeure, Foxcatcher, Frances McDormand, Frances O'Connor, Game Of Thrones, Gett: The Trial Of Viviane Amsalem, Gina Rodriguez, Girls, Gone Girl, Hans Zimmer, Helen Mirren, House Of Cards, How To Get Away With Murder, How To Train Your Dragon 2, Ida, Inherent Vice, Interstealler, Into The Woods, J.K. Simmons, Jake Gyllenhaal, James Spader, Jane The Virgin, Jeffrey Tambor, Jennifer Aniston, Jessica Chastain, Jessica Lange, Joanne Froggatt, Joaquin Phoenix, Johann Johannsson, John Legend, Jon Voight, Julia Louis-Dreyfus, Julianna Marguiles, Julianne Moore, Kathy Bates, Keira Knightley, Kevin Spacey, Lana Del Ray, Lena Dunham, Lenny Kaye, Leviathan, Liev Schreiber, Lorde, Louis C.K., Maggie Gyllenhaal, Maps To The Stars, Mark Ruffalo, Martin Freeman, Matt Bomer, Matthew McConaughey, Meryl Streep, Michael Keaton, Michelle Monaghan, Nightcrawler, Noah, Nurse Jackie, Olive Kitteridge, Orange Is The New Black, Patricia Arquette, Patty Smith, Pride, Quvenzhane Wallis, Ralph Fiennes, Reese Witherspoon, Richard Linklater, Ricky Gervais, Robert Duvall, Robin Wright, Rosamund Pike, Ruth Wilson, Selma, Sia, Silicon Valley, St Vincent, Steve Carell, Still Alice, Tangerines, Taylor Schilling, The Affair, The Blacklist, The Book Of Life, The Boxtrolls, The Good Wife, The Grand Budapest Hotel, The Hundred-Foot Journey, The Hunger Games: Mockingjay Part 1, The Imitation Game, The Judge, The Knick, The Lego Movie, The Missing, The Normal Heart, The Theory Of Everything, Transparent, Trent Reznor, True Detective, Uzo Aduba, Viola Davis, Wes Anderson, Whiplash, Wild, William H. Macy, Woody Harrelson
The 2015 Golden Globes nominations are now in… here they are. BEST MOTION PICTURE – DRAMA Boyhood Selma The Imitation
Biography Comedy Drama Family/Kids Film Film Genre Home Page Music / Musicals
Saving Mr. Banks (2014)
January 6, 2014 December 25, 2013 Dave Andy McPhee, Annie Rose Buckley, B.J. Novak, Bradley Whitford, Claire Bocking, Colin Farrell, David Griffiths, Dendrie Taylor, Emma Thompson, Ginger Pauley, Greg King, Jason Schwartzman, John Lee Hancock, Kathy Baker, Kelly Marcel, Kimberly D'Armond, Kristopher Kyer, Lily Bigham, Lynly Ehrlich, Melanie Paxson, Melissa Bickerton, Michelle Arthur, Paul Giamatti, Rachel Griffiths, Ronan Vibert, Ruth Wilson, Saving Mr. Banks, Saving Mr. Banks review, Sue Smith, Thomas R. Waters, Tom Hanks, Victoria Summer
Summary: When Walt Disney's daughters begged him to make a movie of their favourite book, P.L. Travers' "Mary Poppins," he
Film Film News News
The Lone Ranger – TV Spots
June 23, 2013 Dave Armie Hammer, Barry Pepper, Gore Verbinski, Helena Bonham Carter, James Badge Dale, Johnny Depp, Ruth Wilson, The Lone Ranger, The Lone Ranger TV Spots, Tom Wilkinson, William Fichtner
Disney have released some new television spots/ads for the release of 'The Lone Ranger'. The film which stars Johnny Depp,
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February 14, 2013 June 1, 2013 Dave Aaron Taylor-Johnson, Alexandra Roach, Alicia Vikander, Anna Karenina, Anna Karenina Review, Antony Byrne, Aryhan Galieva, Beatrice Morrissey, Bill Skarsgard, Bodil Blain, Buffy Davis, Byran Hands, Cara Delevigne, Carl Grose, Cecily Morrissey, David Wilmot, Domhnall Gleeson, Emerald Fennell, Emily Watson, Eric MacLennan, Eros Vlahos, Freya Galpin, Giles King, Guro Nagelhus Schia, Henry Lloyd-Jones, Hera Hilmar, Holliday Grainger, Joe Wright, Jude Law, Jude Monk McGowan, Keira Knightley, Kelly Macdonald, Kenneth Collard, Kyle Soller, Leo Tolstoy, Luke Newberry, Marine Battier, Matthew Macfadyen, Max Bennett, Michelle Dockery, Nicholas Blatt, Octavia Morrissey, Olivia Williams, Osar McNamara, Paul Ham, Pip Torrens, Raphael Personnaz, Ruth Wilson, Sam Cox, Steve Evets, Susanne Lothar, Tannishtha Chatterjee, Theo Morrissey, Thomas Howes, Tillie-Bett Grant, Tom Stoppard
Summary: The third collaboration of Academy Award nominee Keira Knightley with acclaimed director Joe Wright, following the award-winning boxoffice successes

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Celebrating Snow White's 80th Anniversary, the second release in the Christmas Series features the princess with skin as white as snow. She is accompanied with a Christmas Dopey to complete the two piece set. Both of the figurines are made from resin. Each piece is hand painted and slight colour variations are to be expected which makes each piece unique. Supplied in branded gift box. Not a toy or children's product. Intended for adults only.


text:
Carson Kitaif
Photo by David Ammon
Carson Kitaif, a Tylertown native and Mississippi State senior wildlife, fisheries and aquaculture major, became enraptured with raptors at age 12. These birds of prey include the barred owl, pictured here alongside Kitaif.
"I saw a raptor in action when I was young, and it hooked me," he said.
Now, a decade later, he is a certified Master of Falconry, able to legally keep birds of prey—as it is illegal to trap raptors without permits—and also take apprentice falconers under his wing and guide them through the intricacies and dedication the sport requires.
Raptors, the birds of prey associated with falconry, require near-constant supervision and training in order to build a beneficial relationship.
"Having a raptor is like having a child, no doubt. It requires 365 days a year—an everyday commitment," Kitaif said. This obligation is why Kitaif, upon deciding to attend MSU for its focus on wildlife and agriculture, also decided not to keep a bird while in school.
"I couldn't have given a bird the dedication it needed while I was at Mississippi State," Kitaif said.
Despite being the responsibility equivalent to that of children, the relationship with falcons is much less warm.
"There's absolutely a mutual trust and respect, but it's purely utilitarian. My bird uses me for food, and I use her for the hunt. It may seem cold, but honestly, the fact that the bird will always be one step away from rewilding itself is incredible. Just getting to witness the rawness of nature in action 10 feet away is enough to keep me hooked," Kitaif said.
Another aspect to falconry is the genuine desire to see these birds succeed. Young birds of prey have a staggeringly low survival rate. A certified falconer cares for and trains them, knowing the birds will eventually leave the falconer with newfound confidence and competence.
Other falconers focus on the rescue and rehabilitation of birds, no doubt to the joy of the owl pictured above. She was found injured in November and, thanks to the efforts of dedicated wildlife conservationist and MSU Associate Professor Scott Rush, is now healthy and being utilized for educational programs emphasizing society's closeness to nature.
Kitaif, having chosen both MSU and his major from a love of agriculture and wildlife, is excited to continue working closely with nature.
"I'm hoping to do some international research and maybe go to Mongolia or central Asia where there are bears or snow leopards. I know there's an irony with the birds, but I'm also a big cat guy," Kitaif said.
Even with his shifting wildlife focus from birds of prey to bigger mammals in Mongolia, Kitaif assures falconry will never leave him.
"I think falconry will be a part of my life for a long time. There is something surreal about being involved in it, witnessing the predator/prey experience first-hand, as it has taken place for thousands of years," Kitaif said.
By Reagan Poston on Friday, October 25, 2019 - 2:18 pm
Suggest an Our People
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Finally a job board for everyone; for students and alumni of every College, University, Graduate School, High School, and Vocational and Adult Education School in the United States. CollegeJobBoard is designed to help students and alumni find internships, temporary jobs, part-time jobs and full-time jobs.
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text:
Q: Localization Strategy We're currently debating two strategies of localization:
A. Have an XML file for the structure of the business objects with a Localized key to a separate CSV file for the translation.
for eg.  /Resources/Schema.xml

in a separate CSV file: we have all key/value pairs for the translations:
/Resources/Localized.txt
Model_Title, Title, Title (in French), ...
This way, when the structure changes, we just change XML once while the LocalizedKey's are in place.
B. Have separate XML files for each language based on Culture.
eg. have two files:
/Resources/en-US/US-Localized.xml
/Resources/fr-AU/AU-Localized.xml
This way, they will have same schema but separate files. Therefore the user would have to make sure that the schemas are the same as they would need to change it twice as opposed to Option #1 where they can just change it once.
However, the readability here is much better since the user would not have to track the key the make the changes.
What are your thoughts/ideas on the strategies I suggested?
Thanks,

A: It is not clear about the environment -- web? desktop? internal enterprise integrated something-or-other? Is there any particular reason you aren't using whatever i18n framework your tool chain supports (gettext, .NET resource files...)?
In general I'd say you want to separate out resources by culture (but to be honsest, fr_AU should be rare) to have better maintainability and do not have to load the entire file for all per-culture-versions in many situations. This is especially true if your number of supported languages/cultures goes into the dozens or more.
However, it would be important to accommodate XML schema changes. The XML could be auto-generated, from simpler structures (key-value, either in a database or files) and validated via a common schema.
This is whether (as commenters noted) you are providing localized products or customers can create their own localizations.

A: In general, you should consider existing tools, rather than start from the scratch.
In .net we are using Data Driven ASP.NET Localization
Resource Provider and Editor Created by rick strahl

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text:
Ilga Kreituse, amb nom de soltera Ilga Grava, (Tērvete, municipi de Tērvete, 5 de juliol de 1952) és una historiadora i política letona, Presidenta del Saeima entre 1995 i 1996.

Biografia
Nascuda a Tērvete el 5 de juliol de 1952, poc després es va traslladar a viure amb la seva família a Jurmala. El 1970 es graduà a l'escola secundària de Jurmala núm. 3. De 1972 a 1977 estudià a la facultat d'Història i Filosofia de la Universitat de Letònia. De 1977 a 1980 cursà el post-grau a la Universitat Estatal de Moscou. El 1982, defensà la tesi "Sobre els mètodes matemàtics de la història de la ciència". També va treballar com a professora a l'Institut Estatal de Riga núm. 1.

Fou escollida com a diputada a la 5a legislatura del Parlament letó (Saeima) pel Partit de Centre Democràtic (DCP). Aquest partit es fusionà el juny de 1995 amb Saimnieks per esdevenir el Partit Democràtic «Saimnieks» (DPS) i concórrer a les eleccions legislatives d'aquell any. En els comicis fou primera força i Kreituse va ser escollida Presidenta del Saeima. Entre desembre de 1996 i setembre de 1997 fou vicepresidenta. El 1996 es va postular a les eleccions presidencials i va obtenir el segon lloc, darrere del reelegit Guntis Ulmanis. Al setembre de 1996, abandonà el partit però no va renunciar a l'acta de diputada de la 6a legislatura. Després de la divisió del Partit Democràtic «Saimnieks», es van convertir, amb el seu marit Aivars Kreituss, en la líder del Partit dels Treballadors, de recent formació. A les eleccions legislatives de 1998, es va presentar en coalició amb la Unió Demòcrata Cristiana i el Partit Verd de Letònia, però la llista no va obtenir el suport necessari per a entrar al Parlament. Escollida per a l'Ajuntament de Riga l'any 2001, va renunciar en favor d'Aivars Kreituss. A les eleccions de 2002, juntament amb el Partit Socialdemòcrata Obrer Letó, no van obtenir els vots necessaris per passar la barrera del 5% necessari i obtenir representació al Parlament letó.

Referències

Persones de Zemgale
Polítics letons
Historiadors europeus
Diputats del Saeima
Presidents del Saeima
Alumnes de la Universitat de Letònia
Distingits amb l'Orde de les Tres Estrelles

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text:
DISCLAIMER: All opinions in this column reflect the views of the author(s), not of EURACTIV Media network.
By Dan Wolff and Laure Roux
18-04-2018 (updated: 21-12-2022 )
Rail freight struggles to remain competitive and continues to lose market shares. It suffers from remaining barriers to market entry and from the strong competition of heavy goods vehicles. [Photographer]
As the EU moves into the era of "smart mobility", Dan Wolff wonders whatever happened to the idea of multimodal transport. Just because road freight innovates faster than other modes doesn't mean policymakers should give up on multimodality, he argues.
Dan Wolff is managing partner at Eurotran, an advisory firm specialised in EU transport policies. Laure Roux is a consultant at Eurotran.
Back in 2001, the European Commission was all about modal shift from road to greener transport modes. In the 2010s, multimodality and intermodality (i.e. the combination and optimal use of various modes) became the new EU religion for transport. It seems that we are now moving to an era of smart mobility, regardless of how many modes are involved.
Does the beginning of multi-speed Europe mean the end of multimodal transport?
Rail freight struggles to remain competitive and continues to lose market shares. It suffers from remaining barriers to market entry and from the strong competition of heavy goods vehicles. While the EU average modal share of rail freight reaches 18%, the road sector still reigns supreme with 75% despite being responsible for over 70% of transport greenhouse gas emissions.
This is far from the objective set by the 2011 White Paper on Transport, which called for shifting "30% of road freight over 300 km to multimodal transport by 2030, and more than 50% by 2050". It certainly does not help deliver the Paris climate deal either.
Why? One of the possible reasons is the lack of European Commission initiatives to make sure that multimodality becomes a reality.
The main EU legal instrument that directly supports multimodal transport is the 23 year-old Combined Transport Directive. On the funding side, private investments will slowly but surely replace EU subsidies. This approach is definitely not adapted to railways where return on investment is uncertain, if not non-existent.
Similarly, the long-awaited European Commission strategy on decarbonising the transport sector could have sent a strong signal in favour of rail, but the Commission chose to focus on the road sector instead. This legislative vacuum is difficult to understand, given the interest in multimodality that can be observed in the European Parliament. In January 2017, in their resolution on Logistics in the EU and multimodal transport in the new TEN-T corridors, MEPs called for multimodal transport solutions and pointed the key role of multimodality in freight logistics.
Some governments are also pushing for multimodal transport for instance through public funding schemes to boost the shift of freight traffic from road to rail. This is the case of the Italian scheme Ferrobonus, approved by the European Commission in November 2016. But let's face it: a number of member states have also given up on a true support to rail freight, in particular to secondary lines. Multi-speed Europe could be worrying from that point of view if one considers that it would weaken the capacity of EU institutions to impose pro-rail measures on all member states.
EU Commissioner for Transport, Violeta Bulc, is a true believer in decarbonisation through innovation. It is a fact that the "innovative" decarbonisation potential of the road sector is growing fast, and has already proved its worth. The first cross-border truck platooning trial was deemed a success.
If new technologies empower the road sector to self-correct its negative externalities, the rationale behind multimodality would be severely dented.
Policymakers should not give up on multimodality just because road innovates faster than other modes. However, the future of rail freight also lies in its own ability to innovate (interoperability, automation, signalling, noise mitigation…).
inter-modal transport
modal shift
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Home » Alternative » Whatever It Takes – Imagine Dragons
Ringtone Whatever It Takes – Imagine Dragons
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"Whatever It Takes" is a song by American rock band Imagine Dragons. The song was released on May 8, 2017, as a promotional single through Kidinakorner and Interscope. It later became the third single from the band's third studio album, Evolve, on October 6, 2017.
The track was also the official theme song for WWE's PPV event Battleground, in addition to its inclusion in the video game Madden NFL 18. The song was sent to US contemporary hit radio on February 13, 2018. Listen and download ringtone Whatever It Takes – Imagine Dragons free for iPhone & Android.
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I'm ready for whatever it takes
I do what it takes
>>> Download more: Ringtone Without Me – Halsey
Whatever It Takes – Imagine Dragons
Zero – Imagine Dragons
Alternative Algebra
Peer Pressure – James Bay Feat Julia Michaels
Quimby Percussions
Human – Rag'n'Bone Man
Tera Hone Laga
Without Me – Halsey
Venice Bitch – Lana Del Rey
Someone You Loved – Lewis Capaldi
When I Was Older – Billie Eilish
Sit Next To Me – Foster The People
Come Out And Play – Billie Eilish
Bad Guy – Billie Eilish
Say Amen (Saturday Night) – Panic! At the Disco