In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only  previous terms of the sequence appear in the equation, for a parameter  that is independent of ; this number  is called the order of the relation. If the values of the first  numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation.

In linear recurrences, the th term is equated to a linear function of the  previous terms. A famous example is the recurrence for the Fibonacci numbers,

where the order  is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on . For these recurrences, one can express the general term of the sequence as a closed-form expression of . As well, linear recurrences with polynomial coefficients depending on  are also important, because many common elementary and special functions have a Taylor series whose coefficients satisfy such a recurrence relation (see holonomic function).

Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of .

The concept of a recurrence relation can be extended to multidimensional arrays, that is, indexed families that are indexed by tuples of natural numbers.

Definition
A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form

where

is a function, where  is a set to which the elements of a sequence must belong. For any , this defines a unique sequence with  as its first element, called the initial value.

It is easy to modify the definition for getting sequences starting from the term of index 1 or higher.

This defines recurrence relation of first order. A recurrence relation of order  has the form

where  is a function that involves  consecutive elements of the sequence.
In this case,  initial values are needed for defining a sequence.

Examples

Factorial
The factorial is defined by the recurrence relation

and the initial condition

This is an example of a linear recurrence with polynomial coefficients of order 1, with the simple polynomial

as its only coefficient.

Logistic map
An example of a recurrence relation is the logistic map:

with a given constant ; given the initial term , each subsequent term is determined by this relation.

Fibonacci numbers
The recurrence of order two satisfied by the Fibonacci numbers is the canonical example of a homogeneous linear recurrence relation with constant coefficients (see below).  The Fibonacci sequence is defined using the recurrence

with initial conditions

Explicitly, the recurrence yields the equations

etc.

We obtain the sequence of Fibonacci numbers, which begins
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

The recurrence can be solved by methods described below yielding Binet's formula, which involves powers of the two roots of the characteristic polynomial ; the generating function of the sequence is the rational function

Binomial coefficients
A simple example of a multidimensional recurrence relation is given by the binomial coefficients , which count the ways of selecting  elements out of a set of  elements.
They can be computed by the recurrence relation

with the base cases . Using this formula to compute the values of all binomial coefficients generates an infinite array called Pascal's triangle. The same values can also be computed directly by a different formula that is not a recurrence, but uses factorials, multiplication and division, not just additions:

The binomial coefficients can also be computed with a uni-dimensional recurrence:

with the initial value  (The division is not displayed as a fraction for emphasizing that it must be computed after the multiplication, for not introducing fractional numbers).
This recurrence is widely used in computers because it does not require to build a table as does the bi-dimensional recurrence, and does involve very large integers as does the formula with factorials (if one uses  all involved integers are smaller than the final result).

Difference operator and difference equations

The  is an operator that maps sequences to sequences, and, more generally, functions to functions. It is commonly denoted  and is defined, in functional notation, as

It is thus a special case of finite difference.

When using the index notation for sequences, the definition becomes

The parentheses around  and  are generally omitted, and  must be understood as the term of index  in the sequence  and not  applied to the element

Given sequence  the  of  is

The  is
 A simple computation shows that

More generally: the  th difference is defined recursively as  and one has

This relation can be inverted, giving

A  of order  is an equation that involves the  first differences of a sequence or a function, in the same way as a differential equation of order  relates the  first derivatives of a function.

The two above relations allow transforming a recurrence relation of order  into a difference equation of order , and, conversely, a difference equation of order  into recurrence relation of order . Each transformation is the inverse of the other, and the sequences that are solution of the difference equation are exactly those that satisfies the recurrence relation.

For example, the difference equation

is equivalent to the recurrence relation

in the sense that the two equations are satisfied by the same sequences.

As it is equivalent for a sequence to satisfy a recurrence relation or to be the solution of a difference equation, the two terms "recurrence relation" and "difference equation" are sometimes used interchangeably. See Rational difference equation and Matrix difference equation for example of uses of "difference equation" instead of "recurrence relation"

Difference equations resemble to differential equations, and this resemblance is often used to mimic methods for solving differentiable equations to apply to solving difference equations, and therefore recurrence relations.

Summation equations relate to difference equations as integral equations relate to differential equations. See time scale calculus for a unification of the theory of difference equations with that of differential equations.

From sequences to grids
Single-variable or one-dimensional recurrence relations are about sequences (i.e. functions defined on one-dimensional grids). Multi-variable or n-dimensional recurrence relations are about -dimensional grids. Functions defined on -grids can also be studied with partial difference equations.

Solving

Solving linear recurrence relations with constant coefficients

Solving first-order non-homogeneous recurrence relations with variable coefficients
Moreover, for the general first-order non-homogeneous linear recurrence relation with variable coefficients:

there is also a nice method to solve it:

Let

Then

If we apply the formula to  and take the limit , we get the formula for first order linear differential equations with variable coefficients; the sum becomes an integral, and the product becomes the exponential function of an integral.

Solving general homogeneous linear recurrence relations
Many homogeneous linear recurrence relations may be solved by means of the generalized hypergeometric series. Special cases of these lead to recurrence relations for the orthogonal polynomials, and many special functions. For example, the solution to

is given by

the Bessel function, while

is solved by

the confluent hypergeometric series. Sequences which are the solutions of linear difference equations with polynomial coefficients are called P-recursive. For these specific recurrence equations algorithms are known which find polynomial, rational or hypergeometric solutions.

Solving first-order rational difference equations

A first order rational difference equation has the form . Such an equation can be solved by writing  as a nonlinear transformation of another variable  which itself evolves linearly.  Then standard methods can be used to solve the linear difference equation in .

Stability

Stability of linear higher-order recurrences
The linear recurrence of order ,

has the characteristic equation

The recurrence is stable, meaning that the iterates converge asymptotically to a fixed value, if and only if the eigenvalues (i.e., the roots of the characteristic equation), whether real or complex, are all less than unity in absolute value.

Stability of linear first-order matrix recurrences

In the first-order matrix difference equation

with state vector  and transition matrix ,  converges asymptotically to the steady state vector  if and only if all eigenvalues of the transition matrix  (whether real or complex) have an absolute value which is less than 1.

Stability of nonlinear first-order recurrences
Consider the nonlinear first-order recurrence

This recurrence is locally stable, meaning that it converges to a fixed point  from points sufficiently close to , if the slope of   in the neighborhood of  is smaller than unity in absolute value:  that is,

A nonlinear recurrence could have multiple fixed points, in which case some fixed points may be locally stable and others locally unstable; for continuous f  two adjacent fixed points cannot both be locally stable.

A nonlinear recurrence relation could also have a cycle of period  for .  Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function

with  appearing  times is locally stable according to the same criterion:

where  is any point on the cycle.

In a chaotic recurrence relation, the variable  stays in a bounded region but never converges to a fixed point or an attracting cycle; any fixed points or cycles of the equation are unstable.  See also logistic map, dyadic transformation, and tent map.

Relationship to differential equations
When solving an ordinary differential equation numerically, one typically encounters a recurrence relation. For example, when solving the initial value problem

with Euler's method and a step size , one calculates the values

by the recurrence

Systems of linear first order differential equations can be discretized exactly analytically using the methods shown in the discretization article.

Applications

Mathematical biology
Some of the best-known difference equations have their origins in the attempt to model population dynamics. For example, the Fibonacci numbers were once used as a model for the growth of a rabbit population.

The logistic map is used either directly to model population growth, or as a starting point for more detailed models of population dynamics. In this context, coupled difference equations are often used to model the interaction of two or more populations. For example, the Nicholson–Bailey model for a host-parasite interaction is given by

with  representing the hosts, and  the parasites, at time .

Integrodifference equations are a form of recurrence relation important to spatial ecology. These and other difference equations are particularly suited to modeling univoltine populations.

Computer science
Recurrence relations are also of fundamental importance in analysis of algorithms. If an algorithm is designed so that it will break a problem into smaller subproblems (divide and conquer), its running time is described by a recurrence relation.

A simple example is the time an algorithm takes to find an element in an ordered vector with  elements,  in the worst case.

A naive algorithm will search from left to right, one element at a time. The worst possible scenario is when the required element is the last, so the number of comparisons is .

A better algorithm is called binary search. However, it requires a sorted vector. It will first check if the element is at the middle of the vector. If not, then it will check if the middle element is greater or lesser than the sought element. At this point, half of the vector can be discarded, and the algorithm can be run again on the other half. The number of comparisons will be given by

the time complexity of which will be .

Digital signal processing
In digital signal processing, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in infinite impulse response (IIR) digital filters.

For example, the equation for a "feedforward" IIR comb filter of delay  is:

where  is the input at time ,  is the output at time , and  controls how much of the delayed signal is fed back into the output.  From this we can see that

etc.

Economics

Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics.  In particular, in macroeconomics one might develop a model of various broad sectors of the economy (the financial sector, the goods sector, the labor market, etc.) in which some agents' actions depend on lagged variables.  The model would then be solved for current values of key variables (interest rate, real GDP, etc.) in terms of past and current values of other variables.

See also

 Holonomic sequences
 Iterated function
 Orthogonal polynomials
 Recursion
 Recursion (computer science)
 Lagged Fibonacci generator
 Master theorem (analysis of algorithms)
 Circle points segments proof
 Continued fraction
 Time scale calculus
 Combinatorial principles
 Infinite impulse response
 Integration by reduction formulae
 Mathematical induction

References

Footnotes

Bibliography

 Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 1990. . Chapter 4: Recurrences, pp. 62–90.

 chapter 7.
  Chapter 9.1: Difference Equations.

  at EqWorld - The World of Mathematical Equations.
  at EqWorld - The World of Mathematical Equations.

External links

  OEIS index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)

Algebra

Combinatorics

Recurrence relation


Martin "Marty" Taccetta (born May 2, 1951) is an imprisoned New Jersey mobster who was the alleged boss of the Jersey Crew, a powerful faction of the Lucchese crime family.

Early life
Born in Newark, New Jersey in 1951, Martin Taccetta is the son of Anthony Taccetta, a self-employed building materials supplier who allegedly belonged to the Lucchese family. Martin's older brother Michael "Mad Dog" Taccetta is a capo in the Lucchese family who later served as boss of the Jersey Crew. During the early 1960s, Martin Taccetta, his brother, and his cousins belonged to a tough street gang in Newark, New Jersey. Martin and brother Michael played on the same Little League baseball team as did their cousins, Daniel and Thomas Ricciardi and future mob associate Robert Spagnola.

He is the cousin of mobsters Michael Perna, and Daniel, Joseph and Thomas Ricciardi. Michael is also the uncle of Joseph Perna born c.a. 1969, John G. Perna born c.a. 1977 and Ralph M. Perna Junior born c.a. 1972 who are the sons of his cousin Ralph V. Perna and his wife Roseanne. They all followed their father and uncle into organized crime. He is also the uncle of Carlo Taccetta, the son of his brother Michael, and Pamela Abdy, the Hollywood film producer whose credits include "Garden State" that was awarded an Independent Spirit Award for Best First Feature in 2005 as well as three other movies including "Man on the Moon" in 1999.

By the late 1960s, both Martin and Michael Taccetta started working with Lucchese mobster Anthony "Tumac" Accetturo in illegal gambling and loansharking operations. At that time, the boss of the Lucchese family was Gaetano "Tommy" Lucchese. In 1976, the Taccetta brothers and Accetturo all became full members, or "made men", in the Lucchese family, which was now run by new boss Anthony "Tony Ducks" Corallo, Accetturo's old mentor.

The Jersey crew
During the late 1970s and early 1980s, the Jersey crew grew in power, wealth, and independence. Based in Northern New Jersey, the crew had 20 members with its own family-like hierarchy. When Accetturo was indicted on state charges during this period, he moved to Florida. While in exile, Accetturo continued to run the Jersey crew through his underboss, Michael Taccetta, with Martin as second in command.

Michael and Martin Taccetta operated large gambling and loansharking operations around Newark, New Jersey, as well as drug trafficking and money laundering activities through their legitimate business, Taccetta Group Enterprises. When Philadelphia crime family boss Angelo "Gentle Don" Bruno was murdered on March 12, 1980, Accetturo took advantage of the ensuing chaos to establish a new Lucchese crew in Philadelphia.

In 1984, the Jersey leadership murdered James "Jimmy Sinatra" Craporatta, a contractor and Lucchese associate. When Craporatta refused to share the proceeds of a video gaming operation he controlled, the Jersey mobsters beat him to death with metal head golf clubs.

The longest trial in history
In 1985, Taccetta and 19 co-defendants were indicted in New Jersey on 76 counts of labor racketeering, illegal gambling, loansharking, extortion, drug trafficking, money laundering, conspiracy and murder for hire. These indictments were the result of a four-year investigation of the Jersey crew. Much of the evidence came from wiretapped conversations at the Hole-in-the-Wall, a restaurant frequented by mob figures in Newark, New Jersey.

In 1986, when the trial started, the Lucchese family in New York underwent a momentous leadership change with the conviction of Corallo. Facing a life sentence in prison, Corallo designated Lucchese capo Victor Amuso as the new acting boss of the family. Almost immediately, a dispute occurred between Accetturo and Amuso over the independence of the Jersey crew, and the large increase in tribute that Amuso expected from them.

On August 26, 1988, after a 21-month trial that was the longest in U.S. history, all 20 defendants were acquitted. Prosecutors were stunned by the verdict. The general consensus was that jurors did not trust the government witnesses, many of whom were themselves criminals testifying in return for reduced sentences. As the jurors left the courtroom, the defendants cheered and clapped for them.

Lucchese family conflicts
With the end of the racketeering trial, the Taccettas and the Jersey crew were shaken by conflict. During the trial, the strong relationship between the Taccetta brothers and Anthony Accetturo turned into a bitter rivalry. Michael Taccetta was jealous of the power that Anthony Accetturo Jr. was gaining in the family, along with a feeling that Anthony Accetturo Sr. was not showing Michael Taccetta sufficient respect. The Jersey crew soon split into Taccetta and Accetturo factions. In addition to the crew rivalries, family boss Amuso had finally lost patience with the entire New Jersey crew.

In the Fall of 1988, the dispute with Amuso turned to crisis when he demanded that henceforth the Jersey crew turn over 50% of their earnings to him. Both Accetturo and the Taccettas refused his demand. An enraged Amuso summoned the entire Jersey crew to a meeting in New York. Fearful of being the killed, the crew boycotted the meeting and all went into hiding. At this stage, Amuso gave his famous order "Whack Jersey", kill the entire Jersey crew, especially Accetturo. Over the next 12 months, as Amuso calmed down, Jersey crew members started returning to the family. During this period, Martin and Michael Taccetta reaffirmed their loyalty to Amuso.

To weaken the Accetturo faction in New Jersey, Michael Taccetta told Amuso that Accetturo was plotting with a Colombian drug gang to assassinate the Lucchese family leadership.  Amuso agreed to the murder contract and allowed the Taccetta brothers to assume control of the Jersey crew.

Weakened crew
Michael was now leader of the Jersey crew, with Martin Taccetta serving as his consigliere. However, the long racketeering trial and the conflict with Amuso had hurt the crew, and their legal problems were not over. Michael Taccetta was soon convicted of tax evasion and sent to prison in Minnesota. At this point, Martin took over the crew.

On December 5, 1989, Martin Taccetta was indicted in California for fraud over the operation of an adult film company. The company had bought video equipment on credit, sold the equipment, and then closed without paying the suppliers.

Also in 1989, Taccetta was indicted in New Jersey on conspiracy to defraud the Joint Underwriters Association of more than $400,000 in phony auto insurance claims. During this time period, Martin Taccetta began cooperating with Gambino crime family captain Thomas "Tommy" Gambino on activities in New Jersey.

Imprisonment
In 1991, Accetturo and the Taccetta brothers were indicted on charges of racketeering, narcotics, illegal gambling, extortion, loansharking, conspiracy and the 1984 Craporatta murder. Reacting to the indictments and the murder contract from Amuso, Accetturo decided to become a government witness and testified against both Martin and Michael Taccetta. In 1993, Martin Taccetta was acquitted of murder, but convicted of racketeering charges and, along with his brother, was sentenced to 30 years to life in prison.

In 2001, Martin Taccetta's lawyers uncovered a Federal Bureau of Investigation (FBI) memo that said their confidential informants cleared Taccetta from the Craporatta murder. For some reason, this memo was not produced during Taccetta's trial. The lawyers were planning to use this information to appeal his conviction.

On December 8, 2005, after serving ten years in prison, Martin Taccetta was released on appeal. A state Superior Court judge ruled he had not received adequate legal representation prior to his 1993 trial.  Taccetta had rejected a plea bargain deal from the government based on inadequate advice from his attorney.

New indictment
On May 8, 2008, Martin Tacchetta was indicted along with 23 other New Jersey gangsters for racketeering. Taccetta's charges included conspiring to use extortion to collect a loan from an unidentified debtor and for conspiring in 2007 with Gambino mobster Andrew Merola to extort $20,000 from contractors to allow non-union labor on a construction project in Morristown, New Jersey.

In August 2009, the New Jersey Supreme Court overturned the Superior Court ruling and reinstated Martin Taccetta's 1993 racketeering sentence. New Jersey State Attorney General Anne Milgram made the following statement,

For decades Marty Taccetta was one of the most prolific members of organized crime, whose power and control over the Lucchese family threatened the residents of New Jersey and illegally impacted commercial activity. We are gratified that the Supreme Court's decision will ensure that he will be returned to state prison to serve the remainder of his sentence of life plus 10 years for racketeering and extortion.

On June 10, 2010, U.S. Attorney Paul J. Fishman moved to dismiss the pending counts against Taccetta "because further prosecution of these charges is not in the interests of the United States at this time," according to a court order.

As of November 2011, Martin Taccetta is serving his life sentence at the New Jersey State Prison in Trenton, New Jersey. He will be eligible for parole after serving 30 years in prison.

On September 30, 2015 Martin Taccetta was sentenced to eight years in prison after pleading guilty to racketeering in Operation Heat.

In popular culture

The 2006 Sidney Lumet film Find Me Guilty is loosely based on the Taccetta brothers' 1988 racketeering trial that lasted 21 months.

Further reading
Raab, Selwyn. The Five Families: The Rise, Decline & Resurgence of America's Most Powerful Mafia Empire. New York: St. Martins Press, 2005.
Rudolph, Robert C. The Boys from New Jersey: How the Mob Beat the Feds. New York: William Morrow and Company Inc., 1992.

References

External links
http://www.nj.com/sopranos/stories/mafiosi_20020410sl.html
http://www.nj.com/sopranos/ledger/index.ssf?/sopranos/ledger/index.ssf?/sopranos/stories/mafiosi_20030723sl_ricciardi.html
"Reputed crime family underboss summoned to court in Newark" BY PETER J. SAMPSON The Record Thursday, December 10, 2009

1951 births
American gangsters of Italian descent
Gangsters from Newark, New Jersey
Lucchese crime family
Lucchese crime family New Jersey faction
Living people

Martin Taccetta


George G. Robertson is an American information visualization expert and senior researcher, Visualization and Interaction (VIBE) Research Group, Microsoft Research. With Stuart K. Card, Jock D. Mackinlay and others he invented a number of Information Visualization techniques.

Biography
Robertson has worked as a faculty member of the Computer Science Department at Carnegie-Mellon University, a senior scientist at Thinking Machines, and a senior scientist at Bolt Beranek and Newman.
Before his current position at Microsoft he was principal scientist at Xerox PARC. Since 1996 he is a principal researcher in the Visualization and Interaction group at Microsoft Research.

He is an ACM Fellow and manages a project on 3D User Interfaces and Information Visualization. He is associate editor of the Journal of Information Visualization.

Robertson's research interests are "user interfaces, 3D interactive animation, 3D graphics, information visualization, intelligent information access, multimedia, and hypermedia", see also: Connection Machine and Gender HCI

Publications
Articles, a selection:
 1972. "XCRIBL - A Hardcopy Scan Line Graphics System for Document Generation". With Raj Reddy, W. Broadley, Lee D. Erman, R. Johnsson, J. Newcomer, and J. Wright. In: Inf. Process. Lett. 1(6): 246-251.
 1978. "An Extensible File System for Hydra". With Guy T. Almes. In: ICSE 1978: 288-294.
 1981. "Accent: A Communication Oriented Network Operating System Kernel". With: Richard F. Rashid, in: SOSP 1981: 64-75.
 1987. "Parallel Implementation of Genetic Algorithms in a Classifier Rystem". in: Proceedings of the 2nd International Conference on Genetic Algorithms, July 1987: 140-147.
 1988. "Population Size in classifier Systems". in: ML 1988: 142-152.
 1989. "The cognitive coprocessor architecture for interactive user interfaces". With S. K. Card, and J. D. Mackinlay. In: Proceedings of the ACM SIGGRAPH Symposium on User Interface Software and Technology, pages 10–18. ACM Press, Nov 1989.
 1991. "Cone Trees: Animated 3D Visualizations of Hierarchical Information". With Jock D. Mackinlay and Stuart K. Card. In: Robertson, Scott P., Olson, Gary M. and Olson, Judith S. (eds.) Proceedings of the ACM CHI 91 Human Factors in Computing Systems Conference April 28 - June 5, 1991, New Orleans, Louisiana. pp. 189–194.
 1993. "Information Visualization Using 3D Interactive Animation". With Stuart K. Card, and Jock D. Mackinlay. In: Communications of the ACM, 36(4), April 1993. pp. 57–71.
 1993. "The document lens". With J. D. Mackinlay. In: Proceedings of the ACM Symposium on User Interface Software and Technology. ACM Press, Nov 1993.

References

External links
 George G. Robertson Homepage at Microsoft research.
 List of publications at www.informatik.uni-trier.de.

Living people
Carnegie Mellon University alumni
Human–computer interaction researchers
Information visualization experts
Year of birth missing (living people)
Scientists at PARC (company)

George G. Robertson


The Air Support Division (ASD) is the police aviation division of the Los Angeles Police Department (LAPD). It is the largest municipal airborne law enforcement organization in the United States and operates from the LAPD Hooper Heliport.

While originally devoted to aerial traffic enforcement, the ASD has grown to support a wide variety of police activity. Today, its operations are divided between Air Support To Regular Operations (ASTRO) and Special Flight Section (SFS). The Air Support Division currently operates sixteen helicopters and 1 fixed wing aircraft of two different helicopter brands, and maintains the largest municipal police aviation unit around the world, in addition to having the world's largest roof-top airport and world's busiest heliport.

History

The ASD was established as the LAPD Helicopter Unit in 1956 with one Hiller UH-12C three-seat helicopter. They added a second helicopter in 1963 and a third in 1965. The city operated Bell 47G and 47J model helicopters. In 1968, the unit received its first turbine-powered helicopter, the Bell 206A JetRanger, which significantly decreased police response times. With a major expansion in 1974, the Helicopter Unit was renamed the Air Support Division. At that time, the ASD grew to fifteen helicopters and one Cessna 210 crewed by 77 sworn personnel. In 1976, the ASD added the Special Flight Section (SFS), a unit dedicated to supporting undercover police operations. In this support role, SFS is a significant contributor to narcotics and serialized criminal investigations. In 1989, the ASD added its first Aerospatiale AS350 B1. The city replaced all but four of the Bell 206 JetRangers and retired the older piston models. Today, LAPD air units provide aerial surveillance for vehicle pursuits, robberies, large crowd demonstrations, drug interdiction, and search and rescue missions. Air units are automatically requested when initiating a traffic stop on a suspect with known wants or warrants that are a felony in order to limit the potential for a pursuit. Aircraft will not fly during poor weather due to aviation safety.

Organization
The Air Support Division consists of 88 sworn personnel and 16 helicopters (which include one Bell 206B3 JetRangers, seven Airbus AS-350B2, seven Airbus H125, and one Bell 412), and one Beechcraft King Air 200 twin-engined aircraft.

Two officers with at least three years of patrol car service fly in each air unit. They are armed and able to land and make arrests in areas not accessible by other means.

The city of Los Angeles briefly flew a fleet of Bell 407s in the late 1990s as a replacement for the AS-350B1s. However, in 2000 the LAPD started replacing the 407s with more powerful AS-350B2s. Two of the 407s were sold to the General Services Department, which uses the helicopters on flights for the Department of Water and Power.

Fleet
7 Eurocopter AS350 B2 A-Star
7 Airbus H125
1 Bell 206 JetRanger
1 Bell 412
1 Beechcraft King Air 200

Accidents and incidents

On November 30, 1964, Sergeant Norman D. Piepenbrink was killed in a helicopter accident.

On August 30, 1966, Policemen Larry Amberg and Alex N. Ilnicki, were on traffic patrol in Air 1 (Bell 47G, reg. N1162W) flying in the vicinity of Dodger Stadium, when a media helicopter (Bell 47G, reg. N1157W) was also in the area reporting on freeway traffic conditions. Air 1 and the media helicopter collided, resulting in the deaths of both officers and the occupants of the media helicopter. Policeman Ilnicki had about 401 hours of total flight time and 236 hours in type at the time of the crash

On May 29, 1974, Commander Paul J. Gillen was killed when his helicopter crashed.

On June 11, 1976, Officer Jeffrey B. Lindenberg was killed when the Bell 47G-5 helicopter (U.S. reg. N7085J) he was training in lost power and crashed while landing. Lindenberg was practicing simulated urban high-rise rooftop landings at an off-site pad on top of a small mountain near the Los Angeles Zoo in the hills above Hollywood. On short final approach, the engine lost power and the helicopter impacted  short of the pad. The helicopter rolled down the mountain side . Lindenberg was killed and another officer was seriously injured. Lindenberg had been with the agency for seven years. Lindenberg was an experienced instrument rated pilot with 3575 hours of total flight time and 426 in type.

On March 1, 1983, Reserve Officer Stuart Taira was killed as a result of a police helicopter crash. Taira, an observer for the helicopter unit, and two other officers were conducting aerial patrols following a tornado. In between patrols the officers were dispatched to investigate a report of a burglar on a roof. As the helicopter took off it struck a power line, causing it to crash. The officers survived the initial impact and Taira was able to exit the aircraft. Taira then returned to the aircraft in an attempt to rescue his two partners. One of the helicopter's rotors struck Taira in the head, killing him. Taira was posthumously awarded the department's Medal of Valor.

On June 13, 1991, Officers Gary Alan Howe and Charles Randall Champe were killed when they experienced an in-flight engine failure which caused their helicopter to crash into a parking lot near Normandie Elementary School. They were flying an AS350B1 helicopter (U.S. reg. N214LA).

See also
 LAPD Hooper Heliport

References

External links
 LAPD Air Support Division

Air Support Division
Police aviation units of the United States

LAPD Air Support Division


was the pseudonym for a noted artist and epicure during the early to mid-Shōwa period of Japan. His real name was , but he is best known by his artistic name, Rosanjin. A man of many talents, Rosanjin  was also a calligrapher, ceramicist, engraver, painter, lacquer artist and restaurateur.

Biography
Rosanjin was born in what is now part of Kita-ku, Kyoto, as the younger son of the head priest of Kamigamo Shrine. At the age of six, he was apprenticed to (and adopted by) Fukuda Takeshi, a Kyoto woodblock engraver, after his father committed suicide on finding out that the child was not his own son. At the age of ten, while still in elementary school, he was also working at a local Chinese herbalist. In 1903, Rosanjin moved to Tokyo with the intent of studying Japanese calligraphy, winning first prize in a contest by the Japan Art Academy the following year. In 1905, he was accepted as an apprentice by noted calligrapher Okamoto Ippei, who sent him to northern China from 1908-1910 to study calligraphy and the art of Seal cutting.

In 1915, Rosanjin moved to Kyoto and Kanazawa, where he first began experimenting with decorated ceramics and developing his aesthetic theories on the relationship between food and the design of the ceramics on which it was served. In 1919, he returned to Tokyo, where he opened an art gallery and in 1920 he founded the Bishoku Club (Gourmet's Club) on the second floor of his gallery, where he began serving food on his collection of traditional ceramics. In 1925, the restaurant moved to the Nagatachō neighborhood of Tokyo, where it was renamed the Hoshigaoka-saryo.

However, the Great Tokyo earthquake of 1923 destroyed most of his ceramics collection, so Rosanjin began making pottery to replace it. In 1926, with the assistance of Toyozō Arakawa, he established a kiln in the Yamasaki neighborhood of Kamakura. Rosanjin began by imitating the classic forms of Japanese Mino, Shigaraki, Bizen and Kutani ceramics, and also for classic blue-and-white wares and colored porcelains of Ming period China. However, he often surpassed the classical forms, and became famous for his simple, but daring, original designs, at time incorporating elements of Japanese calligraphy, of which he was also an acknowledged master.

Rosanjin was also noted as a scholar of antique pottery publicizing his work in a privately published periodical, Hoshigaoka, during the 1930s.

In the post-war period, despite some financial difficulties, in 1946 Rosanjin opened a restaurant in the Ginza district of Tokyo called Kadōkadō-byō. It was patronized by the upper levels of the American occupation forces, and helped establish Ryosanjin's reputation overseas. In 1951, noted sculptor Isamu Noguchi and his actress wife Yoshiko Ōtaka accepted an invitation to live on Rosanjin's property in Kamakura, where they stayed for several years. Rosanjin launched a quarterly magazine, Doppo, in 1954. Also in 1954, Rosanjin accepted an invitation by the Rockefeller Foundation to hold a solo exhibition of his works in New York City at the Museum of Modern Art. He continued on to Europe, where he met with Pablo Picasso and Marc Chagall. In 1955, one of his Oribe ware works was designated an Important Cultural Property of Japan by the Japanese government. Rosanjin was designated a Living National Treasure by the Japanese government in 1959, but was one of the very few people to decline the honor.

Rosanjin died in Yokohama in 1959 of cirrhosis of the liver brought about by a liver fluke infection. His grave is at the temple of Saihō-ji in Kyoto. In 2020, the Adachi Museum of Art in Yasugi, Shimane Prefecture opened a dedicated Rosanjin Hall to house the museum's collection of about 500 works by Rosanjin.

In popular culture
In the manga Oishinbo, the character Kaibara Yuzan is modeled after Rosanjin.

Kioicho Fukudaya, a ryōtei in Tokyo awarded two stars by the Michelin Guide, was established in 1939 with Rosanjin's guidance and maintains a significant collection of his service ware, seal engravings, and calligraphy.

The television series "Iron Chef" features the Rosanjin school of cuisine.

References

Further reading
 Ashton, Dore. Noguchi East and West. University of California Press. (1992).
 Tiger, Caroline. Isamu Noguchi. Infobase Publishing (2007)

External links

The Noguchi pages biography
Naoe biographies
Rosanjin Kitaoji Walking the Path of Art
The Noguchi Museum
Museum of Modern Art

1883 births
1959 deaths
Japanese calligraphers
Japanese potters
Food writers
Artists from Kyoto
20th-century ceramists

Rosanjin