Pininfarina S.p. A. is coachbuilder, with headquarters in Cambiano, Italy. It was founded by Battista "Pinin" Farina in 1930. On 14 December 2015, Mahindra Group acquired Pininfarina S.p. A. for about €168 million. Pininfarina is employed by a wide variety of automobile manufacturers to design vehicles; these firms have included long-established customers such as Ferrari, Alfa Romeo, Fiat, GM, Maserati, to emerging companies in the Asian market with Chinese manufactures like AviChina, Changfeng, JAC and VinFast in Vietnam and Korean manufacturers Daewoo and Hyundai. Since the 1980s Pininfarina has designed high-speed trains, trams, rolling stocks, automated light rail cars, people movers, yachts and private jets. With the 1986 creation of "Pininfarina Extra" it has consulted on industrial design, interior design and graphic design. Pininfarina was run by Battista's son Sergio Pininfarina until 2001 his grandson Andrea Pininfarina until his death in 2008. After Andrea's death, his younger brother Paolo Pininfarina was appointed as CEO.

At its height in 2006 the Pininfarina Group employed 2,768 with subsidiary company offices throughout Europe, as well as in Morocco and the United States. As of 2012 with the end of series automotive production, employment has shrunk to 821. Pininfarina is publicly traded on the Borsa Italiana; when automobile designer and builder Battista "Pinin" Farina broke away from his brother's coach building firm, Stabilimenti Farina, in 1928, he founded "Carrozzeria Pinin Farina" with financial help from his wife's family and Vincenzo Lancia. That first year the firm employed built 50 automobile bodies. On 22 May 1930 papers were filed to become a corporation, Società anonima Carrozzeria Pinin Farina headquartered in Turin, Italy, at 107 Corso Trapani. During the 1930s, the company built bodies for Lancia, Alfa Romeo, Isotta Fraschini, Hispano-Suiza, Fiat and Rolls-Royce. With its close relationship with Lancia, the pioneer of the monocoque in automobile design, Farina became the first coachbuilder to build bodies for the new technique known as unibody construction.

This development happened in the mid-1930s when others saw the frameless construction as the end of the independent coachbuilder. In 1939, World War II ended automobile production, but the company had 400 employees building 150 bodies a month; the war effort against the Allies brought work making ambulances and searchlight carriages. The Pinin Farina factory was destroyed by Allied bombers ending the firm's operations. After the war, Italy was banned from the 1946 Paris Motor Show; the Paris show was attended by 809,000 visitors, queues stretched from the main gate all the way to the Seine. Pinin Farina and his son Sergio, determined to defy the ban, drove two of their cars from Turin to Paris, found a place at the entrance to the exhibition to display the two new creations; the managers of the Grand Palais said of the display, "the devil Pinin Farina", but to the press and the public it was the successful "Turin coachbuilder's anti-salon". At the end of 1945 the Cisitalia 202 Coupé was designed.

An elegantly proportioned design with a low hood, it is the car, given credit for establishing Pinin Farina's reputation. The Pinin Farina design was honored in the Museum of Modern Art's landmark presentation "Eight Automobiles" in 1951. A total of 170 Coupés were produced by Pinin Farina; the publicity of the Museum of Modern Art exhibit brought Pinin Farina to the attention of Nash-Kelvinator managers. The subsequent cooperation with Nash Motors resulted in high-volume production of Pinin Farina designs and provided a major entry into the United States market. In 1952, Farina visited the U. S. for the unveiling of his design for the Nash Ambassador and Statesman lines, although they did carry some details of Pinin Farina's design, were designed by Nash's then-new in-house styling staff when the original Farina-designed model proved unsuited to American tastes, exhibiting a popular 1950s appearance called "ponton". The Nash-Healey sports car body was, however designed and assembled in limited numbers from 1952 to 1954 at Pinin Farina's Turin facilities.

Nash advertised its link to the famous Italian designer, much as Studebaker promoted its longtime association with Raymond Loewy. As a result of Nash's million advertising campaign, Pinin Farina became well known in the U. S. Pinin Farina built the bodies for the limited-series Cadillac Eldorado Brougham for General Motors in 1959 and 1960, assembled them and sent them back to the U. S. There were 99 Broughams built in 1959 and 101 in 1960. A similar arrangement was repeated in the late 1980s when Pininfarina designed the Cadillac Allanté at the San Giusto Canavese factory; the car bodies were assembled and painted in Italy before being flown from the Turin International Airport to Detroit for final vehicle assembly. It started in 1951 with a meeting at a restaurant in Tortona, a small town halfway between Turin and Modena; this neutral territory was chosen because neither Farina nor Enzo Ferrari wanted to meet at the other's headquarters. Pinin’s son, Sergio Pininfarina recalled, "It is not difficult to imagine how I felt that afternoon when my father, without taking his eyes off the road for one moment told me his decision as we drove back to Turin: "From now on you'll be looking after Ferrari, from A to Z. Design, technology, construction—the lot!"—I was over the moon with happiness."

"Since that meeting the only road-going production Fer

Perturbation function

In mathematical optimization, the perturbation function is any function which relates to primal and dual problems. The name comes from the fact. In many cases this takes the form of shifting the constraints. In some texts the value function is called the perturbation function, the perturbation function is called the bifunction. Given two dual pairs separated locally convex spaces and. Given the function f: X → R ∪, we can define the primal problem by inf x ∈ X f. If there are constraint conditions, these can be built into the function f by letting f ← f + I c o n s t r a i n t s where I is the indicator function. F: X × Y → R ∪ is a perturbation function if and only if F = f; the duality gap is the difference of the right and left hand side of the inequality sup y ∗ ∈ Y ∗ − F ∗ ≤ inf x ∈ X F, where F ∗ is the convex conjugate in both variables. For any choice of perturbation function F weak duality holds. There are a number of conditions. For instance, if F is proper, jointly convex, lower semi-continuous with 0 ∈ core ⁡ and X, Y are Fréchet spaces strong duality holds.

Let and be dual pairs. Given a primal problem and a related perturbation function the Lagrangian L: X × Y ∗ → R ∪ is the negative conjugate of F with respect to y; that is the Lagrangian is defined by L = inf y ∈ Y. In particular the weak duality minmax equation can be shown to be sup y ∗ ∈ Y ∗ − F ∗ = sup y ∗ ∈ Y ∗ inf x ∈ X L ≤ inf x ∈ X sup y ∗ ∈ Y ∗ L = inf x ∈ X F. If the primal problem is given by inf x: g ≤ 0 f = inf x ∈ X f ~ where f ~

Lisa Brown-Miller

Lisa Brown-Miller is an American female ice hockey player. She won a gold medal at the 1998 Winter Olympics, she graduated from West Bloomfield High School, where she participated in ice hockey and softball. She enjoys mountain biking and water skiing. Played four years of hockey for the Providence Friars women's ice hockey program and graduated in 1988 with a degree in humanities, she earned All-Eastern College Athletic Conference accolades as a sophomore and senior. In addition, she was named the ECAC Player of the Year and American Women's Hockey Coaches' Association Player of the Year following her senior campaign, she finished her career with 154 career points on 62 assists. She played one year of softball at Providence College. A member of the United States Women's National Team since its inception, Lisa Brown-Miller is one of just three players to have appeared on six teams. In 30 games with the national team, she has recorded 25 assists. In addition, she appeared on United States Women's Select Teams in 1993, 1995, 1996 and 1997.

Brown-Miller served as the head coach of the Princeton University women's ice hockey team from 1991–96. During those years, her teams compiled a 60-45-5 overall record. In 1994–95, the Tigers finished the season as Ivy League Co-Champions. In 1991–92, her first season at the helm of the program, Brown-Miller guided the Tigers to the Ivy League Championship and earned Eastern College Athletic Conference Coach of the Year honors, she resigned in 1996 to train full-time with the United States women's program. In 2019 she accepted the position of head coach at Aquinas College in Mi. Brown-Miller now resides in Holland with her two children and Morgan. Both of the Brown-Miller children played hockey, she coaches with the Griffins Youth Foundation in Grand Rapids. She has a dog, a chinchilla, a bunny and the occasional guinea pigs. Miller enjoys photography and hikes in the outdoors. Named the Most Valuable Player of the 1992 United States Women's National Team after scoring nine points in five games bio