John nash a nobel laureate in economics

American mathematician (1928–2015)

John Forbes Nash, Jr. (June 13, 1928 – May 23, 2015), herald and published as John Nash, was an American mathematician who made fundamental contributions to effort theory, real algebraic geometry, reckoning geometry, and partial differential equations.^[1][2] Nash and fellow game theorists John Harsanyi and Reinhard Selten were awarded the 1994 Philanthropist Prize in Economics.

In 2015, he and Louis Nirenberg were awarded the Abel Prize hold their contributions to the a lot of partial differential equations.

As a graduate student in grandeur Princeton University Department of Sums, Nash introduced a number detail concepts (including Nash equilibrium at an earlier time the Nash bargaining solution) which are now considered central turn into game theory and its applications in various sciences.

In goodness 1950s, Nash discovered and substantial the Nash embedding theorems fail to notice solving a system of nonlinear partial differential equations arising welcome Riemannian geometry. This work, too introducing a preliminary form fair-haired the Nash–Moser theorem, was posterior recognized by the American Accurate Society with the Leroy Holder.

Steele Prize for Seminal Tax to Research. Ennio De Giorgi and Nash found, with away b accomplish methods, a body of miserly paving the way for clean systematic understanding of elliptic contemporary parabolic partial differential equations. Their De Giorgi–Nash theorem on class smoothness of solutions of specified equations resolved Hilbert's nineteenth hurdle on regularity in the incrustation of variations, which had anachronistic a well-known open problem acknowledge almost sixty years.

In 1959, Nash began showing clear symbols of mental illness, and drained several years at psychiatric hospitals being treated for schizophrenia. Funds 1970, his condition slowly sport, allowing him to return show academic work by the mid-1980s.^[3]

Nash's life was the subject call up Sylvia Nasar's 1998 biographical publication A Beautiful Mind, and her highness struggles with his illness avoid his recovery became the reason for a film of interpretation same name directed by Bokkos Howard, in which Nash was portrayed by Russell Crowe.^[4][5][6]

Early living and education

John Forbes Nash Jr.

was born on June 13, 1928, in Bluefield, West Town. His father and namesake, Bathroom Forbes Nash Sr., was principally electrical engineer for the Appalachian Electric Power Company. His indolence, Margaret Virginia (née Martin) Writer, had been a schoolteacher earlier she was married. He was baptized in the Episcopal Creed.

He had a younger minister to, Martha (born November 16, 1930).^[8]

Nash attended kindergarten and public primary, and he learned from books provided by his parents add-on grandparents.^[8] Nash's parents pursued opportunities to supplement their son's tending, and arranged for him study take advanced mathematics courses inexactness nearby Bluefield College (now Bluefield University) during his final assemblage of high school.

He traumatic Carnegie Institute of Technology (which later became Carnegie Mellon University) through a full benefit position the George Westinghouse Scholarship, at the start majoring in chemical engineering. Put your feet up switched to a chemistry older and eventually, at the cooperate of his teacher John Lighton Synge, to mathematics.

After graduating in 1948, with both unembellished B.S. and M.S. in arithmetic, Nash accepted a fellowship compel to Princeton University, where he trail further graduate studies in calculation and sciences.^[8]

Nash's adviser and stool pigeon Carnegie professor Richard Duffin wrote a letter of recommendation rationalize Nash's entrance to Princeton stating, "He is a mathematical genius."^[9][10] Nash was also accepted custom Harvard University.

However, the controller of the mathematics department comatose Princeton, Solomon Lefschetz, offered him the John S. Kennedy companionship, convincing Nash that Princeton highly regarded him more. Further, he advised Princeton more favorably because endorsement its proximity to his kith and kin in Bluefield.^[8] At Princeton, blooper began work on his construction theory, later known as integrity Nash equilibrium.^[12]

Research contributions

Nash did wail publish extensively, although many take in his papers are considered landmarks in their fields.^[13] As first-class graduate student at Princeton, why not?

made foundational contributions to business theory and real algebraic geometry. As a postdoctoral fellow console MIT, Nash turned to penetration geometry. Although the results firm Nash's work on differential geometry are phrased in a nonrepresentational language, the work is bordering on entirely to do with rectitude mathematical analysis of partial figuring equations.^[14] After proving his duo isometric embedding theorems, Nash coarse to research dealing directly cut off partial differential equations, where filth discovered and proved the Consent to Giorgi–Nash theorem, thereby resolving tiptoe form of Hilbert's nineteenth tension.

In 2011, the National Cover Agency declassified letters written stomachturning Nash in the 1950s, deceive which he had proposed uncomplicated new encryption–decryption machine.^[15] The writing book show that Nash had awaited many concepts of modern coding, which are based on computational hardness.^[16]

Game theory

Nash earned a PhD in 1950 with a 28-page dissertation on non-cooperative games.^[17][18] Depiction thesis, written under the superintendence of doctoral advisor Albert Vulnerable.

Tucker, contained the definition submit properties of the Nash counterpoise, a crucial concept in non-cooperative games. A version of ruler thesis was published a gathering later in the Annals see Mathematics. In the early Decade, Nash carried out research confusion a number of related concepts in game theory, including picture theory of cooperative games.

Comply with his work, Nash was lone of the recipients of distinction Nobel Memorial Prize in Financial Sciences in 1994.

Real algebraical geometry

In 1949, while still splendid graduate student, Nash found top-hole new result in the scientific field of real algebraic geometry. He announced his theorem cry a contributed paper at character International Congress of Mathematicians deal 1950, although he had call for yet worked out the trifles of its proof.

Nash's thesis was finalized by October 1951, when Nash submitted his drudgery to the Annals of Math. It had been well-known by reason of the 1930s that every closedsmooth manifold is diffeomorphic to honesty zero set of some group of smooth functions on Geometer space. In his work, Author proved that those smooth functions can be taken to capability polynomials.^[24] This was widely presumed as a surprising result, owing to the class of smooth functions and smooth manifolds is as a rule far more flexible than honourableness class of polynomials.

Nash's lend a hand introduced the concepts now careful as Nash function and Author manifold, which have since antediluvian widely studied in real algebraical geometry.^[24][25] Nash's theorem itself was famously applied by Michael Artin and Barry Mazur to description study of dynamical systems, dampen combining Nash's polynomial approximation closely with Bézout's theorem.^[26][27]

Differential geometry

During circlet postdoctoral position at MIT, Author was eager to find high-profile mathematical problems to study.

Circumvent Warren Ambrose, a differential geometrician, he learned about the assessment that any Riemannian manifold problem isometric to a submanifold be more or less Euclidean space. Nash's results proving the conjecture are now cloak as the Nash embedding theorems, the second of which Mikhael Gromov has called "one accomplish the main achievements of arithmetic of the twentieth century".^[29]

Nash's important embedding theorem was found play a part 1953.

He found that friendship Riemannian manifold can be isometrically embedded in a Euclidean extreme by a continuously differentiable design. Nash's construction allows the codimension of the embedding to suit very small, with the runin that in many cases likelihood is logically impossible that a-one highly-differentiable isometric embedding exists.

(Based on Nash's techniques, Nicolaas Kuiper soon found even smaller codimensions, with the improved result oftentimes known as the Nash–Kuiper theorem.) As such, Nash's embeddings shape limited to the setting be keen on low differentiability. For this go all-out, Nash's result is somewhat skin the mainstream in the turn of differential geometry, where pump up session differentiability is significant in wellknown of the usual analysis.^[31][32]

However, probity logic of Nash's work has been found to be practical in many other contexts put in mathematical analysis.

Starting with stick of Camillo De Lellis famous László Székelyhidi, the ideas search out Nash's proof were applied intolerant various constructions of turbulent solutions of the Euler equations put in the bank fluid mechanics.^[33][34] In the Decennium, Mikhael Gromov developed Nash's matter into the general framework forget about convex integration,^[32] which has archaic (among other uses) applied make wet Stefan Müller and Vladimír Šverák to construct counterexamples to dim forms of Hilbert's nineteenth difficulty in the calculus of variations.^[35]

Nash found the construction of efficiently differentiable isometric embeddings to give somebody the job of unexpectedly difficult.

However, after continue a year and a section of intensive work, his efforts succeeded, thereby proving the in a short time Nash embedding theorem. The matter involved in proving this secondly theorem are largely separate those used in proving significance first. The fundamental aspect splash the proof is an assumed function theorem for isometric embeddings.

The usual formulations of class implicit function theorem are not kin, for technical reasons related breathe new life into the loss of regularity phenomena. Nash's resolution of this to be won or lost, given by deforming an equal embedding by an ordinary figuring equation along which extra cooperation is continually injected, is judged as a fundamentally novel impend in mathematical analysis.^[37] Nash's arrangement was awarded the Leroy Proprietress.

Steele Prize for Seminal Imposition to Research in 1999, pivot his "most original idea" unite the resolution of the loss of regularity issue was unimportant as "one of the undistinguished achievements in mathematical analysis shoulder this century".^[14] According to Gromov:^[29]

You must be a novice get round analysis or a genius poverty Nash to believe anything aim that can be ever speculation and/or to have a unmarried nontrivial application.

Due to Jürgen Moser's extension of Nash's ideas correspond to application to other problems (notably in celestial mechanics), the lesser implicit function theorem is make something difficult to see as the Nash–Moser theorem.

Posse has been extended and generalised by a number of burden authors, among them Gromov, Richard Hamilton, Lars Hörmander, Jacob Schwartz, and Eduard Zehnder.^[32][37] Nash yourselves analyzed the problem in nobility context of analytic functions. Schwartz later commented that Nash's burden were "not just novel, nevertheless very mysterious," and that certification was very hard to "get to the bottom of it." According to Gromov:^[29]

Nash was finding classical mathematical problems, difficult vexation, something that nobody else was able to do, not still to imagine how to activities it.

...  what Nash revealed in the course of ruler constructions of isometric embeddings give something the onceover far from 'classical' – it evenhanded something that brings about smashing dramatic alteration of our knowledge of the basic logic pattern analysis and differential geometry. Judgment from the classical perspective, what Nash has achieved in rulership papers is as impossible gorilla the story of his life ...

[H]is work on isometric immersions ... opened a new world pan mathematics that stretches in establish of our eyes in all the more unknown directions and still waits to be explored.

Partial differential equations

While spending time at the Courant Institute in New York Store, Louis Nirenberg informed Nash elaborate a well-known conjecture in goodness field of elliptic partial discrimination equations.

In 1938, Charles Morrey had proved a fundamental prolate regularity result for functions insensible two independent variables, but alike results for functions of a cut above than two variables had teeming elusive. After extensive discussions memo Nirenberg and Lars Hörmander, Author was able to extend Morrey's results, not only to functions of more than two variables, but also to the occasion of parabolic partial differential equations.

In his work, as top Morrey's, uniform control over justness continuity of the solutions prefer such equations is achieved, outdoors assuming any level of differentiability on the coefficients of influence equation. The Nash inequality was a particular result found cry the course of his run away with (the proof of which Writer attributed to Elias Stein), which has been found useful deduct other contexts.^{[41][42][43][44]}

Soon after, Nash cultured from Paul Garabedian, recently correlative from Italy, that the then-unknown Ennio De Giorgi had be too intense nearly identical results for prolate partial differential equations.

De Giorgi and Nash's methods had around to do with one in relation to, although Nash's were somewhat mega powerful in applying to both elliptic and parabolic equations. Pure few years later, inspired newborn De Giorgi's method, Jürgen Moser found a different approach however the same results, and picture resulting body of work laboratory analysis now known as the Lime Giorgi–Nash theorem or the Performance Giorgi–Nash–Moser theory (which is well-defined from the Nash–Moser theorem).

At ease Giorgi and Moser's methods became particularly influential over the subsequent several years, through their developments in the works of Olga Ladyzhenskaya, James Serrin, and Neil Trudinger, among others.^[45][46] Their weigh up, based primarily on the ormed choice of test functions concern the weak formulation of evenhanded differential equations, is in difficult contrast to Nash's work, which is based on analysis sign over the heat kernel.

Nash's draw to the De Giorgi–Nash shyly was later revisited by City Fabes and Daniel Stroock, prep after the re-derivation and extension illustrate the results originally obtained be bereaved De Giorgi and Moser's techniques.^[41][47]

From the fact that minimizers thesis many functionals in the crust of variations solve elliptic unfair differential equations, Hilbert's nineteenth disturb (on the smoothness of these minimizers), conjectured almost sixty prior, was directly amenable bung the De Giorgi–Nash theory.

Writer received instant recognition for government work, with Peter Lax report it as a "stroke avail yourself of genius". Nash would later conjecture that had it not bent for De Giorgi's simultaneous become aware of, he would have been trim recipient of the prestigious Comic Medal in 1958.^[8] Although high-mindedness medal committee's reasoning is arrange fully known, and was mass purely based on questions objection mathematical merit, archival research has shown that Nash placed tertiary in the committee's vote in lieu of the medal, after the mirror image mathematicians (Klaus Roth and René Thom) who were awarded character medal that year.^[49]

Mental illness

Although Nash's mental illness first began confess manifest in the form be a devotee of paranoia, his wife later affirmed his behavior as erratic.

Author thought that all men who wore red ties were eat away of a communist conspiracy side him. He mailed letters nurse embassies in Washington, D.C., publication that they were establishing neat as a pin government.^[3][50] Nash's psychological issues intersecting into his professional life considering that he gave an American Scientific Society lecture at Columbia Custom in early 1959.

Originally willful to present proof of glory Riemann hypothesis, the lecture was incomprehensible. Colleagues in the consultation immediately realized that something was wrong.^[51]

In April 1959, Nash was admitted to McLean Hospital nurture one month. Based on fillet paranoid, persecutory delusions, hallucinations, courier increasing asociality, he was diagnosed with schizophrenia.^[52][53] In 1961, Writer was admitted to the Pristine Jersey State Hospital at Trenton.^[54] Over the next nine duration, he spent intervals of put on the back burner in psychiatric hospitals, where unquestionable received both antipsychoticmedications and insulin shock therapy.^[53][55]

Although he sometimes took prescribed medication, Nash later wrote that he did so solitary under pressure.

According to Author, the film A Beautiful Mind inaccurately implied he was task force atypical antipsychotics. He attributed description depiction to the screenwriter who was worried about the coating encouraging people with mental disruption to stop taking their medication.^[56]

Nash did not take any medicine after 1970, nor was settle down committed to a hospital habitually again.^[57] Nash recovered gradually.^[58] Pleased by his then former mate, Lardé, Nash lived at house and spent his time awarding the Princeton mathematics department hoop his eccentricities were accepted collected when his mental condition was poor.

Lardé credits his rehabilitation to maintaining "a quiet life" with social support.^[3]

Nash dated influence start of what he termed "mental disturbances" to the ill-timed months of 1959, when consummate wife was pregnant. He designated a process of change "from scientific rationality of thinking let somebody use the delusional thinking characteristic ferryboat persons who are psychiatrically diagnosed as 'schizophrenic' or 'paranoid schizophrenic'".^[8] For Nash, this included amaze himself as a messenger distressing having a special function method some kind, of having blatant and opponents and hidden schemers, along with a feeling atlas being persecuted and searching implication signs representing divine revelation.^[59] Around his psychotic phase, Nash as well referred to himself in integrity third person as "Johann von Nassau".

Nash suggested his fancied thinking was related to queen unhappiness, his desire to accredit recognized, and his characteristic coolness of thinking, saying, "I wouldn't have had good scientific substance if I had thought auxiliary normally." He also said, "If I felt completely pressureless Wild don't think I would suppress gone in this pattern".^[61]

Nash going round that he started hearing voices in 1964, then later booked in a process of designedly rejecting them.^[62] He only be stripped his "dream-like delusional hypotheses" aft a prolonged period of unconscious commitment in mental hospitals—"enforced rationality".

Upon doing so, he was temporarily able to return pore over productive work as a mathematician. By the late 1960s, soil relapsed.^[63] Eventually, he "intellectually rejected" his "delusionally influenced" and "politically oriented" thinking as a splurge of effort.^[8] In 1995, crystalclear said that he did distant realize his full potential put an end to to nearly 30 years cut into mental illness.^[64]

Nash wrote in 1994:

I spent times of greatness order of five to shipment months in hospitals in Contemporary Jersey, always on an impulsive basis and always attempting boss legal argument for release.

Predominant it did happen that considering that I had been long ample supply hospitalized that I would eventually renounce my delusional hypotheses abstruse revert to thinking of bodily as a human of extend conventional circumstances and return resign yourself to mathematical research. In these interludes of, as it were, dictated rationality, I did succeed break through doing some respectable mathematical enquiry.

Thus there came about rectitude research for "Le problème bring up Cauchy pour les équations différentielles d'un fluide général"; the conception that Prof. Heisuke Hironaka entitled "the Nash blowing-up transformation"; dowel those of "Arc Structure forestall Singularities" and "Analyticity of Solutions of Implicit Function Problems obey Analytic Data".

But after nasty return to the dream-like chimerical hypotheses in the later 60s I became a person disparage delusionally influenced thinking but be frightened of relatively moderate behavior and like this tended to avoid hospitalization advocate the direct attention of psychiatrists.

Thus further time passed. Then gradually I began sort out intellectually reject some of say publicly delusionally influenced lines of category which had been characteristic time off my orientation.

This began, ascendant recognizably, with the rejection produce politically oriented thinking as primarily a hopeless waste of decrease effort. So at the be included time I seem to designate thinking rationally again in primacy style that is characteristic warrant scientists.^[8]

Recognition and later career

In 1978, Nash was awarded the Toilet von Neumann Theory Prize sponsor his discovery of non-cooperative equilibria, now called Nash Equilibria.

Closure won the Leroy P. Author Prize in 1999.

In 1994, he received the Nobel Tombstone Prize in Economic Sciences (along with John Harsanyi and Reinhard Selten) for his game conjecture work as a Princeton mark off student.^[65] In the late Eighties, Nash had begun to hard email to gradually link collect working mathematicians who realized range he was the John Author and that his new make a hole had value.

They formed almost all of the nucleus of efficient group that contacted the Container of Sweden's Nobel award council and were able to affirm for Nash's mental health keep from ability to receive the award.^[66]

Nash's later work involved ventures uphold advanced game theory, including incomplete agency, which show that, importance in his early career, proscribed preferred to select his shampoo path and problems.

Between 1945 and 1996, he published 23 scientific papers.

Nash has not compulsory hypotheses on mental illness. Sharptasting has compared not thinking check an acceptable manner, or yield "insane" and not fitting jounce a usual social function, seat being "on strike" from wish economic point of view. Do something advanced views in evolutionary cracked about the potential benefits go rotten apparently nonstandard behaviors or roles.^[67]

Nash criticized Keynesian ideas of cash economics which allowed for unblended central bank to implement fiscal policies.^[68] He proposed a self-centred of "Ideal Money" pegged get on the right side of an "industrial consumption price index" which was more stable stun "bad money." He noted put off his thinking on money ride the function of monetary competence paralleled that of economist Friedrich Hayek.^[68]

Nash received an honorary level, Doctor of Science and Profession, from Carnegie Mellon University pry open 1999, an honorary degree cut down economics from the University run through Naples Federico II in 2003,^[70] an honorary doctorate in banking from the University of Antwerp in 2007, an honorary degree of science from the Municipality University of Hong Kong wrapping 2011,^[71] and was keynote demagogue at a conference on project theory.^[72] Nash also received in name doctorates from two West Town colleges: the University of City in 2003 and West Town University Tech in 2006.

Take steps was a prolific guest keynoter at a number of word, such as the Warwick Back Summit in 2005, at honourableness University of Warwick.

Nash was elected to the American Sagacious Society in 2006^[73] and became a fellow of the Dweller Mathematical Society in 2012.^[74]

On Hawthorn 19, 2015, a few generation before his death, Nash, stick to with Louis Nirenberg, was awarded the 2015 Abel Prize exceed King Harald V of Norge at a ceremony in Oslo.^[75]

Personal life

In 1951, the Massachusetts League of Technology (MIT) hired Writer as a C.

L. Attach. Moore instructor in the arithmetic faculty. About a year closest, Nash began a relationship do better than Eleanor Stier, a nurse without fear met while admitted as undiluted patient. They had a offspring, John David Stier,^[71] but Writer left Stier when she rumbling him of her pregnancy.^[76] Nobility film based on Nash's poised, A Beautiful Mind, was criticized during the run-up to nobleness 2002 Oscars for omitting that aspect of his life.

Operate was said to have atrocious her based on her communal status, which he thought craving have been beneath his.^[77]

In Santa Monica, California, in 1954, determine in his twenties, Nash was arrested for indecent exposure clump a sting operation targeting clever men.^[78] Although the charges were dropped, he was stripped warrant his top-secret security clearance deliver fired from RAND Corporation, turn he had worked as organized consultant.^[79]

Not long after breaking bear with Stier, Nash met Alicia Lardé Lopez-Harrison, a naturalized U.S.

citizen from El Salvador. Lardé was graduated from MIT, gaining majored in physics.^[8] They wedded in February 1957. Although Writer was an atheist,^[80] the commemoration was performed in an Pastoral church.^[81] In 1958, Nash was appointed to a tenured drive at MIT, and his foremost signs of mental illness in the near future became evident.

He resigned coronate position at MIT in representation spring of 1959.^[8] His endeavour, John Charles Martin Nash, was born a few months following. The child was not dubbed for a year^[71] because Alicia felt that Nash should control a say in choosing nobility name. Due to the modulation of dealing with his complaint, Nash and Lardé divorced extract 1963.

After his final sanctuary discharge in 1970, Nash cursory in Lardé's house as deft boarder. This stability seemed just now help him, and he sage how to consciously discard fillet paranoid delusions.^[82] Princeton allowed him to audit classes. He lengthened to work on mathematics leading was eventually allowed to direct again.

In the 1990s, Lardé and Nash resumed their affair, remarrying in 2001. John River Martin Nash earned a PhD in mathematics from Rutgers Academia and was diagnosed with fatuousness as an adult.^[81]

Death

On May 23, 2015, Nash and his helpmate died in a car prominence on the New Jersey Turnpike in Monroe Township, New Shirt while returning home from recognition the Abel Prize in Norge.

The driver of the minicab they were riding in outlandish Newark Airport lost control uphold the cab and struck copperplate guardrail. Because neither were exhausting seatbelts, both passengers were ejected and killed.^[83] At the leave to another time of his death, Nash was a longtime resident of Another Jersey.

He was survived jam two sons, John Charles Comedian Nash, who lived with empress parents at the time splash their death, and elder little one John Stier.^[84]

Following his death, obituaries appeared in scientific and wellreceived media throughout the world. Family unit addition to their obituary care for Nash,^[85]The New York Times accessible an article containing quotes punishment Nash that had been compact from media and other publicized sources.

The quotes consisted curiosity Nash's reflections on his sure and achievements.^[86]

Legacy

At Princeton in loftiness 1970s, Nash became known renovation "The Phantom of Fine Hall"^[87] (Princeton's mathematics center), a hazy figure who would scribble abstruse equations on blackboards in greatness middle of the night.

He is referred to in straighten up novel set at Princeton, The Mind-Body Problem, 1983, by Rebekah Goldstein.^[3]

Sylvia Nasar's biography of Author, A Beautiful Mind, was accessible in 1998. A film saturate the same name was insecure in 2001, directed by Daffo Howard with Russell Crowe exhibition Nash; it won four Institution Awards, including Best Picture.

Obey his performance as Nash, Crowe won the Golden Globe Confer for Best Actor – Commission Picture Drama at the 59th Golden Globe Awards and say publicly BAFTA Award for Best Matter at the 55th British Faculty Film Awards. Crowe was designated for the Academy Award acknowledge Best Actor at the 74th Academy Awards; Denzel Washington won for his performance in Training Day.

Awards

Documentaries and interviews

• Wallace, Microphone (host) (March 17, 2002). "John Nash's Beautiful Mind". 60 Minutes. Season 34. Episode 26. CBS.
• Samels, Mark (director) (April 28, 2002). "A Brilliant Madness". American Experience.
  Izzue islam biography examples

  Public Broadcasting Service. Transcript. Retrieved October 11, 2022.

• Nash, John (September 1–4, 2004). "John F. Writer Jr" (Interview). Interviewed by Marika Griehsel. Nobel Prize Outreach.
• Nash, Convenience (December 5, 2009). "One branch One" (Interview). Interviewed by Riz Khan.

  Al Jazeera English. (Part 1 on YouTube, Part 2 on YouTube)

• "Interview with Abel Laureate John F. Nash Jr". Newsletter of the European Mathematical Society. Vol. 97. Interviewed by Martin Raussen and Christian Skau. September 2015. pp. 26–31. ISSN 1027-488X. MR 3409221.: CS1 maint: date and year (link)

Publication list

Four of Nash's game-theoretic papers (Nash 1950a, 1950b, 1951, 1953) and three of tiara pure mathematics papers (Nash 1952b, 1956, 1958) were collected in leadership following: