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# 9.4: RolloMay and Existential Psychology

Rollo May (1909-1994) introduced existentialism to American psychologists, and has remained the best known proponent of this approach in America. Trained in a fairly traditional format as a psychoanalyst, May considered the detachment with which psychoanalysts approached their patients as a violation of social ethics. For example, if a psychoanalyst helps a patient to be the best they can be, and the person happens to earn their living in an unseemly or criminal way, it hardly seems proper (Stagner, 1988). On the other hand, who is to decide which values should be preferred in a particular society? In the pursuit of freedom, May suggested that sometimes individuals might reasonably oppose the standards or morality of their society. Politics, a wonderful topic for lively debates, is dependent on opposing viewpoints. Only when an individual lives an authentic life, however, should their opinion be considered valid, and existential psychology seeks to help individuals live authentic lives.

## A Brief Biography of RolloMay

Rollo Reese May was born on April 21, 1909, in Ohio, and grew up in Marine City, Michigan. He attended Oberlin College in Ohio, graduating in 1930. Having always been interested in art and artistic creativity, he joined with a small group of artists and traveled to Europe, where they studied the local art of Poland. In order to remain in Europe, May took a teaching position with the American College at Salonika in Greece. When not teaching, he traveled widely throughout Greece, Poland, Romania, and Turkey. He attended the summer school taught by Alfred Adler. Deeply impressed by Adler (as Frankl had been), he nonetheless considered Adler’s theories overly simplistic and too general. This may well have been due to his awakening awareness of the tragic side of human life, keeping in mind that much of Europe suffered greatly during the depression between World War I and World War II (Reeves, 1977).

Upon returning to the United States, May worked as a student advisor and the editor of a student magazine at Michigan State University. In 1936, he enrolled at Union Theological Seminary in New York, with the intention of asking, and most likely hoping to find answers to, the ultimate questions about human life. Despite having no particular desire to become a minister, he did serve in a parish in Montclair, New Jersey for a while. While at the seminary, he became a lifelong friend of Paul Tillich, a well-known existential theologian. Tillich, whose classes May regularly attended, introduced May to the works of Kierkegaard and Heidegger. May also met Kurt Goldstein during this time, and became acquainted with Goldstein’s theories of self-actualization and anxiety as a reaction by organisms to catastrophic events. Regarding his time as a minister, May reflected that the only events which seemed to include an element of reality were the funerals (Reeves, 1977).

Shortly after graduating from the seminary, May began writing books on counseling and creative living. He worked as a counselor at the College of the City of New York, and trained as a psychoanalyst at the William Alanson White Institute of Psychiatry, Psychoanalysis, and Psychology in New York. His time at the training institute overlapped with Harry Stack Sullivan being the president of the William Alanson White Foundation, and Erich Fromm as a fellow associate. In 1946, May began a private practice in psychoanalysis, in 1948 he became a faculty member at the institute, and in 1949 he received the first Ph.D. in clinical psychology at Columbia University. His doctoral dissertation was published as The Meaning of Anxiety (May, 1950), a book that heavily cites the work of Freud and Kierkegaard on anxiety, as well as Fromm, Horney, and Tillich (May, 1950; Reeves, 1977).

Similar to Viktor Frankl, May’s life had taken a dramatic turn during this time, an uncontrollable event that threatened his life: May contracted tuberculosis. At the time, there were no effective treatments for this contagious disease, many people died from it, and like many others May had to spend several years at a sanitarium (Saranac Sanitarium in upstate New York). It was during his time in the sanitarium that May theorized about anxiety and came to one of the most important conclusions in his career. He determined that although Freud had done a masterful job of characterizing the effects of anxiety on the individual, it was Kierkegaard who had truly identified what anxiety is: the threat of becoming nothing. From this point on May could clearly be identified as an existential psychologist. He collaborated with Abraham Maslow, Carl Rogers, and Gordon Allport to present a symposium on existential psychology, in conjunction with the 1959 annual convention of American Psychological Association, which led to the publication of a book on the subject (Reeves, 1977).

As May’s career continued, he became a supervisory and training analyst at the William Alanson White Institute, and an adjunct professor of psychology in the graduate school at New York University. He gave a series of radio talks on existential psychology on a Canadian Broadcasting Corporation show, he served as a visiting professor at Harvard and Princeton, and he continued writing. His later books include works on dreams, symbolism, religion, and love. He eventually settled in California, where he died in 1994.

## Anxiety

May considered anxiety to be the underlying cause of nearly every crisis, whether domestic, professional, economic, or political. He described the world we live in as an age of anxiety. Even though May published The Meaning of Anxiety in 1950, it is safe to say that his concerns are even more relevant today, particularly with the advent of the depersonalization of our world due to the computer age (Reeves, 1977). May considered a wide range of theories on anxiety, including philosophers, neurologists (Kurt Goldstein), and the major psychodynamic theorists (including Freud, Adler, Jung, Horney, Sullivan, and Fromm). He came to the conclusion that Freud had done the best job of explaining anxiety, but it was Kierkegaard who best understood anxiety. May was particularly impressed by Kierkegaard’s idea that anxiety must be understood in the context of an orientation toward freedom. Freedom is the goal of personality development, and although this freedom brings with it anxiety, it is through facing this anxiety that the possibility of freedom arises (May, 1950). In praise of Kierkegaard, May wrote:

…Kierkegaard is proclaiming that “self-strength” develops out of the individual’s successful confronting of anxiety-creating experiences; this is the way one becomes educated to maturity as a self. What is amazing in Kierkegaard is that despite his lack of the tools for interpreting unconscious material - which tools have been available in their most complete form only since Freud - he so keenly and profoundly anticipated modern psychoanalytic insight into anxiety; and that at the same time he placed these insights in the broad context of a poetic and philosophical understanding of human experience. (pg. 45; May, 1950)

In defining anxiety, May distinguished between anxiety and fear, and between normal anxiety and neurotic anxiety. According to May, “anxiety is the apprehension cued off by a threat to some value which the individual holds essential to his existence as a personality” (pg. 191; May, 1950). The threat may be either physical or psychological, such as facing death from tuberculosis or being imprisoned in a concentration camp (which, of course, brought the threat of death in addition to the loss of freedom), or the threat may challenge some other value that the individual identifies with their existence or personal identity (such as the loss of a career, a divorce, a challenge to patriotism in time of war, etc.). What differentiates anxiety from fear, is that fear is a reaction to a specific event, whereas anxiety is vague and diffuse. For example, during a robbery you may fear a man with a gun, but in America today many people are anxious about terrorism. No one can tell when or where terrorists may strike, or even whether they will be foreign terrorists (such as in the World Trade Center attacks) or American terrorists (such as the bombing of the federal building in Oklahoma City or the D.C. sniper killings). May carefully pointed out that using the terms “vague” and “diffuse” to describe anxiety should in no way diminish our understanding of the intensity and painfulness that anxiety can bring. Therein lies the difference between normal vs. neurotic anxiety (May, 1950).

Everyone faces challenges in life, but not everyone sees the same challenges as actual threats. Losing one’s job can be an opportunity to begin a new career, perhaps to go back to school to pursue that new career. However, the transition is often difficult, especially when one is used to being the primary wage earner in the family, and also if the family has to cut back on items they can no longer afford. So anxiety would be a reasonable reaction. That anxiety is considered normal if it is 1) not disproportionate to the objective threat, 2) does not involve mechanisms of intrapsychic conflict, and 3) does not require defense mechanisms for its management (May, 1950). Normal anxiety is often overlooked in adults since it is not particularly intense, especially compared to neurotic anxiety, and it can be managed constructively. It does not show itself in panic or other dramatic symptoms. Neurotic anxiety is, simply, the opposite of normal anxiety. It is disproportionate to the objective threat, it does require intrapsychic defense mechanisms, and it results in neurotic symptoms in spite of those defense mechanisms. It is important to keep in mind that we should not consider individuals who suffer from neurotic anxiety as suffering from objective weaknesses, but rather they suffer from inner psychological patterns and conflicts that prevent them from using their powers to cope.

True to his training in psychodynamic theory, May believed that the psychological patterns resulting in the inability to cope have their origin in childhood, particularly due to poor early relations between the infant and its parents, since an infant’s essential values arise from the security patterns established between the infant and its caregivers (as in Erikson’s first psychosocial crisis: trust vs. mistrust, see Chapter 7). One of the most important factors seems to be the infant’s subjective interpretation of rejection by its primary caregiver, and that subjectivity is influenced by expectations that form later in life (e.g., middle- and upper-class children, who expect more support from their parents, are especially prone to react to rejection with neurotic anxiety; May, 1950).

Discussion Question $$\PageIndex{1}$$

May felt that we must understand anxiety in relation to freedom, or rather, as the fear that we will lose our freedom. He said that some of this anxiety is normal, and only in extreme cases does it become neurotic anxiety. What are some of the situations in your life that make you anxious, and how might they be a threat to your personal freedom? Do you think the level of these anxieties is normal, or is it severe enough to perhaps be considered neurotic?

## Culture, Anxiety, and Hostility

May also addressed the effects of culture on anxiety, and the close interrelationship between anxiety and hostility. Culture affects both the kinds and the quantities of anxiety experienced by individuals. Beyond the essential relationship between infant and caregiver, the determinants of personality that each of us consider essential to our existence as a personality are largely cultural. Indeed, even the nature of the infant/caregiver relationship is subject to cultural influence. The amount of anxiety most people are likely to experience is determined, in part, by the stability of the culture. For example, if a culture is relatively stable and unified, there will be less anxiety throughout that culture (May, 1950). Today, however, many societies are in dramatic flux, due in large part to the powerful trend toward globalization.

As psychologists have begun to examine anxiety in different groups around the world, a variety of interesting, and sometimes disturbing, results have been found. Keep in mind, however, that these are generalities, and do not necessarily apply to each individual within any group. Generally, Asians are more anxious than Europeans and White Americans, who are more anxious than Black Americans and Africans, and there may be a neurological basis for these relative anxiety levels (Rushton, 1999). However, when looking at the specific form of anxiety related to taking academic tests, Black Americans and Chilean students demonstrate higher levels of test anxiety than White Americans (Clawson, Firment, & Trower, 1981; Guida & Ludlow, 1989). One suggestion for the higher levels of anxiety among Blacks in America is that our society is much less sociocentric than most African cultures. Thus, Blacks in America, even if they have lived here for generations, still experience the effects of their displacement from Africa when the culture they carried with them is at odds with Western cultural expectations (Okeke at al., 1999), and even more so when an individual seems to be at odds with most members of their own cultural group (Copeland, 2006). Indeed, the greater the discrepancy between one’s individual cultural expectations and the cultural expectations of the majority of society, the greater the anxiety an individual experiences. This is particularly true during attempts at intercultural communication (Matsumoto & Juang, 2004). Any subsequent breakdown of intercultural communication, which is more likely during periods of high anxiety, can either lead to or enhance pre-existing hostility, prejudice, discrimination, and scapegoating (Whitley & Kite, 2006). One important challenge to intercultural communication in psychology is the need for clinical psychologists to recognize the growing number of anxiety disorders unique to non-Western cultures, such as: hwa-bung (Korea), koro (Malaysia and Southern China), nervios (Latin America), dhat syndrome (India), susto (Latin America), and taijin kyofusho (Japan) (Castillo, 1997).

Culture can influence individuals in a wide variety of ways. May (1950) used the example of competitive individual success in the Western world as his main example, which he considered to be the dominant goal in America. There are many negative effects of this competition, including the high incidences of gastric ulcers and heart disease in our society. Less than a decade later, Freidman and Rosenman (1959) published their classic study on the relationship between Type A behavior (studied in highly competitive businessmen) and cardiovascular disease. Subsequent studies have shown that the key component of Type A behavior predictive of heart disease is hostility, which we will discuss in more detail below (Dembrowski et al., 1985; Lachar, 1993; MacDougal et al., 1985). There has also been a great deal of discussion in our society about media influences on body image, the relationship between unreasonable expectations for women to be thin and the incidence of eating disorders in girls and women, and the repression of female sexuality in many cultures. Goldenberg (2005) recently presented an existential perspective on the body itself as a threat. Cultural beliefs often help to overcome fears of mortality by convincing individuals that they are of greater value than other, lower animals. However, despite the beliefs of many that only humans have a soul, our body is still a mortal animal. As a reaction to the anxiety presented by the reality of our mortal body, many people act in a hostile fashion toward their own bodies, ranging from denying themselves healthy physical relationships with others (e.g., sexual repression) to outright self-destructive behavior (e.g., anorexia nervosa). The problem reaches its extreme, however, when one powerful group directs its hostility in an organized fashion toward another group.

The relationship between anxiety and hostility, according to May, involves a vicious circle. Anxiety gives rise to hostility, and hostility gives rise to increased anxiety. But which comes first? May believed that it was anxiety that underlies hostility, and the evidence can be found in clinical cases involving repressed hostility:

Granted the interrelation between hostility and anxiety, which affect is generally basic? There is ground for believing that, even though hostility may be the specific affect present in many situations, anxiety is often present below the hostility…For one example, in some of the psychosomatic studies of patients with hypertension…it has been found that the reason the patients repressed their hostility was that they were anxious and dependent...The hostility would not have to be repressed in the first place except that the individual is anxious and fears counter-hostility or alienation… (pg. 223; May, 1950)

In Reeves’ analysis of May’s theory (1977), Reeves discusses one of the most important social issues to have faced the United States: the civil rights movement of the 1960s. When an individual’s sense of selfhood is challenged by dramatic changes in society, it can be a very painful experience. And one is likely to resent those responsible for those changes. While it is true that many White people in America supported the civil rights movement, White people in the Deep South (and elsewhere, of course) turned their anxiety, and its associated hostility, toward Blacks. It should not be necessary here to describe the many terrible acts of violence that followed. Suffice it to say that the federal government had to use military troops to intervene in some of the worst cases. Today, we face a similar problem in the war on terrorism. Given the often unequal and unfair manner in which globalization brings vastly different cultures into conflict, and the ease with which so many people can travel the globe, perhaps we should not be surprised at the dramatic level of terrorism in the world today.

## Connections Across Cultures: Terrorists and Terrorism

Please allow me to share a little personal history here. When the Iranian revolution under Ayatollah Khomeini overthrew the Shah of Iran, and the revolutionaries captured the American embassy in Tehran and took sixty-six people hostage (fifty-two of those hostages were held for well over a year before being released), I was in the United States Marine Corps Reserve. I received a phone call at 2:00 a.m. on a Friday morning at my apartment in Cambridge, MA. By midnight, that same day, my reserve unit was in Camp Lejeune, NC, with full combat gear, ready to go to war in Iran. We spent the weekend preparing, though President Carter ultimately chose not to send us overseas. Approximately 10 years later, when the first Gulf War erupted after Iraq invaded Kuwait, my sister took part in Operation Desert Storm. As an Air Force nurse, she was sent to England to help prepare a hospital for wounded military personnel being evacuated from the Middle East (fortunately casualties were minimal). I considered re-enlisting in the Marine Corps at that time, since I certainly wasn’t going to sit at home while my own sister “fought” for our country and our allies. Thankfully, that first Gulf War was brief and, seemingly, simple. So I have followed events in the Middle East carefully ever since, and when Al Qaeda attacked us in New York, I saw it as the latest in a continuation of events in my own life since 1979. For people in the Middle East, however, it was a continuation of events that have lasted for thousands of years.

What I believe matters most for Americans today is to begin to make an honest effort to understand terrorism, its causes, its goals, and how best to deal with it around the world. First, we must dispense with misconceptions. Terrorism and Islam are not one and the same. In an insightful and easily readable book entitled Islam versus Terrorism, Firooz Zadeh (2002) discusses how Islam opposes violence and murder, especially of innocent women and children. He also attempts to identify what is and is not terrorism, and in that effort he identifies eight types of terrorism: state terrorism, religious terrorism, criminal terrorism, terrorism by those who are mentally sick, political terrorism, oppositional terrorism, copy cat terrorism, and victim terrorism. According to Zadeh, the highest cost to society results from state terrorism. When the United States supports corrupt, terrorist governments in other parts of the world, our credibility as a nation fighting terrorism is suspect at best. Has this been the case? Yes, and in the worst possible way: we switch sides as it serves our political and economic interests. The United States helped to train Osama bin Laden and the Taliban fighters in Afghanistan when we wanted them to fight the Russians. Now we call them enemies. We provided weapons and training to Saddam Hussein’s army when they were fighting the Iranians, because of the hostages taken in Tehran. Now we have deposed Hussein. We also sold weapons to Iran, and used the money to help support the Contras (freedom fighters or terrorists, depending on your point of view) trying to overthrow the leftist Sandinista government of Nicaragua. Zadeh proposes that people in the Middle East cannot trust the United States, except in one area: our support of Israel. And since other Middle Eastern countries see Israel as the one obstacle to a Palestinian homeland, they disapprove of that support. It does not matter whether the actions of the United States were right or wrong, whether they really were in our best interests or not. What matters is how the rest of the world sees us now, and whether our top government officials are willing to consider how we are viewed globally and to act responsibly in terms of foreign policy in order to ensure what is best for all people around the world. In addressing the Middle East in particular, Fathali Moghaddam wrote:

Islamic communities in many parts of the world are experiencing a profound and historic identity crisis, one tragic manifestation of which is terrorism. In order to understand and avert this destructive trend, we must come to grips with the monumental crisis of identity that is paralyzing moderate movements but energizing fanatic forces in Islamic communities.

…Why do we need to understand how the terrorists see the world? Because this is the best way for us to find an effective means to end terrorism…Seeing the world from the terrorists’ point of view does not mean condoning terrorism; rather, it means better understanding terrorism so as to end it. (pg. ix; Moghaddam, 2006)

As mentioned above, there are many different forms of terrorism, so it is difficult to define exactly what it is. Nevertheless, in an effort to do so, Moghaddam (2005) defines terrorism as “politically motivated violence, perpetrated by individuals, groups, or state-sponsored agents, intended to instill feelings of terror and helplessness in a population in order to influence decision making and to change behavior.” Moghaddam suggests that psychologists need to play an important role in understanding terrorism for two main reasons: the basis for terrorist actions is typically subjectively interpreted values and beliefs, and the actions of terrorists are designed to cause specific psychological experiences, i.e., terror and helplessness. Moghaddam (2005, 2006) proposes a metaphor for how one becomes a terrorist, based on climbing a staircase, in which options are perceived to become more and more limited as one climbs the stairs. The most significant factor is the condition in which many people live on the ground floor, before they even consider climbing that staircase. Many people in this world live in abject poverty, under repressive governments that are unjust. When individuals see no hope within the system, and they lack any political means to effect change, then a path toward terrorism becomes perhaps the only reasonable possibility. Still, very few people are likely to become suicide bombers.

Individuals living in desperate conditions may move to the first floor on the staircase toward terrorism, where they evaluate their perceived options to fight unfair treatment. If there appear to be no options for justice within one’s society, no opportunity to be heard, and no opportunity for personal mobility, the individual may then move to the second floor. Here the individual begins to displace their aggression. This often involves education/propaganda that identifies a clear target, for example the United States, also known as the “Great Satan.” This is the important beginning of an us-versus-them mentality. On the third floor, individuals become morally engaged with the terrorist organization. While we may see terrorists as immoral, they are beginning to believe that they are fighting for a just cause, against the immoral repression of their chosen target. As they move to the fourth floor, they solidify their categorical thinking (the us-versus-them mentality) and begin to see the terrorist organization, and terrorist acts, as legitimate. At this point there is little chance that they can leave the terrorist organization alive. For specific individuals, the training necessary to carry out a terrorist act takes place, often very quickly. Not only does a terrorist need to learn about weapons and tactics, they must also be trained to sidestep the natural, biological inhibition against killing other human beings. Two factors in helping to prepare people to kill are the intense indoctrination in the belief that their actions are for a greater good and secrecy. If an attack is done suddenly and without warning, victims have no opportunity to submit or to beg for mercy. The act occurs before the terrorist might become compassionate as he or she faces their intended victims (Moghaddam, 2005, 2006). Based on this model, Moghaddam proposes four steps that are necessary to stop terrorism by interrupting the formation of new terrorists. First, there must be prevention. Unfortunately, our government has a long history of choosing short-term fixes, rather than long-term preventative measures. Case in point: America’s failure in the war on drugs. Aggressive responses aimed at individuals only provide an opening for someone new to step in and continuing using and/or selling drugs, and the same is true of terrorists. We need to work toward eliminating the pathway to terrorism, so we will not need to use the military and/or FBI to track down individuals (except, of course, in extreme cases such as terrorism that results from psychological disorder - e.g., consider the case of Theodore Kaczynski, the Unabomber). In addition, Moghaddam suggests supporting contextualized democracy, educating against categorical thinking, and promoting interobjectivity and justice. In order for there to be a long-term solution, there must be international dialogue and improved intercultural understanding (Moghaddam, 2005).

Returning to the misconception in the minds of many Americans that terrorism is synonymous with Islam, let’s examine where known terrorist organizations are located around the world. Fairly notorious organizations have come from Northern Ireland (e.g., the Irish Republican Army and the Ulster Defense Association), throughout mainland Europe (e.g., the Red Army Faction in Germany, the Red Brigades in Italy, and Action Directe in France), throughout the Middle East (e.g., Hezbollah in Lebanon, the Palestinian group Hamas, the Stern Gang that fought for the establishment of Israel, and Al Qaeda), Africa, Asia, Latin America, Canada, and the United States (e.g., the Animal Liberation Front, Aryan Nations, the Black Panthers, and the Ku Klux Klan). As of 1999, at least twenty-eight well-organized terrorist groups existed, and when one takes into account factions within those groups and smaller, yet still identifiable, groups, as many as eighty-three terrorists groups have been identified around the world (Henderson, 2001). Some are primarily political, and some are primarily religious. Some are global, and some are more local. They include people and cultures of great diversity: Black, White, Asian, Latin, Catholic, Protestant, Muslim, Hindu, etc. Although terrorism appears to arise out of poverty and desperation, terrorists themselves, or at least the leaders, tend to be better educated than most and they are well versed in propaganda and well trained in weapons and tactics (Moghaddam, 2005; Zadeh, 2002). The only characteristic that all terrorists seem to share is an extreme commitment to violence, which arises out of desperation and perceived injustice, and is viewed as the only means to be heard and to effect change.

So can terrorism effect change, is terrorism effective? One can easily find authors who argue that it does indeed work (Dershowitz, 2002) or that it always fails (Carr, 2002). Alan Dershowitz (2002) argues that the very reason terrorism works is everything we have looked at so far: an effort to understand the root causes of terrorism and the terrorists themselves. Accordingly, he says:

We must take precisely the opposite approach to terrorism. We must commit ourselves never to try to understand or eliminate its alleged root causes, but rather to place it beyond the pale of dialogue and negotiation. Our message must be this: even if you have legitimate grievances, if you resort to terrorism as a means toward eliminating them we will simply not listen to you, we will not try to understand you, and we will certainly never change any of our policies toward you. Instead, we will hunt you down and destroy your capacity to engage in terror. (pp. 24-25; Dershowitz, 2002)

As a case in point, Dershowitz cites the awarding of observer status at the United Nations to the Palestinian Liberation Organization only after Palestinian terrorists began hijacking commercial airliners. Prior to the hijackings, 20 years of pleading their case to the United Nations had little effect. Dershowitz then offers a timeline that appears to clearly establish an effective relationship in which terrorism became more and more effective over time (from 1968-1999) in eliciting international recognition and support for the Palestinian cause. In contrast, Caleb Carr (2002) views terrorism entirely within the discipline of military history. He considers today’s terrorism to be nothing more than a modern permutation of warfare against civilians in order to break their support for either leaders or policies that the terrorists oppose, the origins of which are as old as human conflict itself. Viewing terrorism as warfare has certain interesting implications. Throughout history, those who wage war against civilians ultimately defeated themselves by turning sentiment against them. On 9/11, Al Qaeda attacked civilians to a degree that has not been seen in ages:

…In so doing, the organizers, sponsors, and foot soldiers of every terrorist group involved in the September 11 attacks have unwittingly ensured that their extremist cause will be discredited among many of their sympathizers, disowned by most of their former sponsors, and finally defeated by their enemies: two thousand years of the lessons of terror dictate that this is the ultimate fate that awaits the attackers, no matter how many noncombatants they manage to kill along the way. (pp. 223-224; Carr, 2002).

Carr also addresses the other most important implication of treating terrorism as warfare: it must be met with warfare, but that warfare must not be excessive, such that it might also be viewed as terrorism. If our response to terrorism is excessive military might, then the tide of public opinion can swing back in favor of Al Qaeda, especially in Muslim countries where the United States is not trusted.

Echoing Carr’s concerns about the extent and nature of our military actions in the war on terror, one way in which terrorism might work against us, without seeming to have gained what was intended (if we can even know what was intended), is if our fundamental democratic principles change. In The Lesser Evil: Political Ethics in an Age of Terror, Michael Ignatieff (2004) argues that terrorism must be met with force, and that such force is a lesser evil than the terrorism that necessitated the response. The danger lies in succumbing to the greater evil of seeking revenge. Dershowitz (2002) provides a compelling case for how an amoral society could control and possibly eliminate all terrorism, but America is not an amoral society. Our responses are constrained by the constitution and by the political debate that forms the very basis of our democracy. When we respond to terrorist acts, we must consider what we want that response to accomplish:

Terrorism requires us to think carefully about who we are as free peoples and what we need to do in order to remain so. When we are confronted with terrorist violence, we cannot allow the claims of national security to trump the claims of liberty, since what we are trying to defend is our continued existence as a free people. Freedom must set a limit to the measures we employ to maintain it. (pg. 145; Ignatieff, 2004)

Finally, can the ultimate answer to terrorism be found in promoting democratic governments in every nation? The war on terror has led us to depose both Saddam Hussein in Iraq and the Taliban in Afghanistan, and to replace them with democratically elected governments. Only time will tell whether those governments will survive, but there is reason for caution. Religious turmoil continues in the Middle East. In America, our constitution provides for separation of church and state, and that separation has become an important tradition. But for Muslims, the idea of a secular democracy, one that is not guided by Allah, is simply inconceivable. They are not opposed to democracy per se, indeed it has been argued that Islam is likely to eventually lead to pluralist democracies (Aslan, 2005). But to pressure Islamic countries into accepting the secular democracy that we hold so dear is, according to Robert Shedinger (2004), equivalent to declaring war on Islam. So what appears to be essential to promoting stability in the Middle East, and elsewhere, is an effort to support contextual democracy, that is, forms of democracy that fit with the culture of the people who will create and participate in that democracy (Aslan, 2005; Moghaddam, 2005, 2006; Shedinger, 2004; Zadeh, 2002).

## Integration and the Human Dilemma

In the preface to Man’s Search for Himself (May, 1953), May presents the existential philosophy that there is meaning to be found in challenges and suffering, and that psychologists in particular may find a special opportunity in such circumstances:

When our society, in its time of upheaval in standards and values, can give us no clear picture of “what we are and what we ought to be,”…we are thrown back on the search for ourselves. The painful insecurity on all sides gives us new incentive to ask, Is there perhaps some important source of guidance and strength we have overlooked?…How can anyone undertake the long development toward self-realization in a time when practically nothing is certain, either in the present or the future?…The psychotherapist has no magic answers…But there is something in addition to his technical training and his own self-understanding…This something is the wisdom the psychotherapist gains in working with people who are striving to overcome their problems. He has the extraordinary, if often taxing, privilege of accompanying persons through their intimate and profound struggles to gain new integration. (pg. 7; May, 1953)

Integration, according to May, is similar to Heidegger’s concept of Dasein (being-in-the-world). As conscious, free, and responsible beings our goal should be to separate ourselves from the conformist, automaton masses (the en-soi, according to Sartre) and progressively integrate with others in freely chosen love and creative work (May, 1953), or as Clement Reeves puts it: “To understand and elucidate the specific, distinguishing characteristics of the human being, and to grasp what it is to achieve courageous, decisive, integrated response to the challenge inherent in existence…” (Reeves, 1977). The process of integration is lifelong, and should be appropriate for whatever age each one of us happens to be right now. May suggests that a healthy child of eight, who is fulfilling his capacity of self-conscious choice for a child of eight years old, is more of a person than a neurotic adult who is 30 years old. Likewise, a person who can face death courageously at the age of thirty is more mature than someone 80 years old who “cringes and begs still to be shielded from reality” (May, 1953). Thus, it is important to live each moment with freedom, honesty, and responsibility. If each of us lives within the present moment, working to fulfill our potential, being true to whom we are and the situations within which we live, May proposes that we will experience joy and gratification:

…Does not the uncertainty of our time teach us the most important lesson of all - that the ultimate criteria are the honesty, integrity, courage and love of a given moment of relatedness? If we do not have that, we are not building for the future anyway; if we do have it, we can trust the future to itself. (pg. 276; May, 1953)

One of the challenges to living an integrated life is seen in what May described as the human dilemma (May, 1967). Are we the subject of our lives, or are we an object in our world? When we become absorbed in the details of our responsibilities and actions, when we allow ourselves to be controlled and directed in order to accomplish our assigned tasks, when we become slaves to the clock, doing this and that, going here and there, as others expect us to, we are viewing ourselves as objects. This is reminiscent of what Karen Horney called the tyranny of the should. On the other hand, when we consider our feelings, wishes, and desires, when we are true to ourselves, or living authentically, then we are viewing ourselves as subjects, as active participants in our own lives. According to May (1967), the human dilemma arises out of our capacity to experience ourselves as both subject and object at the same time. But how can opposite poles of the human experience both be true? It is in the process between the two poles that development of human consciousness develops, both deepening and widening that consciousness. This is essentially the same idea, though in different form, used by Heidegger and Sartre in describing the unique nature of human beings. For Heidegger this nothingness was the undefined distinction between Being and beings, for Sartre it was the shell that surrounded the pour-soi.

May believed that existential psychology occupied a space somewhere between the two extremes that existed, and continue to exist, in psychology: behaviorism vs. humanism. May rejected Skinner’s arguments that all human behavior can be understood in terms of stimuli and responses, declaring that there is ample evidence in both clinical practice and everyday life of people being active participants in their view of, actions in, and reactions to their world. He was equally critical of Carl Rogers, believing that humanistic psychologists no longer recognized very real irrational behavior, as well as aggression and hostility (May, 1967). He believed that psychology had become trapped in a misguided desire to define everything scientifically, and according to rules that then determined each psychologist’s view of the world and their patients. As a caution to those psychologists who cannot see beyond their theories, May wrote:

Now I am certainly aware, if I may say so without sounding patronizing, that the compelling need for honesty is one of the motives which leads psychologists to seek quantitative measures…I am also aware that research in our day has to be carefully set up so that the results are teachable and can be built upon by others. The compelling drive to get at the truth is what improves us all as psychologists, and is part and parcel of intellectual integrity. But I do urge that we not let the drive for honesty put blinders on us and cut off our range of vision so that we miss the very thing we set out to understand - namely, the living human being. (pg. 14; May, 1967)

discussion question $$\PageIndex{2}$$

May suggested that we need to separate ourselves from the conformist masses, and then integrate ourselves with others in free and responsible ways. Are you a follower, or a leader? Either way, do you consciously choose the role you play, thereby living an authentic life?

## Love and Intentionality

Love was a very important topic for May. Simply put, “To be capable of giving and receiving mature love is as sound a criterion as we have for the fulfilled personality” (May, 1953). He was certainly not alone. Harry Harlow, best known for his studies on contact comfort, described love as “a wondrous state, deep, tender, and rewarding,” and Abraham Maslow said “We must understand love; we must be able to teach it, to create it, to predict it, or else the world is lost to hostility and to suspicion” (Harlow, 1975; Maslow, 1975). However, there are “a million and one” types of relationships that people call love, so it remains a perplexing issue (May, 1953).

May talked about four types of love in Western tradition: sex, eros, philia, and agape (May, 1969). Sex and eros are closely related, but they are different. Sex is what we also call lust or libido, whereas eros is the drive of love to procreate or create. As changes in society allowed the more open study of sex, prompted by the work of people like Sigmund Freud and Wilhelm Reich, May noted three particular paradoxes. First, our so-called enlightenment has not removed the sexual problems in our culture. In the past, an individual could refrain from sexual activity using the moral guidelines of society as an explanation. As casual sex became common, even expected, individuals had to face expressing their own morality as just that: their own! This also created a new source of anxiety for some, namely the possibility that their personal relationships might carry an expectation of sexual activity, and that if they did not comply they might not be able to continue dating someone they liked. The second paradox is that “the new emphasis on technique in sex and love-making backfires” (May, 1969). Emphasizing technique (or prowess) can result in a mechanistic attitude toward making love, possibly leading to alienation, feelings of loneliness, and depersonalization. This can lead to the anticipatory anxiety described by Frankl. Finally, May believed that our sexual freedom was actually a new form of Puritanism. There is a state of alienation from the body, a separation of emotion from reason, and the use of the body as a machine. Whereas in the Victorian era people tried to be in love without falling into sex, today many people try to have sex without falling in love.

Philia and agape are also related to one another, as with sex and love. Philia refers to feelings of friendship or brotherly love, whereas agape is the love devoted to caring for others. Friendship during childhood is very important, and May believed it was essential for meaningful and loving relationships as adults, including those involving eros. Indeed, the tension created by eros in terms of continuous attraction and continuous passion would be unbearable if philia did not enter into the equation and allow one to relax in the pleasant and friendly company of the object of one’s desires. Harry Harlow, once again, showed that the opportunity to make friends was as essential in the development of young monkeys as it appears to be in humans (cited in May, 1969). In the West, however, given our highly individualistic and competitive society, deep, meaningful friendships seem to be something of the past, especially among men. May cautions, however, that since the evidence shows the importance of friendship during development perhaps we should remember the value of having good friends.

Finally we have agape, a selfless love beyond any hope of gain for oneself. May compared this love to the biological aspect of nature in which a parent will fight to the death in defense of their offspring. With agape, we run the risk of being like God, in the sense that we know others never act without some degree of their own interests in mind. Similarly, we don’t want to be loved in an ethereal sense, or on the other hand only for our body. We want to be loved completely. So, all true love involves some element of the other types of love, no matter how little or how obscured it may be (May, 1969).

Figure $$\PageIndex{1}$$

Agape is exemplified in the bond between a parent and their child.

In the foreword to Love and Will (May, 1969) May acknowledged that some of his readers might find it odd that he combined the two topics in one book, but he felt strongly that the topics belong together. He considered both love and will to be interdependent, they are processes in which people reach out to influence others, to help to mold and create the consciousness of others. Love without will is sentimental and experimental, whereas will without love is manipulative. Only by remaining open to the influence of others can we likewise influence them, so love must have an honest purpose, and purpose must be taken with care.

Will, or will power as it is more commonly known, was one of the earliest subjects in American psychology, having been examined in detail by William James as early as 1890 (see James, 1892/1992) and again in 1897 in The Will to Believe (James, 1897/1992). May considered Sigmund Freud’s greatest discovery to be the uncovering of unconscious desires and motives. Although many people may believe themselves to be acting out of higher ideals, most of us are, in reality, acting according to psychologically determined factors of which we are unaware. Nonetheless, May considered this to be one of the most unfortunate results of Freud’s work. By accepting determinism, we undermine the influence of will and making decisions. As May put it, Freud’s theory suggests that we are “not driving any more, but driven” (May, 1969).

The suggestion that we are no longer in charge of our own lives, that we are driven by psychological determinism, seems strange to those who believe that never before have people had such power, both in terms of individual freedom and in the collective conquest of nature. But May referred to a contradiction in will, the contrast between our feelings of powerlessness and self-doubt and the societal assurances that we can do anything we set our minds to. May believed that we exist in a “curious predicament,” in that the technical wonders that make us feel so powerful are the very same processes that overwhelm us (May, 1969):

Thus, the crisis in will does not arise from either the presence or absence of power in the individual’s world. It comes from the contradiction between the two - the result of which is a paralysis of will. (pg. 189; May, 1969)

Will alone is not the driving force that leads us to responsible and authentic lives. Underlying will is something May called intentionality. Intentionality is the structure that gives meaning to experience, it is both how we perceive the world and how the world can be perceived by us. In other words, through our perceptual processes we influence the world around us; we affect the very things that we perceive. Intentionality is a bridge between subject and object (May, 1969). Compare this once again to the nothingness between beings and Being (à la Heidegger), or between the en-soi and the pour-soi (à la Sartre). Still, our ability to reach and form the very objects that we perceive, in other words, to participate actively in our lives, can be dramatically curtailed by the problem addressed by May early in his career, anxiety:

Overwhelming anxiety destroys the capacity to perceive and conceive one’s world, to reach out toward it to form and re-form it. In this sense, it destroys intentionality. We cannot hope, plan, promise, or create in severe anxiety; we shrink back into a stockade of limited consciousness hoping only to preserve ourselves until the danger is past. (pp. 244; May, 1969)

discussion question $$\PageIndex{3}$$

Consider the different loves in your life. How do they differ? How have they brought meaning to your life? Has your view of what love is changed during your life, in either good or bad ways?

## The Daimonic: Source of Violence and Creativity

The daimonic, according to May, is “any natural function which has the power to take over the whole person” (May, 1969). It can be either destructive or creative, and is often both. In this way it is similar to Jung’s concept of the shadow, and May himself made that comparison (May, 1991; see also Diamond, 1996, Reeves, 1977). In fact, it is the mixture of good and evil in the daimonic that protects us from the dangers of excess, whether excess good or the passivity of feeling powerless. When May did not know whether he would live or die from tuberculosis, he realized that his feelings of helplessness were turning into passivity, and that this was sure to lead to his death (as he had seen with others). He described this experience as the product of his innocence, and that because he was innocent he allowed the bacteria infecting his body to do violence to him. However, when he chose to fight the disease, when he asserted his will to live, he began to make steady progress and, indeed, he recovered. In this sense, May had chosen to allow the daimonic to take over his self in the interest of self preservation. In each instance, how one allows the daimonic to take over is influenced by personal responsibility (Reeves, 1977).

When the daimonic takes over without one having made a responsible choice, however, it can lead to violence toward others. Our lives often involve conflict between those who have power and those who do not. When a person feels powerless, helpless, insignificant, they can lash out under the control of the daimonic. According to May, violence is bred in impotence and apathy (May, 1972). This can be particularly important for those who have little or no advantage in our society. In Power and Innocence (May, 1972), May described a patient who was a young, Black woman. Being both Black and female, born before the civil rights movement, she was about as powerless as one could be in America. Her stepfather had forced her to serve as a prostitute for years. Although quite intelligent, and successful in school and college, she felt so helpless that May described her as having “no active belief that she deserved to be helped.” An important aspect of therapy for this patient was to get in touch with her anger, to get in touch with the violence that had been done to her and that she wished to do to others.

In considering the case of this young woman, May concluded that we must not simply condemn all violence and try to eliminate even the possibility of it. To do so would be to take away a part of full humanity. In this context, May criticizes humanistic psychology and its emphasis on fulfilling self-actualization, an emphasis that May felt moved toward greater moral perfection. However, the recognition that we are not perfect, that each of us has good and evil within, prohibits us from moral arrogance. Recognizing this leads to the restraint necessary for making forgiveness possible.

Our ability to achieve good is dependent on who we are, and who we are is based partly on our own creativity. Since humans are not simply driven by instinct and fixed action patterns, in contrast to every other creature on earth we must create ourselves. This creation must take place within the world that exists around us, and must take into account all of the emotions and predispositions that we do carry with us as biological organisms.

Art - and creative activities of all kinds - can provide comparatively healthy outlets for the constructive expression of anger and rage. Creativity cannot, however, always substitute for psychotherapy. Nevertheless, creativity is at the very core of the psychotherapeutic project: The patient is encouraged to become more creative in psychologically restructuring his or her inner world, and then to continue this creative process in the outer world, not only by accepting and adjusting to reality, but, whenever possible, by reshaping it…

“Creativity” can be broadly defined as the constructive utilization of the daimonic. Creativity is called forth from each one of us by the inevitable conflicts and chaos inherent in human existence… (pp. 255-256; Diamond, 1996)

Pursuing this creativity is not easy, however. We live in a world that is rapidly changing. Since May’s death in 1994 change in the world has probably even accelerated. May asked whether we would withdraw in anxiety and panic as our foundations where shaken, or would we actively choose to participate in forming the future (May, 1975). Choosing to live in the future requires leaping into the unknown, going where others have not been, and therefore cannot guide us. It involves what existentialists call the anxiety of nothingness (May, 1975). Making this bold choice requires courage. One of the reasons we need to be courageous is that we must fully commit ourselves to pursuing a responsible creation of the future, but at the same time we must recognize that sometimes we will be wrong. Those who claim they are absolutely right can be dangerous, since such an attitude can lead to dogmatism, or worse, fanaticism (May, 1975).

Finally, not only must we accept that we might make bad choices, we must also recognize that our creativity is limited. In The Courage to Create (May, 1975), May described having attended a conference where the introductory speaker declared that there is no limit to the possibilities of the human being. Following this statement, the discussion at the conference was a flop. May realized that if there is no limit to what we can accomplish, then there really aren’t any problems any more, we only need to wait until our potentiality catches up with our situation and the problem solves itself. May offered a rather amusing example to clarify this point:

…it is like putting someone into a canoe and pushing him out into the Atlantic toward England with the cheery comment, “The sky’s the limit.” The canoer is only too aware of the fact that an inescapably real limit is also the bottom of the ocean. (pg. 113, May, 1975)

Another inescapable limit is our death. There is no creative act that can change the fact that we will die someday, and that we cannot know when or how it will happen. May believed, however, that these limits are valuable, that creativity itself needs limits. He proposed that consciousness arises from our awareness of these limits, and from the struggle against these limits. May compared this concept to Adler’s theory that much of what we as individuals, and also society as a whole, are arises from our efforts to compensate for inferiority. Thus, our limits lead to what May called a passion for form. In its passion for form, the mind is actively forming and re-forming the world in which we live (May, 1975).

discussion question $$\PageIndex{4}$$

May believed that creatively taking charge of your life required courage. Have you ever had to make a really difficult decision? Did you take the easy way out, or the safe path, or did you make a bold decision that offered great opportunity?

## The Cry for Myth

As a practicing psychoanalyst I find that contemporary therapy is almost entirely concerned, when all is surveyed, with the problems of the individual’s search for myths. The fact that Western society has all but lost its myths was the main reason for the birth and development of psychoanalysis in the first place. (pg. 9; May, 1991)

The preceding quote is how May began The Cry for Myth, the last book of his career (May, 1991). According to May, the definition of a myth is quite simple: it “is a way of making sense in a senseless world.” In addition, myths give substance to our existence. In a healthy society the myths provide relief from neurotic guilt and excessive anxiety, and so a compassionate therapist will not discourage them. In the twentieth century, especially in Western culture, we have lost our myths, and with them we have lost our sense of existence and our direction or purpose in life. The danger in this is that people are then susceptible to cults, drugs, superstition, etc., in a vain effort to replace that purpose (May, 1991).

As we pass through the experiences of our lives, our memory is dependent mainly upon myth. It is well accepted today that human memory is constructive, and influenced by our expectations of memory. As May describes it, the formation of a memory, regardless of whether it is real or fantasy, is molded like clay. We then retain it as a myth, and rely on that myth for future guidance in similar situations. For example, an infant is fed three times a day and put to bed 365 days a year, and yet they remember only one or two of these events from their years of early childhood. For whatever reason, good or bad, these specific events take on mythic proportions and greatly influence the course of our lives. May acknowledges the contribution of Alfred Adler in recognizing the value of these early memories, describing Adler as “a perceptive and humble man, he was gifted with unusual sensitivity for children” (May, 1991). As we have seen, Adler considered the basis for neurosis to be a lack of social interest. In therapy, Adler focused on the “guiding fiction” of a child’s life, something May considered to be synonymous with a “myth.” Since “memory is the mother of creativity,” and memory depends upon myth, May believed that the myths that form the identity of our culture are essential for the formation of our self.

May ends his final book with a chapter entitled The Great Circle of Love. Having covered a variety of famous myths in the book, including Dante’s Divine Comedy, Marlow’s Faust, Captain Ahab in Moby Dick, and Poe’s The Raven, May concludes:

In each of these dramas the liberation of both woman and man is possible only when each achieves a new myth of the other sex, leading to a new significant psychological relationship. They are both then liberated from their previous empty and lonely existence. The woman and the man find their true selves only when they are fully present to each other. They find they both need each other, not only physically but psychologically and spiritually as well. (pg. 288; May, 1991)


text:
Feynman-Kac particle integration with geometric interacting jumps

# Feynman-Kac particle integration with geometric interacting jumps

Pierre Del Moral INRIA Bordeaux Sud-Ouest and University of Bordeaux, France Pierre E. Jacob National University of Singapore, Singapore Anthony Lee University of Warwick, UK Lawrence Murray CSIRO Mathematics, Informatics and Statistics, Perth, Australia Gareth W. Peters University College London, UK
###### Abstract

This article is concerned with the design and analysis of discrete time Feynman-Kac particle integration models with geometric interacting jump processes. We analyze two general types of model, corresponding to whether the reference process is in continuous or discrete time. For the former, we consider discrete generation particle models defined by arbitrarily fine time mesh approximations of the Feynman-Kac models with continuous time path integrals. For the latter, we assume that the discrete process is observed at integer times and we design new approximation models with geometric interacting jumps in terms of a sequence of intermediate time steps between the integers. In both situations, we provide non asymptotic bias and variance theorems w.r.t. the time step and the size of the system, yielding what appear to be the first results of this type for this class of Feynman-Kac particle integration models. We also discuss uniform convergence estimates w.r.t. the time horizon. Our approach is based on an original semigroup analysis with first order decompositions of the fluctuation errors.

Keywords : Feynman-Kac formulae, interacting jump particle systems, measure valued processes, non asymptotic bias and variance estimates.

Mathematics Subject Classification :
Primary: 62L20; 65C05; 60G35. ; Secondary: 60G57, 81Q05; 82C22.

## 1 Introduction

Feynman-Kac formulae are central path integration mathematical models in physics and probability theory. More precisely, these models and their interacting particle interpretations have come to play a significant role in applied probability, numerical physics, Bayesian statistics, probabilistic machine learning, and engineering sciences. Applications of these particle integration techniques are increasingly used to solve a variety of complex problems in nonlinear filtering, data assimilation, rare event sampling, hidden Markov chain parameter estimation, stochastic control and financial mathematics. A detailed account of these functional models and their application domains can be found in the series of research books [3, 10, 21, 27, 34] and, more recently, in [4, 16, 26].

In computational physics, these techniques are used for free energy computations, specifically in estimating ground states of Schrödinger operators. In this context, these particle models are often referred as quantum or diffusion Monte Carlo methods [1, 2, 7, 39]. We also refer the reader to the series of articles [11, 24, 40, 41, 46].

In advanced signal processing, they are known as particle filters or sequential Monte Carlo methods, and were introduced in three independent works in the 90’s [9, 32, 37]. These stochastic particle algorithms are now routinely used to compute sequentially the flow of conditional distributions of the random states of a signal process given some noisy and partial observations [3, 10, 12, 21, 27, 28, 35, 38]. Feynman-Kac formulae and their particle interpretations are also commonly used in financial mathematics to model option prices, futures prices and sensitivity measures, and in insurance and risk models [4, 5, 33, 42, 44, 43]. They are used in rare event analysis to model conditional distributions of stochastic processes evolving in a rare event regime [6, 5, 20].

This article presents geometric interacting jump particle approximations of Feynman-Kac path integrals. It also contains theoretical results related to the practical implementation of these particle algorithms for both discrete and continuous time integration problems. A key result is the presentation of connections between the interacting jump particle interpretations of the continuous time models and their discrete time generation versions. This is motivated by the fact that while the continuous time nature of these models is fundamental to describing certain phenomena, the practical implementation of these models on a computer requires a judicious choice of time discretization. Conversely, as shown in section 2.1 in [25], a discrete time Feynman-Kac model can be encapsulated within a continuous time framework by considering stochastic processes only varying on integer times. Continuous time Feynman-Kac particle models are based on exponential interacting jumps [15, 24, 21, 30, 31, 29, 46], while their discrete time versions are based on geometric type jumps [10, 16, 19]. From a computational perspective, the exponential type interacting jumps thus need to be approximated by geometric type jumps. Incidentally, some of these geometric type interacting jump particle algorithms are better suited to implementation in a parallel computing environment (see section 5.3).

Surprisingly, little attention has been paid to analyze the connections between exponential and geometric type jump particle models. There are references dealing with these two models separately [8, 17, 18, 22, 25, 24, 46], but none provide a convergence analysis between the two. In this paper we initiate this study with a non asymptotic bias and variance analysis w.r.t. the time step parameter and the size of the particle population scheme. Special attention is paid to the stochastic modeling of these interacting jump processes, and to a stochastic perturbation analysis of these particle models w.r.t. local sampling random fields.

We conclude this section with basic notation used in the article. We let be the Banach space of all bounded Borel functions on some Polish111i.e. homeomorphic to a complete separable metric space state space equipped with a Borel -field , equipped with the uniform norm . We denote by the oscillation of a function . We let be the Lebesgue integral of a function with respect to a finite signed measure on . We also equip the set of finite signed measures with the total variation norm , where the supremum is taken over all functions with . We let be the subset of all probability measures. We recall that any bounded integral operator on is an operator from into itself defined by , for some measure , indexed by , and we set . These operators generate a dual operator on the set of finite signed measures defined by . A Markov kernel is a positive and bounded integral operator s.t. . The Dobrushin contraction coefficient of a Markov kernel is defined by , where the supremum is taken over all functions s.t. . Given some positive potential function on , we denote by the Boltzmann-Gibbs transformation defined by .

## 2 Description of the models

### 2.1 Feynman-Kac models

We consider an -valued Markov process , defined on a standard filtered probability space . The set represents the space of càdlàg paths equipped with the Skorokhod topology which turn it into a Polish space. A point represents a sample path of the canonical process . We also let and be the sigma-field and probability measure of the process . Finally, we also consider the -augmentation of so that the resulting filtration satisfies the usual conditions of right continuity and completion by -negligible sets (see for instance [36, 45], and the references therein). We also consider a time inhomogeneous bounded Borel function on .

We let and be the Feynman-Kac measures on defined for any bounded measurable function on , by the following formulae

 Qt(f):=Λt(f)/Λt(1) {\rm with}Λt(f)=E(f(X[0,t]) exp(∫t0Vs(Xs)ds)) (1)

and we let and , respectively, be the -marginals of and .

We consider the mesh sequence , , with time step associated with some integer , and we let and be the Feynman-Kac measures on defined for any bounded measurable function on , by the following formulae

 Q(m)tn(f):=Λ(m)tn(f)/Λ(m)tn(1){\rm with}Λ(m)tn(f)=E(f(X[0,tn]) ∏0≤p

We also denote by and , respectively, the -marginal of and .

• Case (D) : We have and , where , is an -valued Markov chain, and are Borel positive functions s.t. is bounded.

In this case, the marginal and of the Feynman-Kac measures of and on integer times are given for any by the formula

 ηn(f)=γn(f)/γn(1){\rm with}γn(f):=E(f(Xn) ∏0≤p
• Case (C) : The process is a continuous time Markov process with infinitesimal generators defined on some common domain of functions , and . The set is a sub-algebra of the Banach space generating the Borel -field , and for any measurable function the Feynman-Kac semigroup , , defined by

 Qs,t(f)(x)=E(f(Xt) exp(∫tsUr(Xr)dr) ∣∣ Xs=x)

leaves invariant; that is we have that . For any , the mappings and , and their norm as well the norm of the first order derivatives only depend on and on the norms of the functions and their derivatives.

The regularity conditions stated in (C) correspond to time inhomogeneous versions of those introduced in [24]. They hold for pure jump processes with bounded jump rates with , or for Euclidean diffusions on with regular and Lipschitz coefficients by taking as the set of -functions with derivatives decreasing at infinity faster that any polynomial function. These regularity conditions allow the use of most of the principal theorems of stochastic differential calculus, e.g. the “carré du champ”, or square field, operator that characterizes the predictable quadratic variations of the martingales that appear in Ito’s formulae. These regularity conditions can probably be relaxed using the extended setup developed in [21].

We have already mentioned that the particle interpretations associated with the continuous time models (1) are defined in terms of interacting jump particle systems [21, 22, 24, 25]. The implementation of these continuous time particle algorithms is clearly impractical and we therefore resort to the geometric interacting processes associated with the -approximation models defined in (2). These discrete generation interacting jumps models provide new and different types of adaptive resampling procedures, which differ from those discussed in the articles [12, 13], and the references therein.

### 2.2 Mean field particle models

In this section, we provide a brief description of the geometric type interacting jump particle models associated with the -approximation Feynman-Kac model defined in (2). First, if we define

then it is well known that satisfies the following evolution equation

 μ(m)tn+1=ΨGtn(μ(m)tn)Mtn,tn+1. (4)

Further details on the derivation of these evolution equations can be found in [10, 21, 16]. The particle interpretation of this model depends on the interpretation of the Boltzmann-Gibbs transformation in terms of a Markov transport equation

 ΨGtn(μ)=μStn,μ (5)

for some Markov transitions , that depend on the time parameter and on the measure . The choice of these Markov operators is not unique; we refer to [10] for a more thorough discussion of these models. In this article, we consider an abstract general model, and illustrate our study with the following three classes of models.

• Case 1 : We have , for some non negative and bounded function . In this situation, (5) is satisfied by the Markov transition

 Stn,μ(x,dy):=e−Utn(x)/m δx(dy)+(1−e−Utn(x)/m) Ψe−Utn/m(μ)(dy).
• Case 2 : The function is non negative. In this situation, (5) is satisfied by the Markov transition

• Case 3: The Markov transport equation (5) is satisfied by the Markov transition

 Stn,μ(x,dy):=(1−atn,μ(x)) δx(dy)+atn,μ(x) Ψ(eVtn/m−eVtn(x)/m)+(μ)(dy)

with the rejection rate

In these three cases we have the following first order expansion

 Stn,μ=Id+1m ˆLtn,μ+ 1m2 ˆRtn,μ (6)

with some jump type generator and some integral operator s.t. , where the supremum is taken over all and . The jump generators corresponding to the three cases presented above are described respectively in (12), (13), and (14). The proofs of these expansions is rather elementary, and they are housed in the appendix, on page 7.3.

In addition, whenever (5) is satisfied, we have the evolution equation

 μ(m)tn+1=μ(m)tnKn+1,μ(m)tn{\rm with the Markov kernels}Ktn,tn+1,μ=Stn,μMtn,tn+1. (7)

The mean field -particle model associated with the evolution equation (7) is a Markov process in with elementary transitions given by

 P(ξtn+1∈dx | ξtn)=∏1≤i≤NKtn,tn+1,μNtn(ξitn,dxi){\rm with}μNtn=1N∑1≤i≤Nδξitn, (8)

where stands for an infinitesimal neighborhood of the point .

## 3 Statement of the main results

Our first main result relates the Feynman-Kac models (1) and their -approximation measures (2) in case (D) and (C).

###### Theorem 3.1

In case (D), we have

 ν(m)n=νn=γnandμ(m)n=μn=ηn

with the Feynman-Kac measures and defined in (3).

In case (C), we have the first order decomposition

 Λ(m)tn=Λtn+1m rm,tnandQ(m)tn=Qtn+1m ¯¯¯rm,tn

with some remainder signed measures s.t. .

The proof of the theorem is rather technical and it is postponed to the appendix.

The first assertion of theorem 3.1 allows us to turn a discrete time Feynman-Kac model (3) into a continuous time model (1). To be more precise, we have that with the Feynman-Kac semigroup

 Q(m)tp,tn(f)(x):=E(f(Xtn) ∏p≤q

in case (D), for integer times , with , we also have that with the Feynman-Kac semigroup

 Qk,n(f)(x):=E(f(Xn) ∏k≤l

Thus, the normalized Markov kernels also coincide with the Markov kernels . In addition, for any and , we also have the semigroup formulae

 Q(m)k,n(f)(x)=Gk(x)r/m Q(m)k+r/m,n(f)(x){\rm and% }P(m)k,n=P(m)k+r/m,n=Pk,n. (9)

We prove the l.h.s. assertion using the fact that for any and any , with and , we have and

 Q(m)k,n(f)(x) = Gk(x)r/m×E⎛⎝f(Xtnm) ∏k+r/m≤tq

For a Feynman-Kac measure (1) associated with a continuous diffusion style process , it is important to observe that the l.h.s. measure in the -approximation model (2), as defined on a time mesh sequence, can be thought of as a time discretization of the exponential path integrals in the continuous time model (1). Nevertheless, the elementary Markov transitions of the Markov chain are generally unknown. To get some feasible Monte Carlo approximation scheme, we need a dedicated technique to sample the transitions of this chain. One natural strategy is to replace in (2), the reference Markov chain by the Markov chain associated with some Euler type discretization model with time step . The stochastic analysis of these models is discussed in some detail in the articles [17, 18, 15], including first order expansions in terms of the size of the time mesh sequence.

Our second main result is the following non asymptotic bias and variance theorem for the -approximation mean field model introduced in (8).

###### Theorem 3.2

We assume that the Markov transport equation (5) is satisfied for Markov transitions also satisfying the first order decomposition (6).

In case (C), for any function , and any we have the non asymptotic bias and variance estimates

 ∣∣E(μNtn(f))−μtn(f)∣∣≤ctn(f) [1N+1m]

and

 E([μNtn(f)−μtn(f)]2)≤ctn(f) [1N+1m2]

for some finite constant that only depends on and on .

In case (D), for any s.t. , and for any we have the non asymptotic bias and variance estimates

 N ∣∣E(μNn(f))−ηn(f)∣∣≤a(n)andN E([μNtn(f)−μtn(f)]2)≤a(n) (1+1N a(n))

for some some constant

 a(n)≤c ∑0≤k

Under appropriate regularity conditions on the Feynman-Kac model, we can prove that the constant is uniformly bounded w.r.t. the time parameter; that is we have that . For a detailed discussion of these uniform convergence properties w.r.t. the time parameter, we refer the reader to the book [10], and the more recent article [16]. To be more precise, we let be the Feynman-Kac semigroup associated with the flow of measures . In this notation, by proposition 2.3 in [23] we have that the Dobrushin contraction coefficient of the Markov kernel is given by

 β(Pk,n)=supμ1,μ2∈P(E)∥Φk,n(μ1)−Φk,n(μ2)∥tv.

On the other hand, we also have that

 Qk,n(1)(x)=∏k≤l

Using the fact that , we find that

 logQk,n(1)(x)Qk,n(1)(y)=∑k≤l

Assuming that for any and , and

 c1≤Gl(x)≤c2{\rm and}supμ1,μ2∈P(E)∥Φk,l(μ1)−Φk,l(μ2)∥tv≤c3 e−c4(k−l) (10)

for some positive and bounded constants , , we find that

 β(Pk,n)≤c3 e−c4(k−n){\rm and}loggk,n≤2(c2c3/c1)(∑k≤l

This clearly implies that .

For instance, it was proven in [21, 14] that condition (10) is met for time homogeneous models as soon as the Markov transition of the Markov chain satisfies the following mixing condition

 ∃m≥1, ∃ρ>0 : ∀x,y∈EMm(x,% \LARGE.)≥ρ Mm(y,\LARGE.).

It is well known that this condition is satisfied for any aperiodic and irreducible Markov chains on finite state spaces, as well as for bi-Laplace exponential transitions associated with a bounded drift function, and for Gaussian transitions with a mean drift function that is constant outside some compact domain.

The remainder of the article is organized as follows:

Section 4 is concerned with continuous time particle interpretations of the Feynman-Kac models (1). By the representation theorem 3.1, these schemes also provide a continuous time particle interpretation of the discrete time models (3) without further work. In section 4.2, we present the McKean interpretation of the Feynman-Kac models in terms of a time inhomogeneous Markov process whose generator depends on the distribution of the random states. The choice of these McKean models is not unique. We discuss the three interpretation models corresponding to the three selection type transitions presented on page 2.2. The mean field particle interpretation of these McKean models are discussed in section 4.3.

Of course, even for discrete time models (3) these continuous time particle interpretations are based on continuous time interacting jump models and they cannot be used in practice without an additional level of approximation. In this context, when using an Euler type approximation these exponential interacting jumps are replaced by geometric type recycling clocks. These interacting geometric jumps particle models are discussed in section 5, which is dedicated to the discrete time particle interpretations of the Feynman-Kac models presented in (2). In section 5.1, we discuss the McKean interpretation of the Feynman-Kac models in terms of a time inhomogeneous Markov chain model whose elementary transitions depends on the distribution of the random states. Again, the choice of these McKean models is not unique. We discuss the three interpretation models corresponding to the three cases presented on page 2.2. The mean field particle interpretation of these McKean models are discussed on page 29.

Once again, using the representation formulae (2) we emphasize that these schemes also provide a discrete generation particle interpretation of the discrete time models (3). In contrast to standard discrete generation particle models associated with (3), these particle schemes are defined on a refined time mesh sequence between integers. This time mesh sequence can be interpreted as a time dilation. Between two integers, the particle evolution undergoes an additional series of intermediate time evolution steps. In each of these time steps, a dedicated Bernoulli acceptance-rejection trial coupled with a recycling scheme is performed. As the time step decreases to , the resulting geometric interacting jump processes converge to the exponential interacting jump processes associated with the continuous time particle model. The final section, section 6, is mainly concerned with the proof of theorem 3.2.

## 4 Continuous time models

### 4.1 Feynman-Kac semigroups

In case (C) the semigroup of the flow of non negative measures is given for any by the following formulae , with the Feynman-Kac semigroup defined for any by

 Qs,t(f)(x)=E(f(Xt) exp{∫tsVs(Xs) ds} | Xs=x).

This yields , with the nonlinear transformation on the set of probability measures defined for any by

 Φs,t(μs)(f):=μs(Qs,t(f))/μs(Qs,t(1)).

Using some stochastic calculus manipulations, we readily prove that satisfies the following integro-differential equation

 ddtμt(f)=μt(Lt(f))+μt(Vtf)−μt(Vt)μt(f) (11)

for any function . Further details on the derivation of these evolution equations can be found in the articles [24, 22]. The particle interpretation of this model depends on the interpretation of the correlation term in the r.h.s. of (11) in terms of a jump type generator. The choice of these generators is not unique. Next, we discuss three important classes of models. These three situations are the continuous time versions of the three cases discussed on page 2.2.

• Case 1 : We assume that , for some non negative function . In this situation, we have the formula

 μt(Vtf)−μt(Vt)μt(f)=μt(Ut[μt(f)−f])=μt(ˆLt,μt(f))

with the interacting jump generator

 ˆLt,μt(f)(x)=Ut(x) ∫ [f(y)−f(x)] μt(dy). (12)
• Case 2 : When is a positive function, then we have the formula

 μt(Vtf)−μt(Vt)μt(f)=μt(ˆLt,μt(f))

with the interacting jump generator

 ˆLt,μt(f)(x)=∫ [f(y)−f(x)] Vt(y) μt(dy)=μt(Vt) ∫ [f(y)−f(x)] ΨVt(μt)(dy). (13)
• Case 3 : For any bounded potential function we have

 μt(Vtf)−μt(Vt)μt(f)=∫(f(y)−f(x)) (Vt(y)−Vt(x))+ μt(dx) μt(dy)=μt(ˆLt,μt(f))

with , and with the interacting jump generator

 ˆLt,μt(f)(x)=∫ [f(y)−f(x)] (Vt(y)−Vt(x))+ μt(dy). (14)

### 4.2 McKean interpretation models

In the three cases discussed above, for any test functions we have the evolution equation

 ddtμt(f)=μt(Lt,μt(f)){\rm with}Lt,μt:=Lt+ˆLt,μt. (15)

These integro-differential equations can be interpreted as the evolution of the laws, given by , of a time inhomogeneous Markov process with infinitesimal generators that depend on the distribution of the random states at the previous time increment. This probabilistic model is called the McKean interpretation of the evolution equation (15) in terms of a time inhomogeneous Markov process. In this framework, using Ito’s formula for any test function , we have

 dft(¯¯¯¯Xt)=(∂∂t+Lt,μt)(ft)(¯¯¯¯Xt)+d¯¯¯¯¯¯Mt(f) (16)

with a martingale term with predictable angle bracket

 d⟨¯¯¯¯¯¯M(f)⟩t=ΓLt,μt(ft,ft)(¯¯¯¯Xt)dt.

Using the r.h.s. description of in (15), for any we notice that

 ΓLt,μt(f,f)=ΓLt(f,f)+ΓˆLt,μt(f,f).

Next, we provide a description of this Markov process in the three cases discussed above.

• Case 1: In this situation, between the jump times the process evolves as the process . The rate of the jumps is given by the function . In other words, the jump times are given by the following recursive formulae

 Tn+1=inf{t≥Tn : ∫tTnUs(¯¯¯¯Xs) ds≥en}

where , and stands for a sequence of i.i.d. exponential random variables with unit parameter. At the jump time the process jumps to new site randomly chosen with the distribution .

For any we also have that

 ΓˆLt,μt(f,f)(x)=ˆLt,μt((f−f(x))2)(x)=Ut(x) ∫ [f(y)−f(x)]2 μt(dy).

In this situation, an explicit expression of the time inhomogeneous semigroup , , of the process is provided by the following formula

 ¯¯¯¯Ps,t,μs(f)(x) = E(f(¯¯¯¯¯Xt) ∣∣ ¯¯¯¯¯Xs=x) = Qs,t(1)(x) Φs,t(δx)(f)+(1−Qs,t(1)(x)) Φs,t(μs)(f) = Qs,t(f)(x)+(1−Qs,t(1)(x)) Φs,t(μs)(f).

We let and be the Markov transition and the transformation of probability measures defined as and replacing by the integral operator

 Q(m)tn,tn+1(f)(x) = e−Utn(x)/m E(f(Xtn+1) | Xtn=x).

Under the assumptions of theorem 3.1, using elementary calculations we prove that

 ¯¯¯¯P(m)tn,tn+1,μtn=¯¯¯¯Ptn,tn+1,μtn+1m R(m)tn,tn+1,μtn (17)

with some remainder signed measures such that , for some finite constant whose values only depend on the potential function .

• Case 2: In this situation, between jump times the process evolves as the process . The rate of the jumps is given by the parameter . In other words, the jump times are given by the following recursive formulae

 Tn+1=inf{t≥Tn : ∫tTnμs(Vs) ds≥en}

where , and stands for a sequence of i.i.d. exponential random variables with unit parameter. At the jump time the process jumps to new site randomly chosen with the distribution .

For any we also have that

 ΓˆLt,μt(f,f)(x)=ˆLt,μt((f−f(x))2)(x)=∫ [f(y)−f(x)]2 Vt(y) μt(dy).
• Case 3: In this case, between jump times the process evolves as the process . The rate of the jumps is given by the function

 Wt,μt(x) := μt((Vt−Vt(x))+) = μt(Vt 1Vt≥Vt(x))−μt(Vt≥Vt(x)) Vt(x).

In other words, the jump times are given by the following recursive formulae

 Tn+1=inf{t≥Tn : ∫tTnWt,μt(¯¯¯¯Xs) ds≥en}

where , and stands for a sequence of i.i.d. exponential random variables with unit parameter. At the jump time the process jumps to new site randomly chosen with the distribution .

For any we also have that

 ΓˆLt,μt(f,f)(x)=ˆLt,μt((f−f(x))2)(x)=∫ [f(y)−f(x)]2 (Vt(y)−Vt(x))+ μ

format: plain


text:
# Gforce

Model ElementGforce defines a general force and torque acting between two markers.

Gforce

## Description

The force and torque vectors are defined by their three components with respect to a third marker. The components may be defined using MotionSolve expressions or a user-defined subroutine. They may be a function of any system state and time.

## Attribute Summary

Name Property Modifiable by command? Designable?
id Int ()
label Str () Yes
i Reference (Marker) Yes Yes
jfloat Reference (Marker) Yes Yes
rm Reference (Marker) Yes Yes
fx Function () Yes Yes
fy Function () Yes Yes
fz Function () Yes Yes
tx Function () Yes Yes
ty Function () Yes Yes
tz Function () Yes Yes
function Function ("GFOSUB") Yes
routine Routine ()
active Bool () Yes

## Usage

Gforce is available in three flavors.
1. Force and Torque defined in a MotionSolve expression.
Gforce (i=objMarker, j=objMarker, fx=expressionString, fy=expressionString, fz=expressionString,
tx=expressionString, ty=expressionString, tz=expressionString, optional_attributes)
2. Force and Torque defined in a compiled DLL.
Gforce (i=objMarker, j=objMarker, function=userString, routine=string, optional_attributes)
3. Force and Torque Python/Matlab script.
Gforce (i=objMarker, j=objMarker, function=userString, routine=functionPointer, optional_attributes)

## Attributes

Force and Torque defined in a MotionSolve expression.
i
Reference to an existing Marker object.
Specifies the marker at which the force and torque are applied. This is designated as the point of application of the force.
The i attribute is mandatory.
j
Reference to an existing floating Marker object.
Specifies the marker at which the reaction force and torque are applied. This is designated as the point of reaction of the force.
The j attribute is mandatory.
fx
String defining a valid MotionSolve expression.
Specifies the MotionSolve expression that defines the force acting along the x-axis of the reference coordinate system (see rm below). Any valid run-time MotionSolve expression can be provided as input.
The fx attribute is mandatory.
fy
String defining a valid MotionSolve expression.
Specifies the MotionSolve expression that defines the force acting along the y-axis of the reference coordinate system (see rm below). Any valid run-time MotionSolve expression can be provided as input.
The fy attribute is mandatory.
fz
String defining a valid MotionSolve expression.
Specifies the MotionSolve expression that defines the force acting along the z-axis of the reference coordinate system (see rm below). Any valid run-time MotionSolve expression can be provided as input.
The fz attribute is mandatory.
tx
String defining a valid MotionSolve expression.
Specifies the MotionSolve expression that defines the torque acting about the x-axis of the reference coordinate system (see rm below). Any valid run-time MotionSolve expression can be provided as input.
The tx attribute is mandatory.
ty
String defining a valid MotionSolve expression.
Specifies the MotionSolve expression that defines the torque acting about the y-axis of the reference coordinate system (see rm below). Any valid run-time MotionSolve expression can be provided as input.
The ty attribute is mandatory.
tz
String defining a valid MotionSolve expression.
Specifies the MotionSolve expression that defines the torque acting about the z-axis of the reference coordinate system (see rm below). Any valid run-time MotionSolve expression can be provided as input.
The tz attribute is mandatory.
Force and Torque defined in a compiled DLL
i
Reference to an existing Marker object.
Specifies the marker at which the force/torque is applied. This is designated as the point of application of the force.
The i attribute is mandatory.
j
Reference to an existing floating Marker object.
Specifies the marker at which the reaction force/torque is applied. This is designated as the point of reaction of the force.
The j attribute is mandatory.
function
String defining a valid user function MotionSolve expression.
The list of parameters that are passed from the data file to the user-defined subroutine where the Gforce is defined.
The function attribute is mandatory.
routine
String
Specifies an alternative name for the user subroutine. The name consists of two pieces of information, separated by "::". The first is the pathname to the shared library containing the function that computes the response of the user-defined Vforce. The second is the name of the function in the shared library that does the computation.
An example is: routine="/staff/Altair/engine.dll::myGforce"
• "/staff/Altair/engine.dll is the dll
• "myGforce" is the function within this DLL that performs the calculations
The attribute routine is optional.
When not specified, routine defaults to GFOSUB.
Force and Torque defined in a Python function
i
Reference to an existing Marker object.
Specifies the marker at which the torque is applied. This is designated as the point of application of the force.
The i attribute is mandatory.
j
Reference to an existing floating Marker object.
Specifies the marker at which the reaction torque is applied. This is designated as the point of reaction of the force.
The j attribute is mandatory.
function
String defining a valid user function MotionSolve expression.
The list of parameters that are passed from the data file to the user-defined subroutine where the Vtorque is defined.
The function attribute is mandatory.
routine
Pointer to a callable function in Python.
An example is: routine=myGforce
• myGforce is a Python function or method that can be called from wherever the model resides.
The attribute routine is optional.
When not specified, routine defaults to GFOSUB.
Optional attributes - Available to all variants
id
Integer
Specifies the element identification number. This number must be unique among all the Gforce objects in the model.
This attribute is optional. MotionSolve will automatically create an ID when one is not specified.
Range of values: id > 0
label
String
Specifies the name of the Gforce object.
This attribute is optional. When not specified, MotionSolve will create a label for you.
rm
Reference to an existing Marker object.
Specifies the marker in whose coordinate system the torque components are computed. rm can be on any body, including Ground
The rm attribute is optional.
When not specified, rm defaults to the global coordinate system.
active
Bool
Select one from True or False.
• True indicates that the element is active in the model and it affects the behavior of the system
• False indicates that the element is inactive in the model and it does not affect the behavior of the system. It is almost as if the entity was removed from the model, of course with the exception that can be turned "ON" when desirable.
The attribute active is optional. When not specified, active defaults to True.

## Example

1. Gforce defined via expressions.
gfo1 = gforce (label="nonlinear bushing", i=m1801, jfloat=m1901, rm=m1903,
fx="-1e3*DX(1801,1903,1903)",
fy="-1e3*DY(1801,1903,1903)",
fz="-1e3*DZ(1801,1903,1903)",
tx="-5e3*AX(1801,1903) - 2*WX(1801,1903,1903)**,
ty="-5e3*AY(1801,1903) - 2*WY(1801,1903,1903)**3"
tz="-5e3*AZ(1801,1903) - 2*WZ(1801,1903,1903)**3" )
2. Gforce defined in a Python function.
# Define the user subroutine first
def myGfosub (id, time, par, npar, dflag, iflag):
i = par[0]
j = par[1]
k = par[2]
kt= par[3]
ct= par[4]

# Get the state of the bushing
dx = DX(i,j,j)
dy = DY(i,j,j)
dz = DZ(I,j,j)

ax = AX(i,j)
ay = AY(i,j)
az = AZ(i,j)

wx = WX(i,j,j)
wy = WY(i,j,j)
wz = WZ(i,j,j)

# Compute force & torque
fx = -k*dx
fy = -k*dy
fz = -k*dz
tx = -kt*ax - ct*(wx**3)
ty = -kt*ay - ct*(wy**3)
tz = -kt*az - ct*(wz**3)

return [fx, fy, fz, tx, ty, tz]

# Define a bushing with nonlinear damping
gfo2 = Gforce (label="nonlinear bushing", i=m1801, jfloat=m1901, rm=m1903,
function="user(1801,1903,1000.0,5000.0, 2.0)", routine=myGfosub )


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# A contest will consist of n questions, each of which is to

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A contest will consist of n questions, each of which is to [#permalink]

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A contest will consist of n questions, each of which is to be answered either "true" or " false". anyone who answers all n questions correctly will be a winner. what is the least value of n for which the probability is less than 1/1000 that a person who randomly guesses the answer to each question will be a winner?

A. 5
B. 10
C. 50
D. 100
E. 1000

OPEN DISCUSSION OF THIS QUESTION IS HERE: a-contest-will-consist-of-n-questions-each-of-which-is-to-38710.html
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Re: A contest will consist of n questions, each of which is to [#permalink]

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27 Dec 2007, 10:37
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Each question has 2 ways to guess so the probability of guessing correctly is 1/2. The probability of guessing each question correctly for n questions is 1/2^n. What is the smallest number for n that gets the denominator over 1000?

1/2^n = probability

1/2^5 = 1/32
1/2^10 = 1/1024

and you don't need to go any further. Someone guessing on 10 questions has a 1/1024 chance of getting them all correct.

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Re: A contest will consist of n questions, each of which is to [#permalink]

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27 Sep 2009, 11:29
A contest will consist of n questions, each of which is to be answered either "true" or " false". anyone who answers all n questions correctly will be a winner. what is the least value of n for which the probability is less than 1/1000 that a person who randomly guesses the answer to each question will be a winner?

A.5
B.10
C.50
D.100
E.1000

Soln: ans is B

Probability that one question is answered right is 1/2.

Now for minimum number of questions needed to take probability less than 1/1000 is
=> (1/2)^n < 1/1000
n = 10 satisfies this.
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02 May 2011, 09:41
1/2^n < 1/1000

n= 10 is the least value of number of questions. 1/2 is for equal probability of true and false.
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Re: A contest will consist of n questions, each of which is to [#permalink]

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A contest will consist of n questions, each of which is to be answered either "True" or "False." Anyone who answers all n questions correctly will be a winner. What is the least value of n for which the probability is less than 1/1000 that a person who randomly guesses the answer to each question will be winner?

A. 5
B. 10
C. 50
D. 100
E. 1000

The probability to randomly guess the answer for 1 question is 1/2, for 2 questions 1/2*1/2=1/2^2, similarly for n questions the probability is 1/2^n.

We need to find the least value of $$n$$ for which $$\frac{1}{2^n}<\frac{1}{1,000}$$ --> $$2^n>1,000$$ --> $$n_{min}=10$$.

OPEN DISCUSSION OF THIS QUESTION IS HERE: a-contest-will-consist-of-n-questions-each-of-which-is-to-38710.html
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Re: A contest will consist of n questions, each of which is to   [#permalink] 10 Dec 2014, 03:45
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