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# Should we clarify that game-specific tags are for mods?

We have several game-specific tags on the site, such as , and because modding questions for games are considered on-topic here.

We also have , which serves as identifying questions about modding that are not game-specific or for games too unpopular to get game-specific tags on their own.

This means that some of these questions are "double-tagged" with and the game-specific tag, while others are tagged with one or the other.

I propose that we rename game-specific tags to have a "-modding" suffix, so we'd have skyrim-modding and the like. We should rename mods to modding for uniformity, but otherwise leave it alone because it serves a useful purpose.

When we find questions that are double-tagged, we should remove the less-relevant tag.

• Please remember that comments aren't for extended discussion, especially of tangential issues to the post. One can make another meta thread or use the Game Development Chat if necessary.
– user1430
Dec 26, 2018 at 20:26


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# American Institute of Mathematical Sciences

October  2019, 12(6): 1791-1806. doi: 10.3934/dcdss.2019118

## Multi-point Taylor series to solve differential equations

 1 Laboratoire de Mathématiques-Informatique, Université Nangui Abrogoua, Unité de Formation et de Recherche en Sciences Fondamentales et Appliquées, 02 B.P. V 102 Abidjan, Côte d'Ivoire 2 Université de Lorraine, CNRS, Arts et Métiers ParisTech, LEM3, F-57000 Metz, France 3 EDF R & D Saclay, 7 boulevard Gaspard Monge 91120 Palaiseau, France

* Corresponding author: Zézé

Received  November 2017 Revised  February 2018 Published  November 2018

The use of Taylor series is an effective numerical method to solve ordinary differential equations but this fails when the sought function is not analytic or when it has singularities close to the domain. These drawbacks can be partially removed by considering multi-point Taylor series, but up to now there are only few applications of the latter method in the literature and not for problems with very localized solutions. In this respect, a new numerical procedure is presented that works for an arbitrary cloud of expansion points and it is assessed from several numerical experiments.

Citation: Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118
##### References:

show all references

##### References:
Discretization points in the case of three subdomains and two point-Taylor series. The expansion points are $x_i$ and the additional collocation points are $y_j$ (2 per subdomain)
Example 1: $f(x) = g(x) = 1, L = 10$. Comparison of a one-point Taylor series, one-point Taylor series in two subdomains and two-point Taylor series
Example 1, 2-point Taylor series (n = 2). Convergence with the degree p
Example 1, degree p = 4. Convergence with the number expansion points
Example 1, 2-point Taylor series, c $\pm \frac{10}{3}$, p = 6. Distribution of the residual and of the error in the interval
Example 1, 2-point Taylor series, p = 6. Maximal value of the residual and of the error according to the location of the expansion points
Example 2, 4-point Taylor series, expansion points located in ($\pm 0.8; \pm 1.8$). Convergence with the degree
Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 5$, $(L = 10, a = 1)$, $u_{max} = 4.718$
Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 5$, $(L = 10, a = 0.1)$, $u_{max} = 5.0788$
Example 3, Solution with 3 subdomains, Taylor 4-point and $p = 4$, $(L = 10, a = 0.1)$, $u_{max} = 5.109$
Example 3, residual with 3 subdomains, $p = 5$, $n = 4$. Case of a smooth distribution of points
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A search for the production of events containing three W bosons predicted by the standard model is reported. The search is based on a data sample of proton-proton collisions at a center-of-mass energy of 13 TeV recorded by the CMS experiment at the CERN LHC and corresponding to a total integrated luminosity of 35.9 fb$^{-1}$. The search is performed in final states with three leptons (electrons or muons), or with two same-charge leptons plus two jets. The observed (expected) significance of the signal for W$^\pm$W$^\pm$W$^\mp$ production is 0.60 (1.78) standard deviations, and the ratio of the measured signal yield to that expected from the standard model is 0.34 $^{+0.62}_{-0.34}$. Limits are placed on three anomalous quartic gauge couplings and on the production of massive axionlike particles.


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# Convicted but Innocent: Wrongful Conviction and Public Policy

Books

### C. Ronald Huff, Arye Rattner & Edward Sagarin

• Chapters
• Front Matter
• Back Matter
• Subject Index

## Dedication

This book is dedicated to the memory of our friend and colleague, Professor Edward Sagarin, whose ideas and encouragement contributed greatly to this book and to our lives; and to the convicted innocents, whose cause he championed.

C.R.H.
A.R

## Foreword

Of the many troubled social institutions in American life—including public education, welfare, health, and the family—none is in greater difficulty than the criminal justice system. Many billions of federal dollars have been spent on criminal justice agencies and programs since the introduction of the Law Enforcement Assistance Administration in the late 1960s in the “war on crime” stemming from persistent public pressure for greater personal safety. Astronomic sums have been allocated to upgrade law enforcement equipment; to go on-line in this computer age so that law enforcement agencies can more efficiently collect, record, maintain, and share information; to improve forensic science; to recruit and train additional qualified personnel (more than 100,000 new police officers alone in the latest crime bill, 1994–1995); to reduce racial, gender, and ethnic biases in the system; to speed up the criminal justice process; to rewrite the criminal codes and impose draconian sentences on those convicted of drug offenses; to revive the death penalty; to move toward definite (fixed) sentences, of longer duration; and to upgrade the efficiency and effectiveness of the various criminal justice subsystems, including the police, prosecution, courts, and corrections.

Much, much more needs to be done in nearly every facet of the criminal justice enterprise to modernize policing, prosecution, the judicial arm, and the corrections subsystems, including probation, parole, and adult and juvenile prisons and training schools, and to improve efficiency. The corrections system is currently so large that the adult prison system employs more people than any other agency in state government. Significantly, the cost of supporting the swollen criminal justice enterprise is now about half of the total expenditures of large metropolitan communities, especially in highly urban counties. In every legislative session, more criminal justice bills are introduced, and usually left to die without consideration, than bills in any other substantive area. The vastness of the investment in money, personnel, and physical facilities is neither known nor generally understood by the general public, who continue to rank “the crime problem” among their top three concerns—usually as number one.

The criminal justice system has launched massive and costly efforts, especially in the never-ending “war on drugs,” to prevent and control crime; to incapacitate repeat and dangerous offenders; to neutralize urban street gangs in large and small cities and in the prisons; to reduce the incidence of DUI (driving under the influence of alcohol or drugs); to do something about sex offenders, white-collar criminals, and spouse abusers. Efforts have also been made to rehabilitate offenders through K-12 and sometimes postsecondary education programs in prison, vocational training, work, social skills training, counseling, and all the rest. Still, the record remains bleak and the public dubious about the outcome of these efforts and, more generally, of the “war on crime.” It is, of course, true that official Uniform Crime Reports data show that crime peaked in the 1970s, held steady during the first half of the 1980s, and has declined in the past 10 years. The number of all reported “index crimes”—murder and nonnegligent manslaughter, rape, robbery, aggravated assault, burglary, larceny-theft, auto theft, and arson—fell sharply in 1994 and even more sharply in the first half of 1995. Corroboration of this welcome trend is to be found in victimization studies conducted by the Census Bureau and by academic researchers. For some years now, households and business establishments have reported significant declines in the most common personal and property crimes.

At the same time, incarceration rates (adult and juvenile prison and jail populations) have been increasing, with no sign yet that they have begun to level off. Nationally, there are now more than 1 million prison inmates—more than 100,000 in California alone, and Texas and Florida will approach this number before long. In Ohio, as in many other jurisdictions, the number of prisoners has quadrupled since 1970 and doubled since 1980. Where there were once 7 Ohio adult institutions holding 7,700 inmates (1973), at midyear in 1995 there were more than 43,000 convicts in 28 prisons. California's prison system used to take 4% of that state's budget; it now gets 8%—the additional 4% mostly from funds that could have been spent on education. In Ohio, as elsewhere, the increase in the penal population has occurred primarily because of the “war on drugs”; the legislative upgrade in sentences jumped the average time served in confinement in Ohio from 2.2 years to well over 4 years for inmates paroled in the 1980s and remained constant at 1.1 years for those released on definite sentences. Stricter enforcement of the conditions of parole has resulted in an unprecedented jump in parole violators returned to prisons. An assortment of other changes, including the tightening of the system leading to the incarceration of greater numbers of convicted felons and the shrinking in the proportion granted parole, when eligible, accounts for the rest of the increase. The number of lesser offenders being held in jails (awaiting trial or doing time) has also doubled. Finally, the number of juvenile offenders in training schools and private placement has grown substantially. The incapacitation movement has resulted in huge growth in the numbers of blacks and lower-class young adults under correctional supervision and control. More young black males are in penal confinement or under community supervision than are in college. It is estimated that one-fourth of young African American males are under the control of the criminal justice system.

All that having been said, the great revolution in criminal justice in the 20th century has not been in the crime problem but in the criminal justice process. As any student of this process knows, and as many in law enforcement and the prosecutorial and correctional subsystems have yet to accept, court decisions have transformed the face of the process, adding a whole new dimension to it. Never has the term due process been more rigorously implemented in all phases of the justice system—from arrest to final release from state control. Such phrases as Miranda warnings, the exclusionary rule, equity and fairness, arbitrary and capricious decisions, fair trial, and inmate rights, as well as many other concepts and themes, have been forced on the system by federal and higher state courts.

In his book The Limits of the Criminal Sanction, Herbert Packer (1968) describes two very different criminal justice models: the due process model in the United States and the crime control, inquisitorial, or social defense model in continental Europe. Packer compares the European crime control model to an assembly line in which the law enforcement arm is the chief element. The police apprehend factually guilty or probably guilty suspects and the other actors in the system—prosecutors, defense attorneys, and judges—play specialized roles in translating probable guilt into legal guilt. As Packer notes of this model, “The image that comes to mind is an assembly line which moves an endless stream of cases, never stopping, carrying them to workers who stand at fixed stations and who perform on each case as it comes by the same small but essential operation that brings it one step closer to being a finished product or… a closed file.” Further, he notes, ‘The criminal process is seen as a screening process in which each successive stage… involves a series of routinized operations whose success is gauged primarily by their tendency to pass the case along to [successful] conclusion’ (p. 21).

In sharp contrast, in the model if not always in practice, the due process model is supposed to operate “as an obstacle course.” The general premise is that the movement from probable to legal guilt must be cumbersome and difficult so as to ensure near certainty that persons found guilty are indeed guilty and deserving of punishment. In this approach, there can hardly ever be too many legal hurdles in the process of investigation, arrest, trial, plea, conviction, and disposition. In both models, protection of the public is paramount. In the due process model, an equally important goal is the protection of the rights of suspects as well as those found guilty and those already under state control. The due process model can tolerate some false negatives—guilty persons not arrested, convicted, or sentenced—but it cannot readily tolerate false positives—persons wrongly convicted. The social control model can abide false positives but not false negatives. This puts the two concepts at opposite ends of the philosophical spectrum and leads to irreconcilable conclusions as to the operation of the system, the acted on, and the actors.

This, of course, brings us to the subject of the current volume. Wrongful convictions, however infrequent, are anathema to the American due process system and to all who believe in the fairness of our law enforcement and judicial systems and the constitutional protections guaranteed individual citizens. Not only do such wrongful convictions violate trust in our system, but, as the authors painstakingly assert, such convictions undermine public safety by leaving the “true” positives—the guilty—in the community to commit future grave offenses.

As I know from personal contact with all the authors, this book has been a decade or more in the making. The volume expresses their deeply held commitment to due process and the need to protect equally society, the families and loved ones of crime victims, and the false positives from the inevitable errors that all human institutions sometimes make—unintentionally, accidentally, or through failures in the workings of the system known popularly as snafus (and by many less delicate names as well). In a moderate voice, this book examines the painful issues in wrongful conviction. The focus is geared to answering the core questions: How often does it happen? Why does it happen? How does it happen? To whom does it happen? How can it be prevented from happening? How should the wrongfully convicted be made whole again—what constitutes adequate compensation?

In this well-researched and fascinating volume, Huff, Rattner, and Sagarin mix materials from case files in the literature and those reported in numerous research reports and in the media. They present a great deal of information on research studies, both national and international, concerning the accuracy of eyewitness perceptions. Interviews with individuals who have been wrongly convicted and subsequently exonerated, as well as with some of the actors in the system are included, as are trial documents, court transcripts, and media reports on trials. There is no other book available on the “guilty” but innocent with such a broad focus and so much rich detail. It is a good read, indeed.

The authors begin by discussing some celebrated cases—some known to the general reader, but most known only to specialists. These celebrated historic cases include the Salem witchcraft cases; the Alfred Dreyfus accusation, trial, punishment, and exoneration; the recent release after conviction and imprisonment of “Ivan the Terrible” Demjanjuk, and the famous miscarriage of justice in the case of the “Scottsboro Boys.” (As an aside, when I was so young that I could not quite understand the meaning of the Scottsboro boys' trial and conviction, I remember the street demonstrations and torchlight parades in my community and the marchers shouting, “The Scottsboro boys shall not die!” Fortunately, they did not die, even though eight of the nine black youths were initially sentenced to death for the rapes of two white girls—rapes that never occurred and that were later recanted.) Some readers may also remember the Leo Frank conviction and his lynching by a mob while in custody. More recently, there have been the sagas of Randall Adams and the Jacksons. But hardly anyone is likely to know much about Isidore Zimmerman, his 24 years in prison, the commutation of his sentence 2 hours before execution, his 20 subsequent years spent seeking compensation after his prison release, and his death shortly after receiving $1 million for wrongful imprisonment. One of the major contributions of this volume is the attempt to get at some sort of reliable estimate of the incidence of wrongful conviction in the United States per annum. Based on a questionnaire submitted to experienced criminal justice actors in Ohio—police administrators, sheriffs, county prosecutors, public defenders, and criminal court bench judges—0.5% of those convicted of felonies are estimated to be innocent of the crimes charged. The authors are quick to point out that these more than 10,000 convicted innocents (index crimes alone) were not necessarily without previous criminal involvement. Indeed, for some, their prior records may have led to their arrest and conviction. Assuming the accuracy, or even near reliability, of the above estimate, there are many prisoners who have served or are serving time who might well be the subjects of a subsequent book on the innocent but “guilty.” The question, then, is why innocents, whatever their actual number, are found guilty. Why does our cherished due process obstacle course flatten into a straightaway? What happens to all the hurdles? The authors contend that although becoming legally guilty though factually innocent is the result of interaction among several factors, faulty eyewitness identification is the most critical of these variables and alone accounts for as many as half of all erroneous findings of legal guilt. Experiments have shown that race and ethnicity are also crucial in eyewitness mistakes; so are physical conditions such as lighting, angle of sight, and length of observation. These errors, in themselves, are usually insufficient in the face of diligent investigation; however, when coupled with overzealousness, biases, and ineptitude on the part of some police officers, lawyers, and prosecutors, and pressure for convictions from the community and elected officials, erroneous eyewitness identifications are more than enough to convict. Furthermore, in our criminal justice system, where the same actors may participate in many arrests, prosecutions, and trials over a number of years, there tend to develop interactional bonds that are hard to overcome. Thus, as the erroneous conviction case makes its way up the line, there is considerable reluctance to override the “evidence” and judgments made at the preceding levels. Instead of a hurdle, there is instead the “ratification of error.” This is no different from the working codes and internal cultures in other endeavors—medicine (as in malpractice), law, industry. Loyalty tends to override doubt and even experience. Cases are inexorably pushed forward. The authors also include a chapter on the dynamics of obtaining and using doubtful (false) confessions. Faced with the possibility of going to trial for murder, and the uncertainty of the outcome, some defendants conclude (as may their public defenders or even their personal lawyers) that plea-bargained prison stretches are the safer option. There is also the possibility that a suspect's refusal to accept a plea bargain will influence the severity of his or her sentence—if convicted. In the last chapter, the authors suggest public and professional consideration of 14 policy recommendations growing out of this decade of research. I will mention only the last one here, which pertains to irreversible errors. Huff, Rattner, and Sagarin suggest the substitution of life sentences, without the possibility of parole, in all death penalty cases. Some of their other recommendations are no less interesting and controversial. All are worth your consideration and the consideration of lawmakers on the judiciary committees in legislatures around the country. ## Acknowledgments We have received a great deal of valuable assistance and encouragement throughout the course of our research and during the preparation of this manuscript.1 As for the initial idea to conduct a study of wrongful convictions—both the cases and the processes—we are indebted to our friend and colleague the late Professor Edward Sagarin, whom we have also acknowledged in our dedication. Ed contributed more than ideas; he wrote some of the initial material for this book (which we have incorporated almost entirely verbatim), and his enthusiasm and encouragement were always there for us. Since his death, his widow, Gert, has maintained a genuine interest in this book, one of Ed's last writing projects. We are also indebted to the many judges, prosecutors, defense attorneys, and law enforcement officers who have been willing to complete our surveys, and who also (in many cases) have informally shared their experiences and their insights with us over the past decade, both in the United States and in Israel. This book documents the fact that wrongful convictions often result, at least in part, from incompetent, unprofessional, and/or unethical conduct by law enforcement officers, prosecutors, defense attorneys, and even judges. However, we firmly believe that the great majority of those working in the criminal justice system are genuinely concerned about such errors and their implications for justice and for public safety. For their assistance and collaboration in analyzing cases of wrongful conviction and compensation in the state of Ohio, we would like to thank Professor Anand Desai of the School of Public Policy and Management at Ohio State; Howard Ishiyama, a doctoral candidate at that institution; and Jack Jones and Ted Stanich, former graduate research associates. Our appreciation also goes to Martin Yant, a courageous journalist and author who has published many newspaper articles and a book on wrongful conviction, for his assistance in updating and summarizing some of the cases discussed in this book. Marty exemplifies the kind of independent inquiry and dedication to the pursuit of truth that is essential to a free, democratic society. Two distinguished scholars—professors Rita Simon of the American University and Elizabeth Loftus of the University of Washington—reviewed portions of the manuscript prior to its publication and made many useful and insightful suggestions for improvement, which we have attempted to incorporate. We want to recognize their important contributions while at the same time acknowledging that we are solely responsible for the content of the book and the degree to which it meets, or fails to meet, its purpose: describing and analyzing the problem of wrongful conviction and its public policy implications. Ms. Georgia Meyer, administrative secretary at Ohio State's Criminal Justice Research Center, helped immensely in coordinating the production of the final manuscript. She also remained characteristically cheerful in spite of deadline pressures. We owe a special debt of gratitude to some of the convicted innocents and their families for sharing their experiences—injustice, trauma, and, ultimately, vindication. Special recognition goes to Randall Dale Adams, Bradley Cox, Lenell Geter, Todd Neely, and Todd's parents, Edith and Lewis Crosley, for their openness and their willingness to let us try to learn what the experience of wrongful conviction means on a personal and familial level. The insights they provided helped shape much of what is in this book, which is, after all, their story. Finally, we express our love and appreciation to our own families for their patience with us over the past decade as we pursued our analysis of this important problem. We can never compensate them for the time we stole; we can only hope that this book will make a small contribution to the continuing pursuit of social and criminal justice throughout the world. Note 1. We have contributed equally to the research and writing of this book, and our names are therefore displayed alphabetically. ## Introduction The concept of justice is timeless, as is the corresponding concern about convicting an innocent person. Indeed, as Daniel Webster once proclaimed, “Justice, sir, is the greatest interest of man on earth.” This book addresses a specific kind of injustice—wrongful conviction. The topic, in addition to being timeless, is also timely. As we write this, the case of John Demjanjuk, accused of being “Ivan the Terrible,” a Nazi death camp guard who helped in the mass murder of Jews, is being played out on the world's stage. Sharing that international stage within the recent past have been two cases on different continents. First is the wrongful conviction and imprisonment (for 16 years) of the “Birmingham Six,” alleged members of the Irish Republican Army, for the bombing of two British pubs. This case (which inspired the film In the Name of the Father) led to a high-level review of the entire British criminal justice system, which regularly receives scrutiny through a popular television series titled Rough Justice. Second is the tragic wrongful conviction of Lindy and Michael Chamberlain, an Australian couple whose trial and wrongful conviction in the death of their infant daughter inspired the movie A Cry in the Dark. The Chamberlains were awarded$320,000 in compensation by the Australian government.

The wrongful conviction theme is also important in American popular culture, as reflected in the runaway hit movie attraction The Fugitive. In the movie, Harrison Ford plays the role of Dr. Richard Kimble, a physician who is wrongfully convicted of murdering his wife and who claims that a one-armed intruder was the actual murderer. The movie, and the popular former television series of the same name, is loosely based on the actual wrongful conviction of Dr. Sam Sheppard, an Ohio physician imprisoned for life in 1954 for the murder of his wife.

In the pages that follow, we shall attempt to define, describe, and analyze the problem of wrongful conviction and its implications for society. Not everyone will be pleased with our definition, because we do not include a number of cases in which it is clear that injustice occurred. By our restrictive definition of wrongful conviction, even the case of Jesus would not qualify, because Jesus' behavior, in the name of righteousness, did violate the laws of his day, unfair as they were. We focus here on cases in which the convicted persons did not commit the crimes alleged—in other words, they are behaviorally innocent of the crimes.

We present for the reader's consideration survey data concerning the possible magnitude of the problem and its causes; actual cases of wrongful conviction, as well as some in which doubt remains; detailed analyses of each of the major factors associated with wrongful conviction and how these factors interact; and, finally, a discussion of the public policy implications of our research, including some recommendations for reducing the occurrence of this problem, identifying and compensating those who are wrongfully convicted, and eliminating (or greatly curtailing) the possibility of executing innocent persons.

We will demonstrate that the American system of criminal justice is so large and has so many arrests each year that even if the system were 99.3% accurate, it would still generate more than 10,000 wrongful convictions a year for the eight serious index crimes alone (these eight crimes are murder and nonnegligent manslaughter, forcible rape, aggravated assault, robbery, burglary, larceny-theft, motor vehicle theft, and arson). It is likely that the error rate is even higher for less serious crimes, making it highly probable that wrongful conviction affects a great many Americans each year, even though the error rate may be relatively small.

Finally, throughout the volume we shall attempt to dispel a common, but fallacious, assumption about this topic: that wrongful conviction is an issue that should inherently interest “liberals” more than “conservatives” (who are presumed to be more interested in “law and order” and “public safety”). We attempt to refute this assumption by noting (and vividly illustrating, in many of our cases) that every time an innocent offender is wrongfully convicted, the actual offender typically remains free to continue victimizing the public. Thus, this book concerns a problem that affects not only an individual's right to due process and a fair trial, but also a serious public safety concern. The reader will learn about murders, rapes, and other serious crimes that were permitted to occur because the wrong person was arrested for a given crime.

Finally, we present a framework for understanding this problem within the overall context of the American criminal justice system. Drawing on Packer's (1968) contrasting “crime control” and “due process” models, we illustrate how this conceptual framework can be useful for increasing our understanding of the problem of wrongful conviction and for the formulation of public policy recommendations to reduce the frequency with which it occurs. Although no system of justice can be perfect, we believe that if efforts are focused on preventable errors, the U.S. system can substantially reduce the kinds of injustices discussed in the pages that follow. It would be well worth the effort, and it is our hope that this volume may contribute to that end.

• ## References

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(1955). The Dreyfus case: A reassessment. New York: Reynal.
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, , & , (1979). Effects of interpolated mugshot exposure on accuracy of eyewitness identification. Journal of Applied Psychology, 64, 232–237. http://dx.doi.org/10.1037/0021-9010.64.2.232
(1983). The best defense. New York: Random House.
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(1962, March). For whom the chair waits. Federal Probation, pp. 14–25.
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(1980, December 15). Trials—and errors. National Law Journal, p. 1.
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(1987). Wrongful convictions: Societal tolerance of injustice. In & (Eds.), Research in social problems and public policy (pp. 99–115). Greenwich, CT: ML
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, , & (1993). Wrongful conviction in Ohio: Rights and roles. Unpublished manuscript.
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(1964). The innocents. New York: William Morrow. http://dx.doi.org/10.1007/s12142-008-0088-9
(1983). When justice goes wrong: Convicting the innocent. Unpublished doctoral dissertation, Ohio State University.
, , & (1987). Cross-ethnic identification and misidentification. Unpublished manuscript.
(1959). The compulsion to confess. New York: Farrar, Straus & Cudahy. http://dx.doi.org/10.1001/jama.1981.03310310057028
(1976). Compensating the innocent accused. Ohio State Law Journal, 37, 705–728.
(1984). Prisons and the process of justice: The reductionist challenge. London: Heinemann.
(1975). Deviants and deviance: An introduction to the study of disvalued people and behavior. New York: Praeger. http://dx.doi.org/10.1080/01639625.1979.9967510
(1985). Organizational culture and leadership. San Francisco: Jossey-Bass. http://dx.doi.org/10.1016/j.jwb.2012.01.011
(1940). The influence of attitudes upon the remembering of pictorial material. Archives of Psychology, 36, 258–270.
, & (1986). Meta analysis of facial identification studies. Psychological Bulletin, 700(2), 139–156.
, , & (1973). Facial stereotypes of deviants and judgment of guilt or innocence. Social Forces, 51, 427–433.
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(1982). Eyewitness identification: Legal and practical problems. New York: Clark Boardman.
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(1965). Eyewitness identification in criminal cases. Springfield, IL: Charles C Thomas.
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(1909). The observer as reporter: A survey of the psychology of testimony. Psychological Bulletin, 8, 307–309. http://dx.doi.org/10.1037/h0070121
(1977). Did your eyes deceive you? Expert psychological testimony on the unreliability of eyewitness identification. Stanford Law Review, 29, 969–1030. http://dx.doi.org/10.2307/1228141
Wrongful conviction charges haunt Dallas prosecutors. (1989, October 12). Chicago Tribune, p. 25A.
(1955). A generation of vipers. New York: Rinehart.
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(1979). The psychology of eyewitness testimony. New York: Free sPress.
(1993). The Holocaust, the French, and the Jews. New York: Basic Books.

## Cases

Argersinger v. Hamlin, 407 U.S. 25 (1972).

Bennet v. State, 530 S.W.2d 511 (Tenn. 1975).

Blevins v. State, 108 Ga. App. 738,134 S.E.2d (1963).

Escobedo v. Illinois, 378 U.S. 478 (1964).

Gideon v. Wainwright, 372 U.S. 335 (1963).

Gilbert v. California, 388 U.S. 263 (1967).

Ivan (John) Demjanjuk v. State of Israel, Cr.A. 347/88 (1990).

Jackson v. Fogg, 589 F.2d 108 (2d Cir. 1978).

Kirby v. Illinois, 406 U.S. 682 (1972).

Mapp v. Ohio, 367 U.S. 643 (1961).

Miller v. Pate, 386 U.S. 1 (1967).

Miranda v. Arizona, 384 U.S. 436 (1966).

Nell v. Biggers, 409 U.S. 188 (1972).

O'Neil v. Ohio, Ohio Court of Claims, 81-05750 (1984).

People v. Barad, 362 111. 584, 200 N.E. 858 (1936).

People v. Harris, 28 Cal. 3d 935, 623 P.2d 240 (1981).

People v. Johnson, 38 Cal. App. 3d 1,112 Cal. Rptr. 834 (1974).

People v. Kind, 357 III. 133,191 N.E. 244 (1934).

Powell v. Alabama, 287 U.S. 45 (1932).

Roe v. Wade, 410 U.S. 113 (1973).

Simmons v. United States, 390 U.S. 377 (1968).

State v. Engel. 289 N.W.2d 204 (N.D. 1980).

State v. Landeros, 20 N.J. 76,118 A.2d 524 (1955).

State of Delaware v. Bernard T. Pagano, Delaware Superior Court, Wilmington (1979).

State of Israel v. Ivan (John) Demjanjuk, Criminal Case No. 383/86 (1986).

Stovall v. Denno, 388 U.S. 293 (1967).

United States v. Ash, 413 U.S. 300 (1973).

United States v. Garsson, 291 F. 646,649 (S.D. N.Y. 1923).

United States v. Wade 388 U.S. 218 (1967).

United States v. Williams, 592 F.2d. 1277 (5th Cir. 1979).

## About the Authors

C. Ronald Huff is Director and Professor, School of Public Policy and Management, and Director, Criminal Justice Research Center, at the Ohio State University. He has also held faculty positions at the University of California (Irvine) and Purdue University and served as a visiting professor at the University of Hawaii. His publications include more than SO journal articles and book chapters and 10 books, the most recent of which, an all-new second edition of Gangs in America, will also be published by Sage in 1996. He has served as a consultant on crime and public policy to the U.S. Senate Judiciary Committee, the F.B.I. National Academy, the U.S. Department of Justice, five states, and numerous federal, state, and local organizations. His recent honors include the Donald Cressey Award from the National Council on Crime and Delinquency (1992), the Paul Tappan Award from the Western Society of Criminology (1993), and the Herbert Bloch Award from the American Society of Criminology (1994).

Arye Rattner is Professor of Sociology at the University of Haifa in Israel. He served until recently as Chair of the Department of Sociology and Anthropology at the University of Haifa and has served also as the Chair of the Israel Criminology Council. He has published several articles on wrongful conviction and eyewitness identification and has recently published a number of articles that have dealt with attitudes toward the legal system and have attempted to construct models of legal disobedience. He is currently involved in a research project examining how both Jews and Arabs are processed in the criminal justice system in Israel.

Edward Sagarin (deceased) was Professor of Sociology at City College and City University of New York and also served as Distinguished Visiting Professor of Sociology at the Ohio State University. Prior to his death in 1986, he enjoyed a highly productive career in criminology which was recognized by his peers, who elected him President of the American Society of Criminology and designated him Editor-in-Chief of Criminology: An Interdisciplinary Journal, the Society's official journal. He was regarded as an intellectual, a prolific author of books and articles in both criminology and the sociology of deviance, and an outspoken advocate of justice and compassion for those who were disvalued by society. His intellect, energy, and dedication helped inspire many younger scholars, including his coauthors on this volume.


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What's the completeness relation of Bloch band?

Bloch's theorem can be stated as: $$|\Psi_{n\vec{k}}\rangle=e^{i\vec{k}\cdot\vec{r}}|u_{n\vec{k}}(\vec{r})\rangle \tag{1}$$ where $$|\Psi_{n\vec{k}}\rangle$$ is the solution of single electron Schrodinger equation: $$(T+V)|\psi_n(\vec{k})\rangle=H|\psi_{n\vec{k}}\rangle =E_{n\vec{k}}|\psi_{n\vec{k}}\rangle \tag{2}$$ with $$V(\vec{r}+\vec{R})=V(\vec{r})$$ [$$\vec{R}$$ is the translational vector in real space], and $$u_{n\vec{k}}(\vec{r})$$ in Eq.(1) is periodical in $$\vec{R}$$. By multiplying $$e^{-i\vec{k}\cdot\vec{r}}$$ on both sides of Eq.(2), we find: $$\bar{H}|u_{n\vec{k}}(\vec{r})=E_{n\vec{k}}|u_{n\vec{k}}\rangle \tag{3}$$ where $$\bar{H} \equiv e^{-i\vec{k}\cdot\vec{r}} H e^{+i\vec{k}\cdot\vec{r}}$$. From this eigenequation, we can obtain the completeness relation: $$\sum_n |u_{n\vec{k}}(\vec{r})\rangle \langle u_{n\vec{k}}(\vec{r})|=1 \tag{4}$$ and also this completeness relation: $$\int d\vec{k} \sum_n |u_{n\vec{k}}(\vec{r})\rangle \langle u_{n\vec{k}}(\vec{r})|=1 \tag{5}$$ Now my question is which one is correct? It seems that Eq.(4) is usually used in the community of solid-state physics while Eq.(5) is generally used in the community of nonlinear optics.

I start from the other existing answer, but I elaborate. Also in the question, the source of confusion is in part due to the bad notation.

One can consider a wave function which depends on spin $$s$$ and position $$x$$, i.e. $$\psi_{N}(x,s)$$. Now, suppose you can write it as $$\psi_{n\sigma}(x,s)=\phi_{n\sigma}(x)\chi_\sigma(s)$$ with $$N=\{n\sigma\}$$ (this is possible if spin orbit coupling (SOC) is neglected in condensed matter). Then you have $$\begin{eqnarray} \sum_\sigma |\chi_\sigma\rangle\langle\chi_\sigma|=1 \tag{1}\\ \sum_n |\phi_{n\sigma}\rangle\langle\phi_{n\sigma}|=1 \tag{2} \end{eqnarray}$$ and in general $$\begin{eqnarray} \sum_\sigma \sum_n |\psi_{n\sigma}\rangle\langle\psi_{n\sigma}|=1 \tag{3} \end{eqnarray}$$ Notice that, for each spin channel $$\sigma$$ there is a complete basis set generated by the eigenstates of the spin dependent Hamiltonian $$h_\sigma$$

If you see the parallel with block states, you can do the same and define $$\Psi_{nk}(r+R)=u_{nk}(r)e^{ik(r+R)}$$. Let me define $$e_k(x)=e^{ikx}$$ the exponential function. You obtain $$\begin{eqnarray} \sum_n |u_{nk}\rangle\langle u_{nk}|=1 \tag{4}\\ \int dk |e_k\rangle\langle e_k|=1 \tag{5} \end{eqnarray}$$ and $$\begin{eqnarray} \sum_n \int dk |\Psi_{nk}\rangle\langle\Psi_{nk}|=1\tag{6} \end{eqnarray}$$

• +1. Welcome to our community by the way! We hope to see much more of you in the future! I added equation numbers so that people can say "Eq. 5" instead of "the second equation from the bottom" when citing this answer. Jul 10, 2022 at 2:15

"From this eigenequation, we can obtain the completeness relation"

You don't need that eigenequation to obtain the completeness relation. As long as you have a "complete" set of orthonormal vectors in a Hilbert Space (for example $$n$$ linearly independent vectors in an $$n$$-dimensional Hilbert space) the completeness relation holds. More about this has been discussed on the Physics Stack Exchange.

Some Hilbert spaces have a finite dimension (for example, the operators describing spin-1/2 particles) and the completeness relation is $$\sum_n |\psi_n\rangle \langle \psi_n |$$ where the number of terms in the sum is the dimension of the Hilbert space (which is finite, and equal to 2 in the case of spin-1/2 particles, equal to in the case of spin-1 particles, equal to 4 in the case of spin-3/2 particles, etc.). But some Hilbert spaces are used for continuous quantum mechanical variables such as the position $$\hat{x}$$, in which the completeness relation is $$\int |\psi_x\rangle \langle \psi_x|\textrm{d}x$$.

What you're seeing in Eq. 5 is the combination of two completeness relations. For example if you have wavefunctions which depend on spin $$\hat{s}$$ and position $$\hat{x}$$ at the same time, then we have the following two completeness relations:

$$\begin{eqnarray} \sum_s |\psi_{sx}\rangle \langle \psi_{sx}| &= 1 \tag{1}\\ \int |\psi_{sx}\rangle \langle \psi_{sx}| \textrm{d}x&= 1. \tag{2}\\ \end{eqnarray}$$

Combining the two, we get:

$$\begin{eqnarray} \int \sum_s |\psi_{sx}\rangle \langle \psi_{sx}|\textrm{d}x &= 1.\tag{3} \end{eqnarray}$$


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Properties of Convergent Series

# Properties of Convergent Series

We will now look at some very important properties of convergent series, many of which follow directly from the standard limit laws for sequences.

 Theorem 1: Let $\sum_{n=1}^{\infty} a_n$ be convergent to the sum $A$ and let $\sum_{n=1}^{\infty} b_n$ be convergent to the sum $B$. Then the series $\sum_{a_n + b_n}$ is convergent to the sum $A + B$.
• Proof: Suppose that $\sum_{n=1}^{\infty} a_n = A$ and $\sum_{n=1}^{\infty} b_n = B$. Since both of these series are convergent, it follows that their sequences of partial sums are convergent, that is $\lim_{n\to \infty} s_n = A$ and that $\lim_{n \to \infty} s_n' = B$. By limit laws since both of these sequences are convergent it follows that $\lim_{n \to \infty} \left ( a_n + b_n \right ) = A + B$ which implies $\sum_{n=1}^{\infty} a_n + b_n = A + B$. $\blacksquare$
 Theorem 2: Let $\sum_{n=1}^{\infty} a_n$ be convergent to the sum $A$ and let $\sum_{n=1}^{\infty} b_n$ be convergent to the sum $B$. Then the series $\sum_{n=1}^{\infty} a_n - b_n$ is convergent to the difference $A - B$.
• Proof: Suppose that $\sum_{n=1}^{\infty} a_n = A$ and $\sum_{n=1}^{\infty} b_n = B$. Since both of these series are convergent, it follows that their sequences of partial sums are convergent, that is $\lim_{n\to \infty} s_n = A$ and that $\lim_{n \to \infty} s_n' = B$. By limit laws since both of these sequences are convergent it follows that $\lim_{n \to \infty} \left ( a_n - b_n \right ) = A - B$ which implies $\sum_{n=1}^{\infty} a_n - b_n = A - B$. $\blacksquare$
 Theorem 3: Let $\sum_{n=1}^{\infty} a_n$ be a convergent series to the sum $A$ and let $k$ be a constant. Then $\sum_{n=1}^{\infty} ka_n$ is also a convergent series to the sum $kA$.
• Proof: Suppose that $\sum_{n=1}^{\infty} a_n = A$ and let $\{ s_n \}$ be the sequence of partial sums for this series. Then $\lim_{n \to \infty} s_n = A$. Since this sequence is convergent, it follows by the limit laws that $\lim_{n \to \infty} ks_n = kA$ which implies that that $\sum_{n=1}^{\infty} ka_n = kA$. $\blacksquare$
 Theorem 4: Let $\sum_{n=1}^{\infty} a_n$ be a convergent convergent series to $A$ and let $\sum_{n=1}^{\infty} b_n$ be a convergent series to $B$. If $a_n ≤ b_n$ for all $n \in \mathbb{N}$, then $A ≤ B$.
• Proof: Suppose that $\sum_{n=1}^{\infty} a_n = A$ and $\sum_{n=1}^{\infty} b_n = B$ such that $a_n ≤ b_n$ for all $n \in \mathbb{N}$. We know that $\lim_{n \to \infty} s_n = A$ and $\lim_{n \to \infty} s_n' = B$. Since $a_n ≤ b_n$ for all $n \in \mathbb{N}$ we deduce that $s_n = a_1 + a_2 + ... + a_n ≤ b_1 + b_2 + ... + b_n = s_n'$ for all $n \in \mathbb{N}$, and so $A ≤ B$. $\blacksquare$


text:
Express linear inequalities graphically and algebraically

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0/1
Intros
Lessons
1. i) What is an inequality?
ii) How to solve linear inequalities?
iii) How to graph inequalities on a number line?
0/13
Examples
Lessons
1. Express the following inequalities algebraically.

2. Graph each inequality on a number line.
1. $3.6$ > $x$
2. $x \le - 3\frac{4}{7}$
3. $y \le 11$
4. $5$ < $- \frac{1}{6}$x
3. Find the possible values of $x$ on a number line.
1. $x \ge 6$ and $x \le 11$
2. $x$ > $2\frac{3}{7}$ and $x \le 5$
3. $x \ge - 3.6$ and $x \le 0.25$
4. $x$ > $-7$ and $x \le - 3\frac{1}{2}$
4. A Christmas tree must be 2 feet or taller so that the farmers will cut it down and sell it in the market.
1. Use a number line to express the situation.
2. Express the situation in a statement.
0%
Practice
Topic Notes
Linear inequalities can be represented both graphically and algebraically. In this lesson, we will learn how to express linear inequalities as an equation and on a number line.


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Searching....

# 5.3.8 Semi-Empirical Functionals

(February 4, 2022)

The following semi-empirical methods are available in Q-Chem:

• HF-3c: It is an HF-based semi-empirical method which is used with a minimal basis called MINIX. All the elements from H to Xe are supported. For the elements from Rb ($Z=37$) to Xe ($Z=54$), def2-ECP must be used.1100

• PBEh-3c: It is a DFT based method which is used with def2-mSVP basis set.This is available for all the elements from H to Rn. For the elements from Rb ($Z=37$) to Rn ($Z=86$), def2-ECP has to be used. It is a more accurate method than HF-3c as it has correlation,393 although the double-$\zeta$ basis set makes it more expensive for large molecules.

Note:  The HF-3c and PBEh-3c methods were parameterized for use with the aforementioned basis sets. Although Q-Chem does not enforce this, other basis sets should not be used unless there is a good reason for doing so, and only then with caution as the results may not match published accuracy.

Example 5.4  Q-Chem input for HF-3c method.

$rem method hf-3c basis minix ecp def2-ecp$end

$molecule -1 1 I 0.6539056222 0.0300979939 0.0000000000 O -2.8593081518 0.0835488510 0.0000000000 H -1.8947008157 0.2596704547 0.0000000000 H -2.8641221666 -0.8778240298 0.0000000000$end

View output

Example 5.5  Q-Chem input for PBEh-3c method.

$molecule 0 1 O -1.551007 -0.114520 0.000000 H -1.934259 0.762503 0.000000 H -0.599677 0.040712 0.000000 O 1.350625 0.111469 0.000000 H 1.680398 -0.373741 -0.758561 H 1.680398 -0.373741 0.758561$end

$rem METHOD PBEh-3c BASIS def2-mSVP XC_GRID 000099000590 INCDFT 0 INCFOCK 0 SCF_CONVERGENCE 8 THRESH 14 SYMMETRY false SYM_IGNORE true$end

View output


text:
# python re.sub(): difficulty with special characters (maybe?)

I would like to use re.sub to turn this:

string = '\$$2 \\, e^{\\left(2 \\, x\\right)} \\sin\\left(2 \\, x\\right) + 2 \\, e^{\\left(2 \\, x\\right)} \\cos\\left(2 \\, x\\right)\$$'

into this:

'\$$2 \\, e^{2 \\, x} \\sin\\left(2 \\, x\\right) + 2 \\, e^{2 \\, x} \\cos\\left(2 \\, x\\right)\$$'

This is my best attempt, but it does not work:

re.sub(r'(?P<left-edge>e\^{\\left$$)(?P<input>.*)(?P<right-edge>\\right$$})','e^{\g<input>}',string)

Note that <input> needs to handle an arbitrary expression, while <left-edge> and <right-edge> are fixed character strings.

I am assuming it has to do with the special characters involved, but several dozen attempts demonstrate that it is beyond my expertise.

-

Backslashes in regular expressions must be escaped. You used r'' so they do not have to be escaped as characters in Python string, but that's not enough for them to be interpreted as literal \ chars in regexes. Use double backslashes:

re.sub(r'(?Pe\^{\\left()(?P.*)(?P\\right)})','e\^{\\g}',string)

Were it not for r'', they would have to be escaped twice, i.e. quadrupled, in order to satisfy both Python interpreter and regexp engine:

    re.sub('(?Pe\\^{\\\\left()(?P.*)(?P\\\\right)})','e\\^{\\\\g}',string)

(Additionally, you also forgot to escape one of ^ carets. I corrected that in both of my examples.)

-
that's it. much appreciated. –  strangethingintheland Mar 3 '12 at 17:54
the expressions are not even valid regexs. And it doesn't work even if you fix the syntax errors. You need .*? instead of .* as in my answer –  J.F. Sebastian Mar 3 '12 at 18:05

.* matches too much:

print(re.sub(r'e\^{\\left$$(.*?)\\right$$}', r'e^{\1}', s))

### Output

$$2 \, e^{2 \, x} \sin\left(2 \, x\right) + 2 \, e^{2 \, x} \cos\left(2 \, x\right)$$

-
also helpful. thanks. –  strangethingintheland Mar 3 '12 at 18:12


text:
# Revision history [back]

As mfischer points out, there is an open bug report for this issue. There is a patch under review here, but as of this writing (2014-04-03) it has not been merged.

You can work around this particular problem locally by editing your /etc/hosts file and adding something like:

172.16.16.177 glanceslave
172.16.16.231 glancemaster

And then running your command as:

glance-replicator -v livecopy \
glancemaster:9292 -M $PRIMARY_AUTH_TOKEN glanceslave:9292 -S$SECONDARY_AUTH_TOKEN

This avoids the problems caused by the bad regular expression in the glance-replicator code.


text:
Topology problem about Hausdorff space and compactness

Would anyone have ideas on how to solve the following problem?

Let (X, τ) be a Hausdorff space and τ0 = {X\K: so that K is compact in (X, τ)}

Show that:

1) τ0 is a topology of X.

2) τ0 is rougher than τ (i.e. τ0 is a genuine subset of τ).

3) (X, τ0) is compact.

This was a question in a recent exam that I took (I failed). I was especially clueless with 3).

Thanks for any help.

WannabeNewton
This is related to what is called the Alexandroff extension or Alexandroff compactification. You'll be happy (or not happy?) to know that the given problems are actually quite simple to solve. But before anyone can help you, you must show as per forum policy what you have tried thus far.

1 person
But before anyone can help you, you must show as per forum policy what you have tried thus far.

To show that τ0 is a topology of X, I tried to show that arbitrary unions and finite intersections of members of τ0 are also members of τ0 and that X and ∅ are included in τ0.

τ includes all members of τ0, since from K is compact and (X, τ) Hausdorff follows that K is closed and therefore X/K is open in (X, τ).
To show that τ0 ≠ τ, I argued that because (X, τ) is Hausdorff, all members of that space must have disjoint neighbourhoods, which would not be possible if all the open sets were defined as X/K, K compact. Therefore, τ should contain more open sets than τ0.
Thus, τ0 is a genuine subset of (or rougher than) τ.

Apparently the compactness of (X, τ0) follows from the fact that since all its open sets are defined as {X/K: K is compact in (X, τ)}, an open neighbourhood for any x∈(X, τ0) will be X/K whose complement K is compact? From this it would follow that (X, τ0) is compact. I don't know how to prove the lemma, however.

WannabeNewton
To show that τ0 is a topology of X, I tried to show that arbitrary unions and finite intersections of members of τ0 are also members of τ0 and that X and ∅ are included in τ0.
And were you successful? Did you do this part correctly or did you have some concerns regarding this?

τ includes all members of τ0, since from K is compact and (X, τ) Hausdorff follows that K is closed and therefore X/K is open in (X, τ).
This is perfectly fine but let's write it in a more sequential way. Let ##U\in \mathcal{T}_{0}## then ##U = X \setminus K## for some ##K\subseteq X## compact under ##\mathcal{T}##. Since ##X## is Hasudorff under ##\mathcal{T}##, ##K## is closed under ##\mathcal{T}## so ##U = X\setminus K\in \mathcal{T}##. This is exactly what you said but I have just written it in a more sequential manner.

To show that τ0 ≠ τ, I argued that because (X, τ) is Hausdorff, all members of that space must have disjoint neighbourhoods, which would not be possible if all the open sets were defined as X/K, K compact. Therefore, τ should contain more open sets than τ0.
Thus, τ0 is a genuine subset of (or rougher than) τ.
This is actually false. Let ##X## be a finite set with the discrete topology; ##X## is Hausdorff as well as compact. Let ##U\subseteq X## be open then ##X\setminus U## is closed in ##X## and is therefore a compact subset of ##X## so ##U = X \setminus K## for a compact subset ##K \subseteq X## hence if ##U## is open in the discrete topology on ##X## it is also open in the topology ##\mathcal{T}_{0}## on ##X##.

Apparently the compactness of (X, τ0) follows from the fact that since all its open sets are defined as {X/K: K is compact in (X, τ)}, an open neighbourhood for any x∈(X, τ0) will be X/K whose complement K is compact? From this it would follow that (X, τ0) is compact. I don't know how to prove the lemma, however.
Recall that a topological space ##X## is compact if every open cover of ##X## has a finite subcover. So let ##\mathcal{O}## be an open cover of ##X## under ##\mathcal{T}_{0}## and show that it has a finite subcover.

Like WBN already indicated, I think that when the exercise says that "##\mathcal{T}_0## is rougher than ##\mathcal{T}##", then they mean that ##\mathcal{T}_0\subseteq \mathcal{T}##, but that they don't ask for it to be a genuine subset.

Recall that a topological space ##X## is compact if every open cover of ##X## has a finite subcover. So let ##\mathcal{O}## be an open cover of ##X## under ##\mathcal{T}_{0}## and show that it has a finite subcover.

I think you mean this:

Let ##C## be an open cover of (##X##, ##\mathcal{T}_{0}##).

For a compact ##L## there exists a finite subcover {##X##\##K##j: ##j##∈##J##, # ##J## < ∞}, which is a subset of ##C## .

Therefore the cover of the whole space (##X##, ##\mathcal{T}_{0}##) is [##X##\##L##]##⋃##{##X##\##K##j: ##j##∈##J##, # ##J## < ∞}, i.e. [##X##\##L##]##⋃##[##X##\##K##1]##⋃## ... ##⋃##[##X##\##K##n], which is finite.

However, to me this raises a question:

If we choose ##X## = ##ℝ## and ##L## = ##[0,1]##, which is compact.

Now a finite open cover of ##ℝ## would be [##ℝ##\##[0,1]##]##⋃##[##ℝ##\##K##1]##⋃## ... ##⋃##[##ℝ##\##K##n].

But wouldn't this be a contradiction since ##ℝ## isn't compact?

PS. Apologies about the ugly notation. I'm just starting to learn how to add and use the symbols.

I think you mean this:

Let ##C## be an open cover of (##X##, ##\mathcal{T}_{0}##).

For a compact ##L## there exists a finite subcover {##X##\##K##j: ##j##∈##J##, # ##J## < ∞}, which is a subset of ##C## .

And what exactly is ##L##?

Now a finite open cover of ##ℝ## would be [##ℝ##\##[0,1]##]##⋃##[##ℝ##\##K##1]##⋃## ... ##⋃##[##ℝ##\##K##n].

But wouldn't this be a contradiction since ##ℝ## isn't compact?

The space ##\mathbb{R}## under the usual Euclidean topology is not compact. The space ##\mathbb{R}## with ##\mathcal{T}_0## is compact.

And what exactly is ##L##?

Any compact set of (##X##,##\mathcal{T}##). ##X##\##L## will be open in (##X##, ##\mathcal{T}_0##).

The space ##\mathbb{R}## under the usual Euclidean topology is not compact. The space ##\mathbb{R}## with ##\mathcal{T}_0## is compact.

But even in Euclidean topology ##\mathcal{T_E}## one could claim that ##\mathbb{R}## is compact due the fact that its finite open cover is

{##\mathbb{R}##\##[0,1]##, ##U##1, ..., ##U##n}

as

i) ##[0,1]## is compact in (##\mathbb{R}##,##\mathcal{T_E}##) and it will have a finite open cover ##U##1, ..., ##U##n.

and

ii) ##[0,1]## is closed in (##\mathbb{R}##,##\mathcal{T_E}##) and thus its complement ##\mathbb{R}##\##[0,1]## is open.

So what makes the difference with ##\mathcal{T}_0##?

WannabeNewton
Your method is not going to help you prove that the new space is compact. You are approaching it in a wrong way, although your overall idea is in the right ballpark. Let me show you a method that might seem to work and see if you can fix it to get the correct proof. Let ##\mathcal{O}## be an open cover of ##(X,\mathcal{T}_{0})## and let ##U_{0}\in \mathcal{O}##. We know that ##X\setminus U_{0}## is compact in ##(X,\mathcal{T})## by definition. Therefore, since ##\mathcal{O}## is an open cover of ##X\setminus U_{0}## by open subsets of ##(X,\mathcal{T}_{0})##, there exists a finite subcover ##\{U_1,...,U_n\}## of ##X\setminus U_{0}## hence ##\{U_0,U_1,...,U_n\}## covers all of ##X## therefore ##X## is compact. There is a subtle mistake here, that I have purposefully made for you to point out. Can you tell me what it is? If you can find the mistake you can easily fix it and write down the correct proof.

But even in Euclidean topology ##\mathcal{T_E}## one could claim that ##\mathbb{R}## is compact due the fact that its finite open cover is

{##\mathbb{R}##\##[0,1]##, ##U##1, ..., ##U##n}

as

i) ##[0,1]## is compact in (##\mathbb{R}##,##\mathcal{T_E}##) and it will have a finite open cover ##U##1, ..., ##U##n.

and

ii) ##[0,1]## is closed in (##\mathbb{R}##,##\mathcal{T_E}##) and thus its complement ##\mathbb{R}##\##[0,1]## is open.

This is due to a subtle misunderstanding of compactness. What you proved now is that ##\mathbb{R}## has a finite open cover. This is true. In fact, the cover ##\{\mathbb{R}\}## itself is already a finite open cover. But this does not imply compactness. Compactness says that every open cover has a finite subcover. So simply exhibiting an open cover is not enough, you must actually find a finite subcover for every single open cover of the space.

Now, ##\mathbb{R}## is not compact because ##\{(n-1,n+1)~\vert~n\in \mathbb{Z}\}## has no finite subcover.

Your method is not going to help you prove that the new space is compact. You are approaching it in a wrong way, although your overall idea is in the right ballpark. Let me show you a method that might seem to work and see if you can fix it to get the correct proof. Let ##\mathcal{O}## be an open cover of ##(X,\mathcal{T}_{0})## and let ##U_{0}\in \mathcal{O}##. We know that ##X\setminus U_{0}## is compact in ##(X,\mathcal{T})## by definition. Therefore, since ##\mathcal{O}## is an open cover of ##X\setminus U_{0}## by open subsets of ##(X,\mathcal{T}_{0})##, there exists a finite subcover ##\{U_1,...,U_n\}## of ##X\setminus U_{0}## hence ##\{U_0,U_1,...,U_n\}## covers all of ##X## therefore ##X## is compact. There is a subtle mistake here, that I have purposefully made for you to point out. Can you tell me what it is? If you can find the mistake you can easily fix it and write down the correct proof.

I'm sorry, but I simply don't know what the catch is here. Like I showed before with ##ℝ## and ##[0,1]## in Eucledian (or any) topology, this proof doesn't make sense to me and I don't how to correct it to make it work.

I'm just as clueless with the whole thing as I was in the beginning.

WannabeNewton
Can you tell me the definition of compactness that you were taught in class? I don't think I can make the above any easier for you without giving it away I'm afraid so maybe we need to reinforce the concepts in your head.

Can you tell me the definition of compactness that you were taught in class? I don't think I can make the above any easier for you without giving it away I'm afraid so maybe we need to reinforce the concepts in your head.

Okay, I may be getting it now.

##X## will be compact only if there exists a finite subcover for any open cover of ##X##.

Any open cover of the new space will include a member ##U## whose complement ##X##/##U## is always compact and will have a finite subcover {##U##1, ..., ##U##n}. Thus the whole space will always have a finite subcover ##U## ⋃ {##U##1, ..., ##U##n} regardless of the choice of the open cover.

With ##ℝ## and Euclidean topology one can find an open cover of ##ℝ## that has a member whose complement isn't compact and therefore one cannot find a finite open subcover for any open cover of ##ℝ##. Therefore ##ℝ## isn't compact with Euclidean topology.

Or maybe I'm just dancing around the problem?

WannabeNewton
Yes that is correct with regards to what compactness is. Now can you spot the mistake in the proof I gave in post #9?

Yes that is correct with regards to what compactness is. Now can you spot the mistake in the proof I gave in post #9?

You should add ##U_0## ##≠## ##∅## ?

Otherwise, no.

You should add ##U_0## ##≠## ##∅## ?

Otherwise, no.

The concept "open set" is dependent on the topology. So a set might be open in one topology, but maybe not in the other. The same with the notions of compactness.

So, in your proof you should be careful about which sets are open/compact and in which topology.

Your method is not going to help you prove that the new space is compact. You are approaching it in a wrong way, although your overall idea is in the right ballpark. Let me show you a method that might seem to work and see if you can fix it to get the correct proof. Let ##\mathcal{O}## be an open cover of ##(X,\mathcal{T}_{0})## and let ##U_{0}\in \mathcal{O}##. We know that ##X\setminus U_{0}## is compact in ##(X,\mathcal{T})## by definition. Therefore, since ##\mathcal{O}## is an open cover of ##X\setminus U_{0}## by open subsets of ##(X,\mathcal{T}_{0})##, there exists a finite subcover ##\{U_1,...,U_n\}## of ##X\setminus U_{0}## hence ##\{U_0,U_1,...,U_n\}## covers all of ##X## therefore ##X## is compact. There is a subtle mistake here, that I have purposefully made for you to point out. Can you tell me what it is? If you can find the mistake you can easily fix it and write down the correct proof.

Perhaps the mistake here is that you jump into thinking that ##X\setminus U_{0}## is compact in ##(X,\mathcal{T}_{0})## just because it's compact in ##(X,\mathcal{T})## without proof.

First, the open sets of ##\mathcal{T}_{0}## will cover ##(X,\mathcal{T}_{0})##. Since ##\mathcal{T}## contains ##\mathcal{T}_{0}##, an open cover ##\mathcal{O}## of ##(X,\mathcal{T}_{0})## is always an open cover of ##(X,\mathcal{T})## and therefore it will cover ##X\setminus U_{0}## too. Since ##X\setminus U_{0}## is compact in ##(X,\mathcal{T})##, it has a finite subcover ##\{U_1,...,U_n\}## which is a subset of ##\mathcal{O}##. As the sets of ##\{U_1,...,U_n\}## are open in both spaces, ##X\setminus U_{0}## always has a finite subcover in both spaces and therefore is compact in both spaces.

That seems right!

WannabeNewton
Perhaps the mistake here is that you jump into thinking that ##X\setminus U_{0}## is compact in ##(X,\mathcal{T}_{0})## just because it's compact in ##(X,\mathcal{T})## without proof.

First, the open sets of ##\mathcal{T}_{0}## will cover ##(X,\mathcal{T}_{0})##. Since ##\mathcal{T}## contains ##\mathcal{T}_{0}##, an open cover ##\mathcal{O}## of ##(X,\mathcal{T}_{0})## is always an open cover of ##(X,\mathcal{T})## and therefore it will cover ##X\setminus U_{0}## too. Since ##X\setminus U_{0}## is compact in ##(X,\mathcal{T})##, it has a finite subcover ##\{U_1,...,U_n\}## which is a subset of ##\mathcal{O}##. As the sets of ##\{U_1,...,U_n\}## are open in both spaces, ##X\setminus U_{0}## always has a finite subcover in both spaces and therefore is compact in both spaces.
Brilliant! You're done then :)

Brilliant! You're done then :)

OK! Thank you very much for your help, WannabeNewton and micromass.

Topology is still quite tricky business to me, but I hope this exercise prepared me for challenges to come so I can finally pass my course some day.

WannabeNewton


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# Use the Root Test to determine the convergence or divergence of the series. sum_{n=1}^infty(frac{n}{500})^n

Question
Series
Use the Root Test to determine the convergence or divergence of the series.
$$\sum_{n=1}^\infty(\frac{n}{500})^n$$

2021-03-08
Given: $$\sum_{n=1}^\infty(\frac{n}{500})^n$$
Root test: Suppose we have the series $$\sum a_n$$, Define $$L=\lim_{n\to\infty}\sqrt[n]{|a_n|}=\lim_{n\to\infty}|a_n|^{\frac{1}{n}}$$
Then,
1) If L
2) If L>1, the series is divergent.
3) If L=1, the series may be divergent, conditionally convergent, or absolutely convergent.
Now consider,
$$a_n=(\frac{n}{500})^n$$
$$L=\lim_{n\to\infty}|(\frac{n}{500})^n|^{\frac1n}$$
$$=\lim_{n\to\infty}\frac{n}{500}$$
$$=\frac{\infty}{500}$$
$$=\infty>1$$
As L>1, by the Root Test the series is divergent.

### Relevant Questions

Use the Root Test to determine the convergence or divergence of the series.
$$\sum_{n=1}^\infty(\frac{3n+2}{n+3})^n$$
Use the Root Test to determine the convergence or divergence of the series.
$$\sum_{n=2}^\infty\frac{(-1)^n}{(\ln n)^n}$$
Which of the series, and which diverge?
Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
$$\sum_{n=1}^\infty \frac{e^n}{1+e^{2n}}$$
Use the Limit Comparison Test to determine the convergence or divergence of the series.
$$\sum_{n=1}^\infty\frac{n^{k-1}}{n^k+1},k>2$$
Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series.
$$\sum_{n=1}^\infty3^{-n}$$
Use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series.
$$\sum_{n=1}^\infty\frac{1}{\sqrt{n^3+2n}}$$
$$\sum_{n=1}^\infty\sin\frac{1}{n}$$
$$\sum_{n=1}^\infty\frac{n^2}{(n+1)(n^2+2)}$$
$$\sum_{n=1}^\infty\frac{(2n)!}{n^5}$$
$$\sum_{n=1}^\infty\frac{5}{4^n+1}$$


text:
# independence number

Noun.  (context, graph theory) the number of vertices in a maximum independent set of a given graph, often denoted as $alpha = alpha\left(G\right)$.

This is an unmodified, but possibly outdated, definition from Wiktionary and used here under the Creative Commons license. Wiktionary is a great resource. If you like it too, please donate to Wikimedia.

This entry was last updated on RefTopia from its source on 3/20/2012.


text:
# Are there non-trivial locally trivial fiber bundles of Hilbert manifolds if the fiber is infinite-dimensional?

It is well-known (see Kuiper's theorem) that every Hilbert space bundle over a manifold is trivial if the Hilbert space is truly infinite-dimensional.

Does the same hold true if one considers locally trivial fiber bundles instead and the fiber is assumed to be a truly infinite dimensional Hilbert manifold?

-
What do you mean by "truly" infinite dimensional Hilbert manifold? You can easily build a fiber bundle over $S^1$ with fiber $S^1 \times H$, $H$ a Hilbert space, which is non-trivial just by homotopy reasons. – Alberto Abbondandolo May 2 '12 at 13:08


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# Network Traffic Analysis Using tcpdump Introduction to tcpdump

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## Network Traffic Analysis Using tcpdump Introduction to tcpdump

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The objectives of this course are to introduce you to the fundamentals and benefits of using tcpdump as a tool to analyze your network traffic. We’ll start with introducing concepts and output of tcpdump. One of the most important aspects of using tcpdump is being able to write tcpdump filters to look for specific traffic. Filter writing is fairly basic unless you want to examine fields in an IP datagram that don’t fall on byte boundaries. So, that is why an entire section is devoted to the art of writing filters....

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## Nội dung Text: Network Traffic Analysis Using tcpdump Introduction to tcpdump

1. Network Traffic Analysis Using tcpdump Introduction to tcpdump Judy Novak Johns Hopkins University Applied Physics Laboratory jhnovak@ix.netcom.com 1 All material Copyright  Novak, 2000, 2001. All rights reserved. 1
2. Table of Contents Topics Introduction to tcpdump Writing tcpdump Filters Examination of Datagram Fields Beginning Analysis Real World Examples Step by Step Analysis References 2 2
3. Course Objectives • Introduce the fundamentals of tcpdump • Explain how to write tcpdump filters • Examine fields in datagram for uses/misuses • Analyze traffic by placing it in categories • Demonstrate “real-world” analysis using tcpdump • Let you participate in the analysis process 3 The objectives of this course are to introduce you to the fundamentals and benefits of using tcpdump as a tool to analyze your network traffic. We’ll start with introducing concepts and output of tcpdump. One of the most important aspects of using tcpdump is being able to write tcpdump filters to look for specific traffic. Filter writing is fairly basic unless you want to examine fields in an IP datagram that don’t fall on byte boundaries. So, that is why an entire section is devoted to the art of writing filters. Before we start to use tcpdump to analyze traffic, we’ll examine many of the fields found in the IP datagram. This is done to familiarize you with those fields in theory and also how they might be used in practice. We’ll study how and why fields might be changed and for what purpose. Next, we’ll start the basic analysis process by looking at tcpdump output and categorizing the kind of traffic that you can see. Then, we’ll take a look at some real-world examples and of how tcpdump was used on monitored networks to discover what was happening. Next, the analysis process will be inspected step by step often with missteps to get you comfortable with it. As a note, all tcpdump output shown in this course is activity that actually occurred. Source and destination hosts/IP’s have been altered to obfuscate the true identities. 3
4. Overview • Introduction to tcpdump • Writing tcpdump filters • Examination of Datagram Fields • Beginning Analysis • Real World Examples • Step by Step Analysis 4 This page intentionally left blank. 4
5. Introduction to tcpdump • Introduction to tcpdump • Writing tcpdump Filters • Examination of Datagram Fields • Beginning Analysis • Real World Examples • Step by Step Analysis 5 This page intentionally left blank. 5
6. Objectives • Examine the strengths/weaknesses of tcpdump • Organize collection/analysis process of tcpdump data via Shadow • Examine tcpdump output • Standard • Hexadecimal • Length fields and how to convert them to bytes • Application layer • Interpretation of payload/hex output 6 This page intentionally left blank. 6
8. Strengths • Provides audit trail/historical record of network activity • Provides absolute fidelity • Universally available and used A 8 One of the most important parts of an arsenal in your security infrastructure is at least one tool or software package that captures an audit trail or a historical record of the traffic that enters or leaves your network. There will be times when you will be required to examine activity or connections that occurred in your network – not just traffic that caused an alarm to sound. For instance, what if you suspect that your packet filtering router that acts as your perimeter defense was acting strangely after some major network changes were made. You would have to examine the traffic that was allowed into your network to assist in determining the problem. That is where tcpdump is invaluable. Also, many tools - even logs from firewalls will display suspicious traffic, yet only partial data is displayed. What if you get a log of rejected traffic, but it doesn’t display or keep TCP flags? You’ll never know what kind of connection was attempted. tcpdump allows the analyst to examine all the bits and fields that are collected. If nothing is “wrong” with the connection, examination at the bit level is unnecessary. Yet, if you suspect something “foul” with the traffic, you really need access to all the data down to the bit level. And tcpdump is a tool that is universally used and very portable. If you become familiar with this software or its Windows counterpart, windump, it can be used on just about any platform to assist you in analysis of traffic. 8
9. Weaknesses • By default, doesn’t collect all the payload • Does not scale well on large networks • No idea of state • Limited operations • Do-it-yourself interpretations 9 tcpdump will capture 68 bytes of data from the network interface. Some of this data might be used for the link layer frame header. For Ethernet, 14 bytes of the data are used to capture fields like the source and destination MAC address, along with the type of embedded data. That leaves only 54 bytes to capture the IP header and embedded protocol header as well as any data. Most of the time this size will allow you to capture the IP header and embedded protocol header. But, sometimes protocol headers or data will be truncated. And, if you are interested in the data payload, tcpdump is really not the tool to use for this. tcpdump can collect a large volume of data for larger networks. This can be alleviated by not collecting all the data on the network – perhaps omit web traffic (port 80). Or, another way to deal with this is more disk space and faster processors to analyze all the collected data. But, at some point, the volume gets unwieldy. tcpdump blindly collects packet after packet. It has no idea of state or being able to know that a given packet is anomalous because it does not follow the flow of a normal connection. And while tcpdump has some primitive arithmetic operations or ways to manipulate bits, it cannot do complex operations for analyzing data. Finally, while it is an excellent way to collect data, tcpdump does not attempt to make interpretations of what it sees. It does have some integrity checking operations for certain data to make sure that the data is not irregular, but the analyst has to have the training and savvy to interpret the data. For the sophisticated analyst, this is a bonus because she or he can make the correct call. Compare this with a tool that is prone to false positives that gives no way of verifying the alarmed event. But, for an analyst who has little training, tcpdump can be daunting since it does not interpret events. 9
10. tcpdump Versions • tcpdump: Unix version; official current version 3.4 • ftp://ftp.ee.lbl.gov/tcpdump.tar.Z • ftp://ftp.ee.lbl.gov/libpcap.tar.Z • windump: Windows version • http://netgroup-serv.polito.it/windump • http://netgroup-serv.polito.it/winpcap • Collective effort; current version 3.5: www.tcpdump.org • tcpdump-3.5.tar.gz • libpcap-0.5.tar.gz 10 tcpdump is officially supported by the Lawrence Berkeley Labs. The current version is 3.4. There is an effort to improve tcpdump and patch known problems with tcpdump and libpcap that appears to be a collective effort of anyone interested. The software for this effort can be found at www.tcpdump.org. Their current version is 3.5 For the Unix versions of tcpdump, you need to download software known as libpcap that implements a portable framework for capturing low-level network traffic. windump is a Windows variant of tcpdump. It also requires an application program interface to collect the traffic known as winpcap. The unofficial version of tcpdump has some nice enhancements. It decrypts more of the applications at the application layer and has a very nice capability of converting hexadecimal payload to character output. 10
11. tcpdump in Action 0101001110 111010010011000 00100011011 packets Network tcpdump running on a host “sniffing” network packets tcpdump output 07:00:48.036746 ping.net > myhost.com: icmp: echo request (DF) 07:00:48.036776 myhost.com > ping.net: icmp: echo reply (DF) 07:02:12.622460 log.net.3155 > syslog.com.514: udp 101 07:03:01.132414 send.net.32938 > mail.com.25: S 248631:248631(0) win 8760 11 We see on this slide, a host running tcpdump and gathering records from the network interface. We see the records that tcpdump has collected below. tcpdump has a default standard output based on the protocol (TCP, UDP, ICMP) of the record that is displayed. While each of the various protocols has a similar format to the other, they are also distinct in what is displayed. By default, tcpdump will collect and print, in a standard format, all the traffic passing on the network. There are command line options for tcpdump that will alter the default behavior, either by collecting specified records, printing in a more verbose mode, printing in hexadecimal or writing records as “raw packets” to a file instead of printing as standard output. 11
12. Sample tcpdump Output Sample UDP Record 09:39:19.470000 nmap.edu.728 > dns.net.111: udp 56 timestamp source . port dest . port : protocol bytes Sample TCP Record beginning seq # data bytes 09:35:53.660000 nmap.edu.4 > dns.net.111: SF 136747297:136747297(0) win 1028 flags ending seq # 09:32:43.910000 nmap.edu.1171 > dns.net.139: S 2490962508:2490962508(0) win 512 09:32:43.910000 nmap.edu.1173 > dns.net.21: S 62697789:62697789(0) win 512 09:32:43.910000 nmap.edu.1193 > dns.net.22: S 1360146849:1360146849(0) win 512 09:32:43.920000 nmap.edu.1194 > dns.net.1114: S 372884098:372884098(0) win 512 12 Since we’ll review a lot of tcpdump output in this course, here’s a chance to get more comfortable with it. This is sample output from what appears to be an nmap scan; a popular and informative scan. All records have a timestamp. The sensor host (Redhat Linux 5.2) that captured these records has the precision to capture hundredths of seconds although tcpdump allows places for up to millionths. Different protocols will have different representations in tcpdump output. One of the first challenges is to identify the protocol (TCP, UDP, ICMP). Most will be labeled and while TCP isn’t explicitly labeled, it is the only one with flag bits, sequence and acknowledgment numbers to name a few. Some protocols like DNS will be interpreted at the application layer. Because of this, you may not see the normal clues that you are used to. It may not be obvious if it is UDP or TCP so it is important to look for clues as to which it is. In general, tcpdump gives details about the source/host > destination/host. Note that the bytes (0) transferred on SYN packets is normally 0 since they do not carry a payload because this is just part of establishing the three-way handshake. 12
16. Why Shadow? •  (free for all) • Tunable • Customize your own signatures • Change at will • Provides an audit trail of activity to/from network • Provides an intimate view of activity 16 While not the only reason to install and use Shadow, a very compelling reason is the price tag. In many cases, but not this one, you get what you pay for. Shadow is an excellent no-cost traffic analysis tool. Another benefit is that once you master Shadow, you can change it liberally at any time that you want. For instance, if you hear of a new exploit and can fashion a signature with a tcpdump filter, you can modify Shadow instantaneously. Compare this with some intrusion detection systems that do not offer the capability to change filters or signatures. You have to wait for the software company to update the filters when they get around to it and the updates may not include signatures that you would like to see. Also, since you get all the source code with Shadow, you can customize it for your whims and needs. This is highly unusual and allows you to make changes based on your proficiency of the software. Shadow uses tcpdump as its collection software. By default, you will collect most activity going into and out of your network. This can be very beneficial in providing an audit trail of activity in the network. If you ever find yourself in the midst of some kind of incident, this may be a very valuable attribute for an intrusion detection system to have. Finally, some of the more GUI kinds of intrusion detection systems do not allow the user to examine the actual traffic at the IP datagram level. Shadow, by virtue of tcpdump, will allow the user a very intimate view of the data collected. You will maintain fidelity of data and you can use all fields for interpretation and analysis. If the traffic you are analyzing is corrupted in some way, you want to be able to inspect the entire datagram. 16


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# Doubt while computing the animalous dimension of $\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi\bar{\psi}\gamma^{\nu}\partial_{\nu}\psi$

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I am following the conventions of http://isites.harvard.edu/fs/docs/icb.topic1146665.files/III-9-RenormalizationGroup.pdf. Consider the QED Lagrangian

$$\mathcal{L}=-\frac{1}{4}Z_3F_{\mu\nu}^2+Z_2\bar{\psi}i\gamma^{\mu}\partial_{\mu}\psi-Z_2Z_mm\bar{\psi}\psi+Z_eZ_2\sqrt{Z_3}e\bar{\psi}\gamma^{\mu}A_{\mu}\psi+\sum^jC_j\mathcal{O}_j$$

where $\mathcal{O}_j=Z_j\partial^n\gamma^mA_{\mu}\ldots{}A_{\nu}\bar{\psi}\ldots\psi$ are operators with all fields evaluated at the same point that have any number of photons, fermions, gamma matrices, factors of the metric... and analytic dependence on derivatives. Everything is written using renormalized fields. Consider in particular the operator

$$\mathcal{O}=Z\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi\bar{\psi}\gamma^{\nu}\partial_{\nu}\psi$$

I want to get the anomalous dimension of this operator at one loop. I know that in order to do that I need to get $Z$ but I am clueless of how to proceed. I have the feeling that I have to consider a correlation function but I don't know which.  Any indication wouldbe greatly appreciated.

asked Oct 29, 2015

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# Linux下MPI并行编程环境搭建配置

MPI的全称是Message Passing Interface即标准消息传递界面，可以用于并行计算。MPI有多种实现版本，如MPICH， CHIMP以及OPENMPI。这里我们采用MPICH版本。

tar -xzvf soft/mpich-3.0.4.tar.gz
cd mpich-3.0.4/
./configure --prefix=/usr/local/mpich
make && make install

PATH=$PATH:/usr/local/mpich/bin MANPATH=$MANPATH:/usr/local/mpich/man
export PATH MANPATH

cp -r examples/ /usr/local/mpich

mpirun -np 10 ./examples/cpi

Process 0 of 10 is on server150
Process 9 of 10 is on server150
Process 1 of 10 is on server150
Process 4 of 10 is on server150
Process 5 of 10 is on server150
Process 7 of 10 is on server150
Process 2 of 10 is on server150
Process 3 of 10 is on server150
Process 6 of 10 is on server150
Process 8 of 10 is on server150

pi is approximately 3.1415926544231256, Error is 0.0000000008333325
wall clock time = 0.020644

2、复制编译程序到其他机器上面

scp -r mpich server140:/usr/local/
scp -r mpich server151:/usr/local/
scp -r mpich server130:/usr/local/
scp -r mpich server143:/usr/local/

3、

server150:2 #run 2 process
server140:2
server130:2
server143:2
server151:2

mpiexec -n 10 -f servers ./examples/cpi

Process 0 of 10 is on server150
Process 1 of 10 is on server150
Process 4 of 10 is on server140
Process 5 of 10 is on server140
Process 6 of 10 is on server143
Process 7 of 10 is on server143
Process 8 of 10 is on server130
Process 9 of 10 is on server130
Process 2 of 10 is on server151
Process 3 of 10 is on server151
pi is approximately 3.1415926544231256, Error is 0.0000000008333325
wall clock time = 0.018768

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