At the opening session of the 2016 SNAME Maritime Convention in Bellevue, Washington, U.S.A., George A. (Sandy) Thomson has been elected a Fellow of the Society of Naval Architects and Marine Engineers (SNAME).
To be elected, a candidate must also have ten years of active practice following the candidate's first professional membership in the Society. The candidate must be nominated by three Fellows or Members, approved by the Fellows Committee, and finally elected by the Council.
This is quite an honour for Sandy who has a long family history in the maritime industry. Sandy's great, great grandfather was a deep sea captain sailing the Great Lakes before becoming a lighthouse keeper; his uncle was head of Shell Canada when Shell maintained a fleet of steam powered tankers operating on the Great Lakes; while his father owned and operated Thomson-Gordon, a marine and industrial products supplier in southern Ontario.
Sandy attended Northrup University in California obtaining a Mechanical Engineering degree and eventually migrated back to Canada to work in the family business in 1965. His passion for the marine industry redirected the family business from distributing engineering supplies to manufacturing propeller shaft and rudder bearings – thus leading to the development of Thordon Bearings. Sandy is currently owner and Chairman of the Board of Thomson-Gordon Group, the parent of Canada-based Thordon Bearings Inc.
As Sandy has spent many years growing and developing this business around the world, on one of his sales trips to Russia in 1989 he purchased a steam powered tug named Rudokop. He fell in love with this vessel and retrofitted it with Thordon bearings and seals. Rudokop also served as a floating showcase to illustrate the performance of Thordon products and as a training ground for many Thordon employees and Distributors. Sandy captained the tug travelling to over 300 deep sea ports in Eastern and Western Europe.
After the Rudokop sale in 2007, Sandy shifted the company focus to pursuing sales in the commercial ship and naval markets. Thordon currently supplies over 45 navies and has over 550 commercial ships running its COMPAC water lubricated propeller shaft bearing systems.
Sandy's maritime passion, strong leadership and innovation have transformed the company into a successful marine and industrial bearing and seal manufacturer. Thordon is considered a world leader in non-polluting oil and grease free marine bearings and seals.


text:
HomeNewsAbba Kyari Appears In Court, Begs Court For Bail After Claiming Threat To His Life In Kuje Prison
Abba Kyari Appears In Court, Begs Court For Bail After Claiming Threat To His Life In Kuje Prison
Kyari and three of his co-defendants, through their counsel, urged Justice Emeka Nwite to admit them to bail because their lives in the Kuje Correctional Centre.
A suspended Deputy Commissioner of Police (DCP), Abba Kyari has begged the Federal High Court to release him on bail.
Kyari approached the court on Thursday, alongside three of his co-defendants, through their counsel.
He urged Justice Emeka Nwite to admit them to bail because their lives in the Kuje Correctional Centre, where they are being remanded, were unsafe.
At a resumed hearing, Onyechi Ikpeazu, SAN, counsel to Kyari and suspended ACP Sunday Ubia, informed that an application seeking bail for the 1st, 2nd, 4th and 5th defendants had been filed and served on the National Drug Law and Enforcement Agency (NDLEA), NAN reports.
He said the application became necessary due to the nature of the work the defendants had done in the course of policing the country.
Ikpeazu said the defendants are being remanded with criminals in the correctional centre, who through them, their arrests were made possible.
The senior lawyer said their lives are at risk, hence, the need for their bail.
The News Agency of Nigeria (NAN) reports that four suspended police officers including ACP Ubia, ASP Bawa James, Insp Simon Agirigba and Insp. John Nuhu, listed as 2nd to 5th defendant respectively, are being charged by the NDLEA for an alleged drug offence alongside Kyari.
However, while Kyari, Ubia, Agirigba and Nuhu (1st, 2nd, 4th and 5th defendants), sought bail over the alleged threats to their lives, ASP Bawa James, who is the 3rd defendant in the matter, did not.
Meanwhile, counsel for the prosecution (NDLEA), Sunday Joseph, opposed the request that the bail plea is taken, insisting that the matter was scheduled for review of the facts of the 6th and 7th defendants who pleaded guilty to the counts preferred against them.
Chibunna Umeibe and Emeka Ezenwanne (6th and 7th defendants) were the two alleged drug traffickers that were arrested at the Akanu Ibiam International Airport in Enugu.
Umeibe and Ezenwanne, had pleaded guilty to five, six and seven counts preferred against them and are being remanded in Suleja Correctional Centre.

redpajama_set_name: RedPajamaCommonCrawl


text:
module Bullhorn
module Rest
module Entities

module CorporationDepartment
  extend Base

  define_methods
end

end
end
end

redpajama_set_name: RedPajamaGithub


Super Star Mahesh Babu is preparing for the release of his upcoming and 25th movie on May 9th and the makers of the film are now planning the post-production works to be completed by the last week of April. Meanwhile, Mahesh's fans grew restless and started demanding the producers to release an audio single this month or any glimpse of the trailer.
Without denying the fans request, Vamsi Paidipally has apparently decided to release the first song from the audio album on March 29th and the trailer will be out on the eve of Ugadi, April 6th. It is already known that a character introduction teaser has been released earlier.
If all these speculations about the movie promotions will turn into reality? Wait for the official announcement from the makers. Mahesh has paired with Pooja Hegde for the movie. Where Dil Raju, Ashwini Dutt, and PVP were bankrolling this movie together. Devi Sri Prasad is the music director.


text:
Thankfulness: Abundance, Opportunity, Openness
Metallic Money (Gold/Silver) vs. Credit Money: Kno...
What I'm Grateful For
Why Is Debt the Source of Income Inequality and Se...
Take the Money and Run: China's Ill-Gotten Wealth ...
Taking Stock of the 21st Century: What's Fundament...
If You Must Give Gifts, Consider These Everyday Ki...
The Dark Heart of Centralized Power
ObamaCare: The Neutron Bomb That Will Decimate the...
Have a Merry DeGrowth Christmas--Boycott Black Fri...
The American Model of "Growth": Overbuilding and P...
Hey, Is It a Problem That We're All on One Side of...
Have We Lost Our Common-Sense?
How About Ending Social Security and Paying Retire...
Will the Dollar Lose its Reserve Currency Status t...
The Three Types of Politicians
The Big Lie: Lunch (and Debt) Are Free
Our Era's Definitive Dynamic: Diminishing Returns
What Stanley Kauffmann Taught Me About Film
Can We Support 75 Million Retirees in 2020?
Could Bitcoin (or equivalent) Become a Global Rese...
The Generational Injustice of Social (in)Security
The Problem with Pay-As-You-Go Social Programs: Th...
Why Is Predicting Crisis/Reset So Difficult?
Unaffordable Recreation and the Ratchet Effect
Here are ten possible factors in why it's so difficult to predict crisis/reset.
Doom-and-gloomers (myself included) have been wrong for four years. The financial markets continue higher, and the excesses of the status quo continue expanding with little ill effect (so far).
Why is it so difficult to predict the onset of crisis/collapse? The question is equally valid for both bears and bulls; how could all the boosters of housing be so wrong in 2008 when they asserted that "housing is not a bubble"?
I've assembled ten possible factors in why it's so difficult to predict crisis/reset:
1. Everyone in the status quo has a stake in its survival. Every one of us wants to get our social security, our disability, maintain the freedom of a personal vehicle, have access to clean water and all the other goodies, and those becoming (or maintaining) wealthy and powerful in the current system want to retain their wealth and power.
There are titanic forces that will bend whatever needs to be bent to keep their share of the swag flowing to them. This is just as true of the welfare recipient as it is the global corporation or politico. We shouldn't underestimate the power of this desire to maintain the status quo and bend perceptions to make that appear as if it is not just possible but inevitable.
Consider the housing bubble and bust. Look at the forces benefiting from the bubble: the 2/3 of all U.S. households that owned homes, the entire financial/mortgage/Wall Street investment bank complex, mortgage brokers, realtors, the media that profited from housing/mortgage-related adverts, and last but not least the government, which reaped huge gains in property and income taxes as a direct result of the bubble.
On the other side of the bubble: a handful of marginalized analysts and bloggers, virtually none of whom would profit from the bubble's implosion. Guess which side had the momentum to inflate the bubble for years after it became obvious that a vast credit/real estate bubble was already in place?
2. Self-referential systems with numerous feedbacks are inherently difficult to predict because the forces we extrapolate into the future are adapting under various selective pressures that feed back into each other. Innovations that seem small can trigger outsized consequences (for example, the web, fracking, etc.).
In this context, it's worth recalling an anecdote about Bertrand Russell. A young critic detailed a previous position Russell had taken and noted an inconsistency with his current position. Russell declared, "Young man, I changed my mind." We're allowed to do that as new dynamics emerge and what we extrapolated as critical turns out to be less critical than some other factor we dismissed as minor.
3. Systems feed on the herd instinct of humanity. Real estate was visibly in a bubble in 2004, at least in key markets, yet the bubble continued expanding for 3 more years as the herd drew in skeptics. Those of us who declared the bubble in 2004 were wrong for three long years. The dynamics of the herd overpowered rationality and prudence. Humans will thunder over the cliff just like other herding animals. Just keeping the ability to make independent judgments and sort data without ideological filters should be a key goal.
4. There is a body of sociological study that looks at how what we perceive as risks defines our ideological/sociological world view. Risk assessment by groupthink is very powerful: we are moved by what we perceive to threaten our world view. Hierarchical types see different risks than egalitarian types, for example. This line of analysis goes by the academic name of "cultural cognition of risk."
5. The status quo is a dynamic system with many players. As I have often noted, just for one example, the US military/national security state does not necessarily share the same world view and priorities as other chunks of the empire. Various conspiracy theories neatly tie up the entire system with a bow, but I don't think it's that static and simple. If it was, it would be easier to predict. If we look at systems with few feedbacks, i.e. the sort of systems dominated by small cliques of the sort that all conspiracy theories require, we find systems like the former USSR. It imploded because it lacked the intrinsic ability to reform/adapt for systemic reasons.
6. As a result of #1, alternative systems have very little leverage. Why bust our behinds getting a local farmer's market going when every supermarket is bulging with produce and products engineered to satisfy our reward centers? Everything is an uphill battle to reach the critical 4% threshold of influence in terms of establishing alternative systems. Our own personal resilience only goes so far, but getting people to invest in systems beyond themselves is difficult for any number of reasons, including active suppression by those benefiting from the status quo. The need for real alternatives just isn't strong enough to change world views.
These systems are still nascent. We're still feeling our way forward in revolutionary alternatives (such as "accredit yourself" as an alternative to $100K college degrees in theater studies).
7. Nobody fully understands these complex systems such as the reserve currency, even though the systems have been functioning for decades. If we add up certain dynamics, we would feel very confident in saying this system should not exist--it should implode right now. Yet it continues on year after year. It is not just being perverse--it's very difficult to understand these systems because of the self-referential feedbacks and the motivations of the players to keep it going by whatever means are at hand.
I have been looking at the USD reserve currency for years and am humbled to realize nobody really has a firm grasp of all its dynamics. (At least I haven't found any such source.) There are widely disparate descriptions of its mechanisms and costs/problems, none of which totally accounts for its continued resilience.
8. I tend to think John Michael Greer's concept of catabolic collapse is likely to be the most correct in terms of predicting future dynamics. Things keep following the same vectors for all the reasons stated above until something gives and they reset at a lower level of complexity/energy consumption. Everyone with a stake in the current system takes a hit to their desires but they still retain a meaningful share of the swag. Those who lose their share are too marginalized to threaten the majority who still gain by participating in the status quo.
This stairstep-down process can continue for quite some time, Rome being a pretty good example and various corporate/nation-state failures being more recent examples.
9. It is fairly self-evident that we are in an unprecedented era--just looking at energy, debt and the Internet is enough to reach that conclusion. This means the past is not a very reliable guide. As a result, many people look to behavioral models of economics to explain everything in terms of human emotions and cognitive deficiencies. This also has limits, as systems include forces that may originate in human psychology but psychology and cognitive flaws do not account for the system's full dynamics.
10. New models of doing things are emergent, but that is not a passive process. Some individuals actually have to make this happen by thinking things out, proposing systems, setting up a network of like-minded people, etc. etc. etc. My goal is to part of the process of building alternative structures at least conceptually so people who are completely wedded to the status quo have some alternative framework to grasp when the stairstep down finally breaches their confidence/faith that the system is eternally sustainable as-is.
This occurs when they discover the status quo has deemed them inconsequential enough that their share of the swag can be reduced with no negative consequences to those still at the trough.
The Nearly Free University and The Emerging Economy:
The Revolution in Higher Education
Reconnecting higher education, livelihoods and the economy
With the soaring cost of higher education, has the value a college degree been turned upside down? College tuition and fees are up 1000% since 1980. Half of all recent college graduates are jobless or underemployed, revealing a deep disconnect between higher education and the job market.
It is no surprise everyone is asking: Where is the return on investment? Is the assumption that higher education returns greater prosperity no longer true? And if this is the case, how does this impact you, your children and grandchildren?
We must thoroughly understand the twin revolutions now fundamentally changing our world: The true cost of higher education and an economy that seems to re-shape itself minute to minute.
The Nearly Free University and the Emerging Economy clearly describes the underlying dynamics at work - and, more importantly, lays out a new low-cost model for higher education: how digital technology is enabling a revolution in higher education that dramatically lowers costs while expanding the opportunities for students of all ages.
The Nearly Free University and the Emerging Economy provides clarity and optimism in a period of the greatest change our educational systems and society have seen, and offers everyone the tools needed to prosper in the Emerging Economy.
Read the Foreword, first section and the Table of Contents.
print edition (list $20, now $18)
Kindle edition: list $9.95
Things are falling apart--that is obvious. But why are they falling apart? The reasons are complex and global. Our economy and society have structural problems that cannot be solved by adding debt to debt. We are becoming poorer, not just from financial over-reach, but from fundamental forces that are not easy to identify. We will cover the five core reasons why things are falling apart:
2. Crony capitalism
We are not powerless. Once we accept responsibility, we become powerful.
Kindle: $9.95 print: $24
Thank you, Mary S. ($5/month), for your wondrously generous subscription to this site-- I am greatly honored by your support and readership.


text:
Edward Roy Perl (October 6, 1926 – July 15, 2014) was an American neuroscientist whose research focused on neural mechanisms of and circuitry involved in somatic sensation, principally nociception. Work in his laboratory in the late 1960s established the existence of unique nociceptors. Perl was one of the founding members of the Society for Neuroscience and served as its first president. He was a Sarah Graham Kenan Professor of Cell Biology & Physiology and a member of the UNC Neuroscience Center at the University of North Carolina School of Medicine.

Early life and military training

Perl was born in Chicago, Illinois to John and Blanche Perl, natives of Hungary and Czechoslovakia, respectively.  As a child, Perl was fascinated by electricity, which led to an interest in electronics, radio, and the sciences.  In college at the University of Chicago, Perl focused on physics and engineering, but a conversation with his father, who was a physician and surgeon, convinced him to pursue a career in medicine as a means of studying human physiology.

While in college, Perl was accepted into the U.S. Navy's Officer Training Program.  He served as a medical trainee at the V-12 Navy College Training Program at the Great Lakes Naval Station (Chicago) in the summer of 1945 and began studies at the University of Illinois School of Medicine (Chicago) in the fall of 1945, at which time he was discharged into the naval reserves with the end of WWII.  Perl earned his Bachelor of Science degree in 1947 and his M.D. in 1949.

Early research career

Perl's first exposure to neuroscience came at the University of Illinois School of Medicine's Illinois Neuropsychiatric Institute (Chicago), where he worked for a time as a part-time graduate student in Warren S. McCulloch's laboratory and where he met, among other notables of the time,  Elwood Henneman, whose experiments on spinal reflexes and supraspinal control of motor function were to influence Perl's later research path.  A project undertaken in the laboratory of cardiac physiologist William V. Whitehorn in the late 1940s led to Perl's first scientific paper, published in Science in 1949. The principles behind the device Perl designed for this project became the foundation for impedance cardiography.  This work earned Perl a master's degree in 1951.

In the summer of 1948 Perl had served as a clerk on the Harvard Medical Service of Boston City Hospital, where his interactions with neurologist and neuroscience researcher Derek Denny-Brown steered Perl toward a career in neurophysiology.  Perl began a postdoctoral fellowship in the laboratory of Philip Bard in the Department of Physiology at Johns Hopkins University in the fall of 1950; there he met neuroanatomist Jerzy Rose and neurophysiologist Vernon Mountcastle, who would become a lifelong mentor in surgical and electrophysiological recording techniques.  During this time Perl became interested in how the activity of C-fiber afferent neurons was transferred to the cerebral cortex, a project that proved difficult, but which influenced his interest in these unmyelinated afferent fibers and their then-presumed participation in the detection and transmission of pain and temperature sensations to the brain.

Perl was called to active duty as a naval physician in January, 1952, and served as a medical officer at the Walter Reed Army Institute of Research, where he joined a neuroscience research group led by David McKenzie Rioch and staffed by Robert Galambos, Michael Fuortes, Walle Nauta, and David Whitlock.

First faculty positions

In 1954 Perl accepted a faculty position at the State University of New York, College of Medicine, Syracuse (now known as SUNY Upstate Medical University), where he not only researched crossed spinal reflexes, but also renewed his interest in the activity of C-fiber afferent fibers and their projections to the spinal cord.

Perl left SUNY-Syracuse in 1957 to join the University of Utah's Department of Physiology, then led by neurophysiologist Carlton C. Hunt.  Perl's work at Utah focused on spinothalamic somatosensory pathways and the interactions between primary afferent neurons and spinal dorsal column nuclei.

Beginning in 1962, Perl spent a year in Yves Laporte's laboratory at the Faculté de Médecine in Toulouse, France.  His time in Europe enabled him to meet with and observe French neurophysiologists Paul Bessou, Albert Fessard, Denise Albe-Fessard, Pierre Buser, Jean-Marie Besson, and Hungarian neuroanatomists János (John) Szentágothai and Miklós Réthelyi.  A visit with neurophysiologist Ainsley Iggo at the University of Edinburgh proved vital in allowing Perl expertise in recording the electrical activity of C-fibers.  He would repeat trips to France over the coming decades in order to engage in collaborative research with European colleagues.

Documentation of nociceptors

Perl's return to the University of Utah in 1963 marked the beginning of a research interest in primary afferent neurons, which evolved into a focus on nociceptors.  Paul Bessou visited Perl's laboratory and the two documented the activities of mechanoreceptive primary afferent neurons, whose thinly myelinated afferent fibers were responsive to non-noxious mechanical stimulation.  Pioneering experiments in cats with then-graduate student Paul Richards Burgess demonstrated the existence of a class of thinly myelinated primary-afferent fibers that only responded to noxious (nociceptive) mechanical stimulation; Burgess and Perl (1967) described in depth this class of high-threshold mechanoreceptor, using nociceptor, a term coined by Charles Sherrington in 1906, to identify these neurons. The work by Burgess and Perl represents the first thorough documentation of a large sample of nociceptors, primary afferent neurons that detect stimuli capable of causing tissue injury and transmit information about these insults centrally.  (Previous work by Ainsley Iggo had provided a small sample of primary afferent fibers that are now understood to have been C-polymodal nociceptors.)

Perl extended these studies to primate, showing the existence of high-threshold mechanoreceptors in squirrel monkey.  Further experiments with Bessou not only revealed the complexity of myelinated and unmyelinated nociceptors, but also thoroughly documented the existence and properties of C-polymodal nociceptors, which respond to a variety of noxious stimuli.  Later experiments with Lawrence Kruger and Mathius "Skip" J. Sedivec in Perl's laboratory in the late 1970s (after Perl had moved to the University of North Carolina) examined the fine structure of peripheral endings of cutaneous high-threshold mechanoreceptors (nociceptors) in cats.  In parallel with this work at the University of North Carolina, Perl and co-workers sought to correlate how stimulating individual cutaneous nociceptors in awake human volunteers is experienced by these subjects; the resulting study proved the linkage between activation of identified nociceptors and the subjective experience of pain in humans.

Central projections of nociceptors and nociceptive spinal circuitry

While still at the University of Utah, Perl and Burgess Christensen, then a post-doctoral fellow, determined that the marginal zone (lamina I) of the dorsal horn of the spinal cord contained neurons that were responsive to different kinds of noxious and innocuous stimuli from the periphery.  Experiments with Takao Kumazawa in the late 1960s into the mid-1970s confirmed in monkey observations about unmyelinated primary afferent fibers and their central projections that had earlier been seen in cats.  These studies made clear that areas of the superficial dorsal horn served as integration sites for nociceptive and non-nociceptive information received from the periphery.

Perl continued this work after he left the University of Utah to become chair of the Department of Physiology at the University of North Carolina in 1971.  In the mid-1970s Alan R. Light, Miklós Réthelyi, and Daniel Trevino joined Perl's laboratory to further map the central terminations of thinly-myelinated primary afferent neurons, to study their synaptic morphologies, and to characterize neurons in the dorsal horn of the spinal cord that were responsive to activity of these fibers.  In a methodological tour de force, Perl worked with Yasuo Sugiura and Chong Lee in the mid-1980s to physiologically characterize and label (with Phaseolus vulgaris leucoagglutinin) unmyelinated C-fibers.  These studies revealed for the first time a functional organization to the central termination pattern of unmyelinated afferents with different response profiles to skin stimulation.  Experiments performed by Christopher Honda, Siegfried Mense, and Perl in the early 1980s demonstrated that neurons located in specific areas of the cat thalamus were responsive to noxious stimulation of the skin of the hindlimb.  As a whole, studies in the Perl laboratory in the 1970s and 1980s helped clarify a specific pattern of somatosensory (principally nociceptive) input to the spinal cord and brain and established the foundation for a circuitry devoted to the processing of noxious stimuli from the periphery.

The last decades of work in the Perl laboratory were principally devoted to characterizing the functional organization of the superficial dorsal horn of the spinal cord and understanding how spinal neurons located within these regions interact with one another to process signals arising from the periphery.  These experiments involved recording from neurons responsive to various types of primary afferent input and correlating these functional signatures with morphological features of the spinal neurons in question.  This work in part resulted in the systematic categorization by Timothy Grudt and Perl of functionally characterized spinal neurons based on their morphological features and location within the dorsal horn.  Experiments with Yan Lu and Jihong Zheng were aimed at a better understanding of connections between spinal neurons and how afferent input from the periphery is modulated by these connections.  Perl's experiments with Adam Hantman focused on a unique, homogeneous population of Green fluorescent protein (GFP)-expressing neurons in the spinal substantia gelatinosa of a transgenic mouse.  Hantman and Perl physiologically characterized these neurons, showing them to be inhibitory in nature and responsive only to unmyelinated afferents with a conduction velocity at the high end of the C-fiber range; they also demonstrated the highly specific connections of these GFP-expressing neurons with other types of neurons in the substantia gelatinosa.

Founding of the Society for Neuroscience

At the suggestion of neurophysiologist Ralph W. Gerard, whose idea it was to establish a Society for Neuroscience, Perl chaired a committee of fellow neuroscientists in 1969, the aim of which was to lay the groundwork for the function of the nascent society.  As a founding member, Perl was elected president, but chose to take the title of acting president (1969–1970) until a president could be elected democratically by a representative membership. He felt that it was important for the society to attract young investigators who are active in the laboratory.

Awards and honors

Among other recognitions for his contributions to neuroscience, Perl was awarded the Bristol-Myers Squibb Award for Distinguished Research on Pain in 1991 and the Ralph W. Gerard Prize in Neuroscience in 1998.  He was elected a Fellow in the American Academy of Arts and Sciences in 1992.

Establishment of the Perl-UNC Prize

In 2000 Perl endowed a national prize to be given annually to investigators who have made significant contributions to neuroscience through outstanding discoveries or seminal insights. In establishing the Perl-UNC Neuroscience Prize, Perl noted that "the prize allows me to acknowledge the University of North Carolina for the opportunities it has given me" and, further, that it would be a tribute to the strength of the neuroscience research program at the University.  As of 2014, six recipients of the Perl-UNC Prize have gone on to win Nobel Prizes in either Physiology/Medicine (Linda Buck, Richard Axel, May-Britt Moser, Edvard Moser) or Chemistry (Roger Tsien, Roderick MacKinnon).

Further reading

 Edward Perl. Edward R. Perl,  The History of Neuroscience in Autobiography, Volume 3. Ed. Larry R. Squire. San Diego: Academic Press, 2001. pp. 366–413.
 Edward R. Perl. Ideas about Pain, a Historical View. Nature Reviews Neuroscience 8: 71-80, 2007.
 Edward Perl. Pain Mechanisms: a Commentary on Concepts and Issues.  Progress in Neurobiology 94: 20-38, 2011.

References

External links
 Edward R. Perl's entry at Neurotree.
 Perl-UNC Prize at UNC Neuroscience Center.

1926 births
2014 deaths
American neuroscientists
University of Illinois alumni
University of Chicago alumni

redpajama_set_name: RedPajamaWikipedia


Airloan is customer focused with knowledgeable team members and pilots that are passionate about general aviation. We work directly with customers throughout the entire process to ensure customer satisfaction. All loans are handled in a timely manner, allowing for flexibility to meet our customers' unique needs.


Vincent Jean Mpoy Kompany (French pronunciation: [vɛ̃sɑ̃ kɔ̃pani]; Dutch pronunciation: [vɪnsɛnt kɔmpani]; born 10 April 1986) is a Belgian professional footballer who plays as a centre back and captains the English club Manchester City and also plays Belgium national football team. In the 2011–12 season he was awarded the captaincy of Manchester City, leading his club to win the Premier League that season, their first league title in 44 years. He is also chairman of Belgium D3B Division club BX Brussels.
Kompany began his professional career at Anderlecht, spending three years at the Belgian club before moving to Bundesliga club Hamburg in 2006. In the summer of 2008, at the age of 22, he completed a transfer to his current club, Premier League side Manchester City. He has since established himself as an integral part of the Manchester City squad and is one of the bargain buys of the revolutionised City era, blossoming into one of the Premier League's best centre-backs. Kompany was included in the Premier League Team of the Year for two years in a row in 2011 and 2012 as well as being included in the 2014 team, and won the Premier League Player of the Season in 2012. Regarded as one of the best defenders in the world, he was listed 23rd in "The 100 Best Footballers in the World" by The Guardian.


text:
PHOENIXPRIDELGBTCENTER.org
LGBT in Phoenix Area
How to Date in Phoenix?
What is LGBT?
What To Do Around Phoenix?
Phoenix Scenic Guide
Phoenix is the capital of the state of Arizona—the ninth-largest" city in the United States. Located in the Sonoran Desert, Phoenix has a warm, dry climate that is perfect for outdoor activities like hiking and horseback riding. Being the "Valley of the Sun," it is also a hotbed of innovation. Phoenix is home to the largest airport in the state, an assortment of Fortune 500 companies, and numerous well-known companies in the tech sector, including Intel, Motorola, and Adobe.
Phoenix, like most other major cities, has a thriving gay scene. From the numerous bars and clubs to the myriad of LGBT-friendly restaurants and other entertainment options, it's no wonder that Phoenix regularly makes lists of the best places to live for LGBT individuals. Phoenix also has two major LGBT communities located in midtown Phoenix, one in the Central District and the other in the West Valley.
All Things Phoenix
Phoenix Restaurants
Copyright © All rights reserved.Theme Kiddie Care by Sensational Theme


text:
\section{Introduction}
\label{s1}

The properties of $\ensuremath{\mathcal{PT}}$-symmetric Hamiltonians have been observed in a wide
variety of laboratory experiments \cite{R1,R2,R3,R4,R5,R6,R7,R8,R9,R10}. For a
$\ensuremath{\mathcal{PT}}$-symmetric Hamiltonian having an unbroken $\ensuremath{\mathcal{PT}}$ symmetry a linear
$\ensuremath{\mathcal{PT}}$-symmetric operator $\ensuremath{\mathcal{C}}$ exists that obeys the following three algebraic
equations:
\begin{eqnarray}
\ensuremath{\mathcal{C}}^2&=&1, \label{e1}\\
\left[\ensuremath{\mathcal{C}},\ensuremath{\mathcal{PT}}\right]&=&0, \label{e2}\\
\left[\ensuremath{\mathcal{C}},H\right]&=&0. \label{e3}
\end{eqnarray}
Constructing the $\ensuremath{\mathcal{C}}$ operator is the key step in showing that time evolution
for a non-Hermitian $\ensuremath{\mathcal{PT}}$-symmetric Hamiltonian is unitary \cite{R11,R12}.

The $\ensuremath{\mathcal{C}}$ operator for a few nontrivial quantum-mechanical models has been
calculated exactly \cite{R13,R14,R15,R16,R17} by solving (\ref{e1})-(\ref{e3}).
However, in general this system of equations is extremely difficult to solve
analytically. Therefore, in most cases a perturbative approach has been adopted
for the solution of these equations \cite{R18,R19,R20,R21,R22}.

The standard approach to solving (\ref{e1})-(\ref{e3}) has been to express the
$\ensuremath{\mathcal{C}}$ operator in a simple and natural form as an exponential of a Dirac
Hermitian operator $Q$ multiplying the parity operator $\ensuremath{\mathcal{P}}$:
\begin{equation}
\ensuremath{\mathcal{C}}=e^Q\ensuremath{\mathcal{P}}.
\label{e4}
\end{equation}
Note that $e^{Q/2}$ is precisely the metric operator $\eta$ discussed in
Refs.~\cite{R23,R24,R25,R26,R27,R28}. This operator can be used to construct a
similarity transformation that maps the non-Hermitian Hamiltonian $H$ to an
isospectral Hermitian Hamiltonian \cite{R29}. If we seek a solution for $\ensuremath{\mathcal{C}}$ in
the form (\ref{e4}), we find that (\ref{e1}) and (\ref{e2}), which can be
thought of as {\it kinematical} equations because they hold for all choices of
$H$, imply that $Q(p,q)$ is an odd function of the momentum operator $p$ and an
even function of the coordinate operator $q$ \cite{R11,R12}. The problem is then
reduced to finding the solution to (\ref{e3}), which can be thought of as a {\it
dynamical} equation because it refers to the Hamiltonian $H$.

It is difficult to find a closed-form analytical solution to (\ref{e3}).
However, in the past this equation has been solved perturbatively as follows:
Express the Hamiltonian in the form $H=H_0+\epsilon H_1$ and treat $\epsilon$ as
a small parameter. Then, seek $Q$ as a formal perturbation series in {\it odd}
powers of $\epsilon$:
\begin{equation}
Q(p,q)=\sum_{j=0}^\infty\epsilon^{2j+1}Q_{2j+1}(p,q).
\label{e5}
\end{equation}

The obvious question to ask is, Why do only odd powers in $\epsilon$ appear in
the perturbation series (\ref{e5})? The explanation that has been given in the
past is that even powers of $\epsilon$ are excluded from the series because
$Q(p,q)$ is required to be odd in $p$ and even in $q$. The reasoning goes as
follows: In the quantum-mechanical cases that have been studied so far, such as
$H=H_0+i\epsilon q$ and $H=H_0+i\epsilon q^3$, the unperturbed Hamiltonian $H_0=
\mbox{$\textstyle{\frac{1}{2}}$} p^2+\mbox{$\textstyle{\frac{1}{2}}$} q^2$ is the harmonic-oscillator Hamiltonian. If there were a
term $Q_0\epsilon^0$ in the series (\ref{e5}), then $Q_0$ would satisfy the
commutation relation $[Q_0,H_0]=0$. The vanishing of this commutator implies
that $Q_0$ is a function of $H_0$, and thus it is an {\it even} function of $p$,
which shows that $Q_0=0$. Once it is established that $Q_0=0$, it is relatively
easy to show (see Ref.~\cite{R15}, for example) that $Q_{2j}=0$ ($j=1,\,2,\,3,\,
\ldots$). We show in this paper that this argument is actually incorrect; there
are indeed solutions to the commutator equation $[Q_0,H_0]=0$ that are {\it odd}
in $p$ -- infinitely many such solutions, in fact. It is precisely because of
the existence of these odd-$p$ solutions that the $\ensuremath{\mathcal{C}}$ operator is nonunique.

It has recently become clear that the nonuniqueness of the $\ensuremath{\mathcal{C}}$ operator has
important implications for the mathematical and physical interpretation of
$\ensuremath{\mathcal{PT}}$-symmetric quantum mechanics \cite{R30,R31,R32}. In Ref.~\cite{R31} it
is shown that if the $\ensuremath{\mathcal{C}}$ operator is nonunique, then it is unbounded, and in
this paper we verify this result explicitly.

The paper \cite{R33} is relevant because it discusses for the first time the
existence of multiple (nonunique) solutions to the commutator equation
(\ref{e3}). In Ref.~\cite{R33} it is shown that the {\it inhomogeneous}
commutator equation $[Q_1,H_0]=2H_1$ has an infinite number of particular
solutions, which differ from one another by solutions to the associated {\it
homogeneous} commutator equation $[X,H_0]=0$, and it recognizes that the
solutions $X$ are not necessarily functions of $H_0$ only. The importance of
nonunique solutions to commutator equations is also central to Ref.~\cite{R34},
where a particular time-operator solution $\Theta$ to the inhomogeneous
commutator equation $[\Theta,H]=i$ is called {\it minimal} and a classification
of the infinite number of associated nonminimal solutions is given.

The approach used in the current paper is based on the recognition that finding
multiple solutions for $Q_1$ in (\ref{e5}) is not the only way to demonstrate
nonuniqueness. Here, we introduce a clearer and more fundamental way to explain
the nonuniqueness of the $\ensuremath{\mathcal{C}}$ operator. We show that a more general way to
represent $Q$ is by the expansion
\begin{equation}
Q(p,q)=\sum_{j=0}^\infty\epsilon^jQ_j(p,q)
\label{e6}
\end{equation}
in which all nonnegative integer powers of $\epsilon$ appear. An advantage of
this new representation is that in the limit $\epsilon\to0$ we obtain an
infinite class of {\it exact} $\ensuremath{\mathcal{C}}$ operators for the quantum
harmonic-oscillator Hamiltonian $H_0=\mbox{$\textstyle{\frac{1}{2}}$} p^2+\mbox{$\textstyle{\frac{1}{2}}$} q^2$. Then, once we have
$Q_0$ for the harmonic-oscillator case, we can straightforwardly generalize this
result and verify that the operator $\ensuremath{\mathcal{C}}$ is nonunique.

This paper is organized as follows: In Sec.~\ref{s2} we show how to construct
exact and explicit closed-form solutions that are odd in $p$ and even in $q$
to the homogeneous commutator equation $[Q_0(p,q),H_0]=0$. Our result is that
there is a unique bounded $\ensuremath{\mathcal{C}}$ operator and a nonunique infinite class of
unbounded $\ensuremath{\mathcal{C}}$ operators for the quantum harmonic oscillator. The techniques
used in Sec.~\ref{s2} involve the formal summation of infinite series of
singular operators. However, in Sec.~\ref{s3} we verify the validity of the
formal calculations done in Sec.~\ref{s2} by applying powerful summation
techniques that are used to regulate divergent Feynman integrals. This
verification leads us to conjecture that it may be possible to apply the
principles of summation theory to extend and generalize the rigorous notions of
Cauchy sequences and completeness expansions, which are used in mathematical
Hilbert-space theory, to divergent sequences and series of vectors. Next, in
Sec.~\ref{s4} we develop the formal machinery needed to determine the higher
coefficients $Q_1$, $Q_2$, $\ldots$, in the expansion (\ref{e6}), and in
Sec.~\ref{s5} we concentrate on calculating $Q_1(p,q)$ for the specific case
$H_1=iq$. Finally, in Sec.~\ref{s6} we make some brief concluding remarks.

\section{Solutions to $[Q_0(p,q),H_0]=0$ that are odd in $p$ and even in $q$}
\label{s2}

A powerful strategy for solving operator equations of the form
\begin{equation}
[Q_0(p,q),H_0]=0
\label{e7}
\end{equation}
is to represent the solution $Q_0(p,q)$ as an infinite series of totally
symmetric operator basis functions $T_{m,n}$. The operators $T_{m,n}$ are
described in detail in Refs.~\cite{R34,R35,R36,R37}. However, to make the
presentation in this paper self-contained, we recall that for $m,n\geq0$ the
operator $T_{m,n}$ is defined as a symmetric average over all orderings of $m$
factors of $p$ and $n$ factors of $q$:
\begin{eqnarray}
T_{1,1}&=&\mbox{$\textstyle{\frac{1}{2}}$}(pq+qp),\nonumber\\
T_{1,2}&=&\textstyle{\frac{1}{3}}(pqq+qpq+qqp),\nonumber
\end{eqnarray}
and so on. The operators $T_{m,n}$ obey simple commutation and anticommutation
relations:
\begin{eqnarray}
\left[p,T_{m,n}\right]&=&-inT_{m,n-1},\nonumber\\
\left[q,T_{m,n}\right]&=&imT_{m-1,n},\nonumber\\
\left\{p,T_{m,n}\right\}&=&2T_{m+1,n},\nonumber\\
\left\{q,T_{m,n}\right\}&=&2T_{m,n+1},\nonumber\\
\left[p^2,T_{m,n}\right]&=&-2inT_{m+1,n-1},\nonumber\\
\left[q^2,T_{m,n}\right]&=&2imT_{m-1,n+1},
\label{e8}
\end{eqnarray}
where the curly brackets indicate anticommutators. The operator $T_{m,n}$ can be
re-expressed in Weyl-ordered form \cite{R36}:
\begin{equation}
T_{m,n}=\frac{1}{2^m}\sum_{k=0}^m\binom{m}{k}p^kq^np^{m-k}
=\frac{1}{2^n}\sum_{k=0}^n\binom{n}{k}q^k p^m q^{n-k},
\label{e9}
\end{equation}
where $m,\,n=0,\,1,\,2,\,3,\,\cdots$. Introducing the Weyl-ordered form of
$T_{m,n}$ allows one to extend the operators $T_{m,n}$ either to negative values
of $n$ by using the first sum or to negative values of $m$ by using the second
sum. The commutation and anticommutation relations in (\ref{e8}) remain valid
when $m$ is negative or when $n$ is negative.

To find solutions that are odd in $p$ and even in $q$ to the commutator equation
(\ref{e7}), we take $Q_0(p,q)$ to have the general form
\begin{equation}
Q_0^{(\gamma)}(p,q)=\sum_k a_k^{(\gamma)}T_{2\gamma+1-2k,2k},
\label{e10}
\end{equation}
where $\gamma=0,\,\pm1,\,\pm2,\,\ldots$ is a parameter. Substituting (\ref{e10})
into (\ref{e7}), we obtain the following two-term recursion relation for the
coefficients $a_k^{(\gamma)}$:
\begin{equation}
a_{k+1}^{(\gamma)}(k+1)-(\gamma-k+1/2)a_k^{(\gamma)}=0\quad(k=0,1,2,\ldots).
\label{e11}
\end{equation}

This recursion relation is self-terminating; that is, if we choose $a_{-1}^{(
\gamma)}=0$, then $a_0^{(\gamma)}$ is an arbitrary constant, $a_k^{(\gamma)}$
vanishes for $k<0$, and $a_k^{(\gamma)}$ for $k>0$ is determined in terms of
$a_0^{(\gamma)}$ as the solution to the recursion relation (\ref{e11}):
\begin{equation}
a_k^{(\gamma)}=a_0^{(\gamma)}(-1)^k\frac{\Gamma(k-\gamma-1/2)}{k!\Gamma(-\gamma-
1/2)}\quad(k=0,1,2,\ldots).
\label{e12}
\end{equation}
The series (\ref{e10}) with coefficients (\ref{e12}) can be summed as a binomial
expansion:
\begin{equation}
\sum_{k=0}^\infty a_k^{(\gamma)}x^{2k}=a_0^{(\gamma)}\left(1+x^2\right)^{\gamma+1/2}.
\label{e13}
\end{equation}
Thus, for each $\gamma\geq0$ the odd-$p$ and even-$q$ one-parameter family
of solutions to the homogeneous commutator equation (\ref{e7}) is
\begin{equation}
Q_0^{(\gamma)}=\frac{a_0^{(\gamma)}}{2^{2\gamma+2}}\left\{\ldots\left\{\left\{
\left(1+q\frac{1}{p}q\frac{1}{p}\right)^{\gamma+1/2}+\left(1+\frac{1}{p}q\frac{
1}{p}q\right)^{\gamma+1/2},p\right\},p\right\}\dots,p\right\}_{(2\gamma+1)~{\rm
times}},
\label{e14}
\end{equation}
where we have used the identity \cite{R34}
\begin{equation}
T_{-n,n}=\frac{1}{2}\left(q\frac{1}{p}\right)^n+\frac{1}{2}\left(\frac{1}{q}p
\right)^n.
\label{e15}
\end{equation}

As stated in Sec.~\ref{s1}, we can see that while $Q_0^{(\gamma)}$ commutes with
the Hamiltonian $H_0=\mbox{$\textstyle{\frac{1}{2}}$} p^2+\mbox{$\textstyle{\frac{1}{2}}$} q^2$, it is {\it not} a function of $H_0$
because by construction it is odd in $p$. Furthermore, while the construction of
the solutions in (\ref{e14}) involves series in inverse powers of $p$, these
solutions are well behaved as $p\to0$. To see explicitly the oddness in $p$ we
display the solution corresponding to $\gamma=0$:
\begin{equation}
Q_0^{(0)}=\frac{1}{4}a_0^{(0)}\left(\sqrt{1+q\frac{1}{p}q
\frac{1}{p}}\,\,p+p\,\sqrt{1+q\frac{1}{p}q\frac{1}{p}}+\sqrt{1+\frac{1}{p}q
\frac{1}{p}q}\,\,p+p\,\sqrt{1+\frac{1}{p}q\frac{1}{p}q}\right).
\label{e16}
\end{equation}
In the classical limit for which $p$ and $q$ become commuting numbers, this
solution becomes
\begin{equation}
Q_{0,{\rm classical}}^{(0)}=a_0^{(0)}{\rm sgn}(p)\sqrt{p^2+q^2};
\label{e17}
\end{equation}
the oddness in $p$ is evident.

It is important to point out that for the harmonic oscillator, which corresponds
to $\epsilon=0$, the metric operator $\eta=e^{Q_0}$ is just unity when $Q_0=0$.
Thus, the metric operator is bounded of this special case. However, for $Q_0$ in
(\ref{e16}), the metric operator is no longer bounded, but rather for large $q$
it behaves like $e^q$ and for large $p$ it behaves like $e^p$. Needless to say,
since there is an infinite number of possible choices for $Q_0$, there is an
infinite number of possible metric operators. Only one of the metric operators
is bounded \cite{R38,R39,R40}.

\section{Using summation techniques to verify results of Sec.~\ref{s2}}
\label{s3}

We observed in Sec.~\ref{s2} that even though solutions $Q_0^{(\gamma)}$ in
(\ref{e14}) were constructed by performing a formal infinite sum over arbitrary
powers of the inverse momentum operator $1/p$, these solutions are well behaved
as $p\to0$. However, the calculations in Sec.~\ref{s2} are certainly not
rigorous. The aim of this section is to provide mathematical support for the
validity of the formulas in (\ref{e14}). Specifically, since the $\ensuremath{\mathcal{C}}$ operator
commutes with the Hamiltonian, the $n$th eigenstate $|\psi_n\rangle$ of the
Hamiltonian must also be an eigenstate of $\ensuremath{\mathcal{C}}$. We expect that the eigenvalue
of $|\psi_n\rangle$ is $(-1)^n$. For this to be true, $|\psi_n\rangle$ must be
an eigenstate of $Q_0^{(\gamma)}$ with eigenvalue $0$ for all $n$:
\begin{equation}
Q_0^{(\gamma)}|\psi_n\rangle=0.
\label{e18}
\end{equation}
In this section we show by explicit calculation in which we use powerful
summation techniques that this is indeed the case. Here, we limit our
calculation to the case $\gamma=0$. We first consider the ground state $|\psi_0
\rangle$ and then generalize to the $n$th eigenstate.

From (\ref{e10}) and (\ref{e12}) we see that $Q_0^{(0)}$ is given by
\begin{equation}
Q_0^{(0)}=\sum_{k=0}^\infty a_k T_{1-2k,2k},\qquad a_k=a_0\frac{(-1)^k\Gamma
(k-1/2)}{2\sqrt{\pi}k!}.
\label{e19}
\end{equation}
Also, the unnormalized eigenfunctions of the harmonic-oscillator Hamiltonian in
coordinate space are given by $\psi_n(q)=H_n(q)e^{-q^2/2}$. While the formal sum
in (\ref{e19}) can be written as (\ref{e16}), it is difficult to use this result
to verify the eigenvalue equation (\ref{e18}). A better strategy is to calculate
the action of each term in the sum in (\ref{e19}) on the eigenstates and then to
perform the summation over $k$.

Because inverse powers of the momentum operator arise in (\ref{e19}), it is most
convenient to work in the momentum representation, where the eigenvalue equation
(\ref{e18}) becomes
\begin{equation}
\langle p|Q_0^{(0)}|\psi_n\rangle=\frac{1}{4}\sum_{k=0}^\infty a_k (-1)^k
\left\{p\left(\frac{1}{p}\partial_p\right)^{2k}+ p\left(\partial_p\frac{1}{p}
\right)^{2k}+\left(\frac{1}{p}\partial_p\right)^{2k}p+\left(\partial_p\frac{1}
{p}\right)^{2k}p \right\}\tilde{\psi}_n(p)=0,
\label{e20}
\end{equation}
where we have used (\ref{e15}) and the third formula in (\ref{e8}). Here,
$\tilde{\psi}_n(p)=(-i)^n\psi_n(p)$, where $\tilde{\psi}$ is the Fourier
transform of $\psi$.

\subsection{Ground state $\psi_0$}
\label{ss3a}

Let we first study the simplest case $n=0$. We can see that the action of each
term in the series expansion (\ref{e20}) on the eigenstate $\psi_0$ produces
more and more negative powers of the momentum $p$:
\begin{eqnarray}
Q_0^{(0)}(p)\tilde{\psi}_0(p)&=&\frac{1}{4}\left\{4p\,a_0-\left(4p+\frac{2}{p^3}
\right)a_1+\left(4p+\frac{12}{p^3}+\frac{48}{p^5}+\frac{90}{p^7}\right)a_2-
\left(4p+\frac{30}{p^3}+\frac{240}{p^5}+\frac{1350}{p^7}+\frac{5040}{p^9}\right)
a_3\right.\nonumber\\
&+&\left.\left(4p+\frac{56}{p^3}+\frac{672}{p^5}+\frac{6300}{p^7}+\frac{47040}
{p^9}+\frac{264600}{p^{11}}+\frac{997920}{p^{13}}+\frac{1891890}{p^{15}}\right)
a_4-\ldots\right\}e^{-p^2/2}.\nonumber\\
\label{e21}
\end{eqnarray}
We can rearrange the series in (\ref{e21}) to read
\begin{equation}
Q_0^{(0)}(p)\tilde{\psi}_0(p)=\frac{1}{4}\left\{4p\sum_{k=0}^\infty(-1)^k a_k-
\sum_{k=0}^\infty\frac{(-1)^k}{p^{4k+3}}a_{k+1}P_{2k}(2p^2)\right\}e^{-p^2/2},
\label{e22}
\end{equation}
where $P_n(y)$ are polynomials:
\begin{equation}
P_{n}(y)=\sum_{\alpha=0}^n \frac{(n+1)!(2\alpha+2)!}{2^\alpha \alpha! (\alpha+2)!(n-\alpha)!}y^{n-\alpha}.
\label{e23}
\end{equation}

To verify that $Q_0^{(0)}(p)\tilde{\psi}_0(p)=0$, we must show that
\begin{eqnarray}
4\pi p\sum_{k=0}^\infty\frac{\Gamma(k-1/2)}{k!}-\frac{1}{p^3}\sum_{\alpha=0
}^\infty\frac{1}{p^{2\alpha}}\frac{(2\alpha+2)!}{\alpha!(\alpha+2)!2^\alpha}
\sum_{k=0}^\infty\frac{\Gamma(2k+1)\Gamma(k+3/2)}{\Gamma(k+1)\Gamma(2k-\alpha+1)
}=0,
\label{e24}
\end{eqnarray}
where we have substituted the above formulas for the polynomials $P_n$ and the
coefficients $a_k$. It is easy to verify that the exact sum of the convergent
series $\sum_{k=0}^\infty\Gamma(k-1/2)/k!$ is zero. However, the series
$\sum_{k=0}^\infty\frac{\Gamma(2k+1)\Gamma(k+3/2)}{\Gamma(k+1)\Gamma(2k-\alpha
+1)}$ is divergent for $\alpha\geq3/2$. Therefore, we must introduce a summation
procedure to make sense of this series.

\begin{figure}[b!]
\begin{center}
\includegraphics[scale=0.50]{fig1.jpg}
\end{center}
\caption{A plot of $g(D)$ in (\ref{e26}) for $0<D<6$. Note that $g(3)$ vanishes
even though the integrand of $I(D)$ in (\ref{e25}) is strictly positive for all
$D$. This shows that the notions of positivity and negativity evaporate in the
case of a divergent integral representation.}
\label{F1}
\end{figure}

Our summation procedure is a discrete variant of dimensional continuation, a
technique that is used to interpret divergent Feynman integrals. To illustrate
our approach, let us consider the following $D$-dimensional integral:
\begin{equation}
I(D)=\int d^Dx\frac{x^2+3}{(x^2+1)^2}.
\label{e25}
\end{equation}
This integral converges for $D<2$ and its exact value is $I(D)=S_D\,g(D)$,
where $S_D=2\pi^{D/2}/\Gamma(D/2)$ is the surface area of a $D$-dimensional
sphere of radius $1$ and
\begin{equation}
g(D)=\frac{(3-D)\pi}{2\sin(\pi D/2)}.
\label{e26}
\end{equation}
Evidently, even though the integral representation for $I(D)$ diverges for
$D\geq2$, the function $I(D)$ is well defined and analytic in $D$ except for
isolated simple poles. [Figure \ref{F1} gives a plot of $g(D)$ for $0<D<6$.]
Observe that $I(D)$ vanishes for $D=3$. This result is surprising and somewhat
counterintuitive because the integrand of $I(D)$ is {\it strictly positive} when
$D=3$. The vanishing of $I(3)$ shows that when an integral representation is
divergent we cannot draw qualitative conclusions regarding the sign of its
value. (Indeed, the Borel sum of the series $1+2+4+8+\dots$ is uniquely $-1$
even though all of the terms in this divergent series are positive!)

We will now show that while the sum over $k$ in (\ref{e24}) diverges for $\alpha
>-3/2$, we can evaluate the sum for $\alpha<-3/2$ and then use analytic
continuation in $\alpha$ to sum the series for $\alpha=0,\,1,\,2,\,3,\,\ldots$.
The surprising and counterintuitive result is that while the summand is
positive for $2k+1>\alpha$, the sum of the (divergent) series {\it vanishes} for
all nonnegative integer values of $\alpha$.

The divergent series to be summed is
\begin{equation}
F_\alpha=\sum_{k=0}^\infty\frac{\Gamma(2k+1)\Gamma(k+3/2)}{\Gamma(k+1)\Gamma(
2k-\alpha+1)}.
\label{e27}
\end{equation}
To perform the sum we first express the coefficients in terms of the beta
function:
\begin{equation}
B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)},
\label{e28}
\end{equation}
whose integral representation is
\begin{equation}
B(x,y)=\int_0^1 dt\,t^{(x-1)}\,(1-t)^{(y-1)}\qquad({\rm Re}(x),~{\rm Re}(y)>0).
\label{e29}
\end{equation}
By multiplying and dividing $F_\alpha$ by $\Gamma(-\alpha)$ we obtain the result
\begin{equation}
F_\alpha=\frac{1}{\Gamma(-\alpha)}\sum_{k=0}^\infty\frac{\Gamma(k+3/2)}{\Gamma(
k+1)}B(2k+1,-\alpha)=\frac{1}{\Gamma(-\alpha)}\sum_{k=0}^\infty\frac{\Gamma(k+3/
2)}{\Gamma(k+1)}\int_0^t dt\,t^{2k}(1-t)^{-\alpha-1}.
\label{e30}
\end{equation}
We then use the binomial expansion
\begin{equation}
\sum_{k=0}^\infty\frac{\Gamma(k+3/2)}{k!}t^{2k}=\frac{\sqrt{\pi}}{2}(1-t^2)^{-3
/2}
\label{e31}
\end{equation}
to show that
\begin{equation}
F_\alpha=\frac{\sqrt{\pi}}{2\Gamma(-\alpha)}\int_0^1 dt(1-t)^{-\alpha-5/2}
(1+t)^{-3/2}=\frac{\sqrt{\pi}\,\Gamma(-\alpha-3/2)}{2\Gamma(-\alpha)}{}_2F_1
\left(\frac{3}{2},1;-\alpha-\frac{1}{2},-1\right).
\label{e32}
\end{equation}
This function vanishes for all nonnegative integer values of $\alpha$ because
the hypergeometric function is finite for these values of $\alpha$. In
Fig.~\ref{F2} we plot $F_\alpha$ for $-5\leq\alpha\leq10$. Observe that
$F_\alpha$ vanishes for $\alpha=0,\,1,\,2,\,\ldots$. Note also that $F_\alpha$
is singular at $\alpha=-3/2$, the value of $\alpha$ for which the series
(\ref{e27}) begins to diverge. Interestingly, this function has {\it double}
poles at the half-odd integers $\alpha=1/2,\,3/2,\,5/2,\,\ldots$. As $\alpha$
increases, the double poles begin to resemble single poles

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.60]{fig2.jpg}
\end{center}
\caption{A plot of $F_\alpha$ in (\ref{e32}) for $-5<\alpha<10$. Note that
$F_\alpha$ vanishes at every nonnegative integer. This happens because $\Gamma(
-\alpha)$ in the denominator is infinite when $\alpha=0,\,1,\,2,\,\ldots$ and
the hypergeometric function is finite. These zeros are all simple zeros, but the
poles at the half-odd integers beginning with $1/2$ are all double poles, as is
verified in Fig.~\ref{F3}.}
\label{F2}
\end{figure}

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.55]{fig3.jpg}
\end{center}
\caption{A blow-up of the region near $\alpha=9.5$ in Fig.~\ref{F2}. In the left
panel is a plot of $(\alpha-9.5)F_\alpha$ for $9.496<\alpha<9.505$ and in the
right panel is a plot of $(\alpha-9.5)^2F_\alpha$ for the same range of
$\alpha$. Note that the left plot has a typical simple-pole behavior at
$\alpha=9.5$ while the graph in the right panel is finite at $\alpha=9.5$.}
\label{F3}
\end{figure}

\subsection{Generalization: eigenvalue equation for $\psi_n$}
\label{ss3b}

In this subsection we study the action of the operator $Q^{(0)}_0$ on the $n$th
eigenstates of the harmonic-oscillator Hamiltonian. We describe first the case
for which $n$ is even. (The odd-$n$ case is treated in an identical fashion
and is considered briefly at the end of this subsection.) For even $n$
\begin{equation}
Q_0^{(0)}\tilde{\psi}_{2n}=\left[-\sum_{k=1}^\infty \frac{(-1)^k }{p^{4k-1}}a_k
J^{(n)}_{2k-1}(p^2)+\sum_{j=0}^n p^{2j+1}\sum_{k=0}^\infty(-1)^k a_k S^{(n)}_{
n-j}(k)\right]e^{-p^2/2}\quad(n=1,2,3,\ldots).
\label{e33}
\end{equation}
Both the sums over $k$ in (\ref{e33}) diverge for $\alpha=0,1,2,\ldots$ except
for $j=n$ in the first series. (The special case $j=n$ gives the convergent
series that was already considered in Subsec.~\ref{ss3a}.) The divergent series
can be evaluated by using the summation procedure that was introduced in the
previous subsection. Following the procedure for the $n=0$ case, we will show
that if we write (\ref{e33}) as
\begin{equation}
Q_0^{(0)}\tilde{\psi}_{2n}=\left[-\ensuremath{\mathcal{N}}^{(n)}+\sum_{j=0}^n p^{2j+1}\ensuremath{\mathcal{M}}^{(n,j)}
\right]e^{-p^2/2},
\label{e34}
\end{equation}
then both $\ensuremath{\mathcal{M}}^{(n,j)}$ and $\ensuremath{\mathcal{N}}^{(n)}$ vanish for $n=1,2,\ldots$, $j=0,1,\ldots
n$.

We first consider the series
\begin{equation}
\ensuremath{\mathcal{N}}^{(n)}= \sum_{k=1}^\infty\frac{(-1)^k }{p^{4k-1}}a_k J^{(n)}_{2k-1}(p^2),
\label{e35}
\end{equation}
where $J^{(n)}_k(x)$ are $k$th order polynomials that can be written as
\begin{equation}
J^{(n)}_k(x)=(2n-1)!!\frac{2^{n+1}}{\sqrt{\pi}}\sum_{\alpha=0}^k\frac{2^\alpha
(k+1)!\Gamma(\alpha+1/2)}{(k-\alpha)!(n+\alpha+1)!}E_{n,\alpha}(k)x^{k-\alpha},
\label{e36}
\end{equation}
and $E_{n,\alpha}(k)$ are polynomials of degree $n$ in the variable $k$. The
first four polynomials are
\begin{eqnarray}
E_{1,\alpha}(k)&=& 2(1-\alpha)k+\alpha^2-4\alpha, \nonumber \\
E_{2,\alpha}(k)&=& (4\alpha-8)k^2-(4\alpha^2-20\alpha+4)k+\alpha^3-7\alpha^2+10\alpha, \nonumber \\
E_{3,\alpha}(k)&=& (24-8\alpha)k^3+(12\alpha^2-72\alpha+24)k^2-(6\alpha^2-48\alpha^2+70\alpha-24)k+\alpha^4-9\alpha^3+20\alpha^2-48\alpha,\nonumber\\
E_{4,\alpha}(k)&=& (16\alpha-64)k^4-(32\alpha^2-224\alpha+96)k^3+(24\alpha^3-216\alpha^2+320\alpha-224)k^2\nonumber \\
&&-(8\alpha^4-80\alpha^3+176\alpha^2-536\alpha+96)k+\alpha^5-10\alpha^4+23\alpha^3-158\alpha^2+216\alpha.
\label{e37}
\end{eqnarray}
In terms of $E_{n,\alpha}(2k-1)=\sum_{\gamma=1}^n e_{\alpha,\gamma}k^\gamma$,
the first term in (\ref{e34}) can be written as
\begin{equation}
\ensuremath{\mathcal{N}}^{(n)}=\frac{2}{\pi}(2n-1)!!\sum_{\alpha=0}^\infty \frac{2^\alpha}{p^{2\alpha
+1}}\frac{\Gamma(\alpha+1/2)}{(n+\alpha+1)!}\sum_{\gamma=1}^n e_{\alpha,\gamma}
\,\ensuremath{\mathcal{N}}_{\alpha,\gamma},
\label{e38}
\end{equation}
where $\ensuremath{\mathcal{N}}_{\alpha,\gamma}$ is given by
\begin{equation}
\ensuremath{\mathcal{N}}_{\alpha,\gamma}=\sum_{k=1}^\infty\frac{\Gamma\left(k-\frac{1}{2}\right)
\Gamma(2k+1)}{\Gamma(k-1)\Gamma(2k-\alpha)}k^\gamma.
\label{e39}
\end{equation}
This is the divergent series that we need to study.

Because $\ensuremath{\mathcal{N}}_{\alpha,\gamma}$ is divergent for $\alpha\geq-5/2-\gamma$, we
evaluate the sum for $\alpha<-5/2-\gamma$ and use analytic continuation in
$\alpha$ to sum the series for $\alpha=0,1,2,\ldots$. Multiplying and dividing
$\ensuremath{\mathcal{N}}_{\alpha,\gamma}$ by $\Gamma(-\alpha-1)$ and using the integral
representation of the beta function $B$, we obtain
\begin{equation}
\ensuremath{\mathcal{N}}_{\alpha,\gamma}=\frac{1}{\Gamma(-\alpha-1)}\int_0^1 dt\,(1-t)^{-\alpha-2}t^2
\sum_{k=0}^\infty \frac{\Gamma(k+1/2)k^\gamma}{\Gamma(k)}t^{2k}.
\label{e40}
\end{equation}
The sum over $k$ in (\ref{e40}) gives
\begin{equation}
\sum_{k=0}^\infty\frac{\Gamma(k+1/2)k^\gamma}{\Gamma(k)}t^{2k}=\frac{\sqrt{\pi}}
{2}t^4\,{}_{\gamma+1}F_\gamma\left(\frac{3}{2},2,\ldots;1,\ldots;t^2\right),
\label{e41}
\end{equation}
where the first dots in the hypergeometric functions stand for $(\gamma-1)$-twos
and the other dots stand for $(\gamma-1)$-ones. For fixed $\gamma$, the
hypergeometric function in (\ref{e41}) can be written as $L_\gamma(t^2)
2^{-\gamma}(1-t^2)^{-3/2-\gamma}$, where $L_\gamma(t^2)=\sum_{\sigma=1}^\gamma
\ell_{\sigma,\gamma}t^{2\sigma}$ is a polynomial of order $\gamma$ in the
variable $t^2$. In terms of $L_\gamma(t^2)$ the series (\ref{e39}) becomes
$\ensuremath{\mathcal{N}}_{\alpha,\gamma}=\sum_{\sigma=1}^\gamma\ell_{\sigma,\gamma}\ensuremath{\mathcal{N}}_{\alpha,
\gamma,\sigma}$, where
\begin{eqnarray}
\ensuremath{\mathcal{N}}_{\alpha,\gamma,\sigma}&=&\frac{\sqrt{\pi}}{2}\frac{1}{\Gamma(-\alpha-1)}
\int_0^1 dt\,t^{4+2\sigma}(1-t)^{-\alpha-\gamma-7/2}(1+t)^{-\frac{3}{2}-\gamma}
\nonumber\\
&=&\frac{\sqrt{\pi}}{2}\frac{B\left(5+2\sigma,-\alpha-\gamma-\frac{5}{2}\right)}
{\Gamma(-\alpha-1)}{}_2F_1\left(\gamma+\frac{3}{2},5+2\sigma;-\alpha-\gamma+2
\sigma+\frac{5}{2};-1\right).
\label{e42}
\end{eqnarray}
The function $\ensuremath{\mathcal{N}}_{\alpha,\gamma,\sigma}$ vanishes for all nonnegative integers
$\alpha$, $\gamma$, and $\sigma$ because the denominator becomes infinite while
the hypergeometric function is finite. The special case $\ensuremath{\mathcal{N}}_{\alpha,1,1}$ is
plotted as a function of $\alpha$ in Figs.~\ref{F4} and Fig.~\ref{F5}. The
vanishing of $\ensuremath{\mathcal{N}}_{\alpha,\gamma,\sigma}$ guarantees that the first sum in
(\ref{e33}) is identically zero.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.60]{fig4.jpg}
\end{center}
\caption{A plot of $\ensuremath{\mathcal{N}}_{\alpha,1,1}$ in (\ref{e42}) for $-6<\alpha<8$. Like the
special case displayed in Fig.~\ref{F2}, $\ensuremath{\mathcal{N}}_{\alpha,1,1}$ vanishes for all
nonnegative integers. However, $\ensuremath{\mathcal{N}}_{\alpha,1,1}$ is different from $F_\alpha$
in that it has simple poles rather than double poles. It is not completely
obvious that $\ensuremath{\mathcal{N}}_{\alpha,1,1}$ vanishes when $\alpha=0,\,1,\,2,\,\ldots$, so in
Fig.~\ref{F5} the behavior of $\ensuremath{\mathcal{N}}_{\alpha,1,1}$ near $\alpha=5$ is blown up.}
\label{F4}
\end{figure}

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.60]{fig5.jpg}
\end{center}
\caption{A blow up of the graph of $\ensuremath{\mathcal{N}}_{\alpha,1,1}$ in Fig.~\ref{F4} near
$\alpha=5$. Observe that there is a zero at exactly $\alpha=5$ as well as an
additional zero near and to the left of $\alpha=5$.}
\label{F5}
\end{figure}

Next, we evaluate the sum over $k$ in the divergent series in the second term in
(\ref{e33}):
\begin{equation}
\ensuremath{\mathcal{M}}=\sum_{k=0}^\infty (-1)^k a_k S^{(n)}_{n-j}(k),
\label{e43}
\end{equation}
where $S^{(n)}_{n-j}(k)=\sum_{\ell=0}^{n-j}s_{n,\ell}k^\ell$ are polynomials of
degree $n-j$ in the variable k. The polynomials are listed below for $n=1,2,3$:
\begin{eqnarray}
S_1^{(1)}(k)&=& 8(8k+1),\quad S^{(1)}_0(k)=16, \nonumber \\
S^{(2)}_2(k)&=&16(68k^2+14k+3),\quad S^{(2)}_1(k)=-64(8k+3),\quad S^{(2)}_0(k)=
64,\nonumber\\
S^{(3)}_3(k)&=&96(192k^2+52k^2+46k+5),\quad S^{(3)}_2(k)=-64(196k^2+142k+45),
\nonumber\\
&& \,\, S^{(3)}_1(k)=384(8k+5),\quad S^{(3)}_0(k)=-256.
\label{e44}
\end{eqnarray}
As noted above, the series for $j=n$ [the highest power in $p$ in (\ref{e33})]
are convergent and their exact sum is zero. For the divergent cases, the series
(\ref{e43}) can be written as $\ensuremath{\mathcal{M}}=\sum_{\ell=0}^{n-j}s_{n,\ell}\ensuremath{\mathcal{M}}_\ell$, where
$\ensuremath{\mathcal{M}}_\ell$ is the divergent series
\begin{equation}
\ensuremath{\mathcal{M}}_\ell=\sum_{k=0}^\infty\frac{\Gamma(k-1/2)}{k!}k^{\ell}.
\label{e45}
\end{equation}
This series can be rewritten in the form
\begin{equation}
\ensuremath{\mathcal{M}}_\ell=-\frac{1}{2\sqrt{\pi}}\int_0^1 dt(1-t)^{-3/2} t^{-3/2} \sum_{k=0}^\infty t^k k^\ell.
\label{e46}
\end{equation}

Summing the series in (\ref{e46}) over $k$, we obtain $\sum_{k=0}^\infty t^k
k^\ell=(1-t)^{-\ell-1}\sum_{\sigma=0}^\ell r_{\ell,\sigma}t^\sigma$. Thus,
$\ensuremath{\mathcal{M}}_\ell$ can be written as $\ensuremath{\mathcal{M}}_\ell=\sum_{\sigma=0}^\ell r_{\sigma,\ell}
\ensuremath{\mathcal{M}}_{\ell,\sigma}$, where
\begin{equation}
\ensuremath{\mathcal{M}}_{\sigma,\ell}=-\frac{1}{2\sqrt{\pi}}\int_0^1 dt(1-t)^{-5/2-\ell}
t^{-3/2+\sigma}=-\frac{1}{2\sqrt{\pi}}B(\sigma-1/2,-\ell-3/2),
\label{e47}
\end{equation}
which vanishes at zero for $(\sigma,\ell)=0,\,1,\,2,\ldots$.

Finally, we consider the case of odd $n$. For this case we get
\begin{equation}
Q_0\tilde{\psi}_{2n+1}=\left[\sum_{j=0}^{n+1}p^{2j}\sum_{k=0}^\infty (-1)^k
a_k V^{(n)}_{n-j}(k)+\sum_{k=0}^\infty \frac{(-1)^k }{p^{4k+2}}a_k O^{(n)}_{2k}
(p^2)\right]e^{-p^2/2}.
\label{e48}
\end{equation}
Here, $V^{(n)}_{n-j}(k)$ are polynomials of degree $n-j$ in the variable k:
\begin{eqnarray}
V_1^{(1)}(k)&=&16k,\quad V^{(1)}_0(k)=-8, \nonumber \\
V^{(2)}_2(k)&=&16(14k^2-k),\quad V^{(2)}_1(k)=-48(4k+1), \quad V^{(2)}_0(k)=32,\nonumber \\
V^{(3)}_3(k)&=&128(26k^3-4k^2+3k),\quad V^{(3)}_2(k)=-32(124k^2+58k+15),
\nonumber\\
&&\,\, V^{(3)}_1(k)=640(2k+1),\quad V^{(3)}_0(k)=-128.
\label{e49}
\end{eqnarray}
Also, the polynomials $O^{(n)}_k(x)$ can be written as
\begin{eqnarray}
O^{(n)}_k(x)=\frac{2^{n+3}}{\sqrt{\pi}}(2n+1)!!\sum_{\alpha=0}^k \frac{2^\alpha\Gamma(\alpha+3/2)(k+2)!}{(k-\alpha)!(n+\alpha+2)!}U_n^{(\alpha)}(k),
\label{e50}
\end{eqnarray}
where $U_n^{(\alpha)}(k)$ are polynomials of degree $n$ in the variable $k$. The
first four such polynomials are
\begin{eqnarray}
U^{(\alpha)}_1(k)&=&1,\nonumber \\
U^{(\alpha)}_2(k)&=&2k-\alpha+3,\nonumber \\
U^{(\alpha)}_3(k)&=&4k^2-(4\alpha-12)k+\alpha^2-56\alpha+12,\nonumber\\
U^{(\alpha)}_4(k)&=&8k^3-(12\alpha-36)k^2+(6\alpha^2-30\alpha+76)k-\alpha^3+6\alpha^2-29\alpha+60 .
\label{e51}
\end{eqnarray}
As in the case of even $n$, we can verify that the sum of both series in
(\ref{e48}) vanish for all $n$.

This completes the verification that the eigenfunction $\psi_n$ of the
harmonic-oscillator Hamiltonian is also an eigenfunction of the $Q_0$ operator
with eigenvalue $0$. Thus, $\psi_n$ is an eigenfunction of $\ensuremath{\mathcal{C}}=e^{Q_0}\ensuremath{\mathcal{P}}$ with
eigenvalue $(-1)^n$.

\section{Calculation of $Q$ to first order in $\epsilon$}
\label{s4}

The general approach in this paper is to find an operator $\ensuremath{\mathcal{C}}=e^Q\ensuremath{\mathcal{P}}$ for the
$\ensuremath{\mathcal{PT}}$-symmetric Hamiltonian $H=H_0+\epsilon H_1$, where $Q$ has the
power-series expansion (\ref{e6}) in the parameter $\epsilon$ and this expansion
has a nonvanishing zeroth-order term; that is, $Q_0\neq0$. In this section we
concentrate on the formal problem of determining the first-order coefficient
$Q_1$ once $Q_0$ is given.

The coefficient $Q_1$ satisfies the equation
\begin{equation}
\left[e^{Q_0+\epsilon Q_1},H_0\right]=\epsilon\left\{e^{Q_0},H_1\right\},
\label{e52}
\end{equation}
which follows immediately from (\ref{e3}). If we expand this equation to
first order in $\epsilon$, we obtain
\begin{equation}
Z+\frac{1}{2}(Q_0Z+ZQ_0)+\frac{1}{6}(Q_0^2Z+Q_0ZQ_0+ZQ_0^2)+\frac{1}{24}(Q_0^3Z
+Q_0^2ZQ_0+Q_0ZQ_0^2+ZQ_0^3)+\ldots=\left\{e^{Q_0},H_1\right\},
\label{e53}
\end{equation}
where
\begin{equation}
Z\equiv\left[Q_1,H_0\right].
\label{e54}
\end{equation}

Recall that $Q_0$ is a solution to $[Q_0,H_0]=0$, which is a {\it homogeneous}
equation. Thus, any parameter $\mu$ times $Q_0$ is also a solution. Our approach
will now be to make the replacement $Q_0\to\mu Q_0$ in (\ref{e53}) and to treat
$\mu$ as a small perturbation parameter. We can thus expand $Z$ as
\begin{equation}
Z=\sum_{n=0}^\infty Z_n\mu^n
\label{e55}
\end{equation}

To zeroth order in $\mu$ we obtain the result $Z_0=2H_1$. To first order in
$\mu$ we obtain $Z_1=0$, and in fact we find that $Z_{2j+1}=0$ for $j=0,\,1,\,2,
\,\ldots$. The general result for $n\geq2$ can be given in terms of Bernoulli
numbers $\ensuremath{\mathcal{B}}_n$:
\begin{equation}
Z_n=\frac{2\ensuremath{\mathcal{B}}_n}{n!}\left[Q_0,\ldots\left[Q_0,\left[Q_0,H_1\right]\right]\ldots
\right]_{n~{\rm times}},
\label{e56}
\end{equation}
where $\ensuremath{\mathcal{B}}_0=1$, $\ensuremath{\mathcal{B}}_1=-\mbox{$\textstyle{\frac{1}{2}}$}$ (which is not used in the above formula), and
$$\ensuremath{\mathcal{B}}_2=\frac{1}{6},~\ensuremath{\mathcal{B}}_3=0,~\ensuremath{\mathcal{B}}_4=-\frac{1}{30},~\ensuremath{\mathcal{B}}_5=0,~\ensuremath{\mathcal{B}}_6=\frac{1}{42}
,~\ensuremath{\mathcal{B}}_7=0,~\ensuremath{\mathcal{B}}_8=-\frac{1}{30},~\ensuremath{\mathcal{B}}_9=0,~\ensuremath{\mathcal{B}}_{10}=\frac{5}{66},~\ensuremath{\mathcal{B}}_{11}=0,~
\ensuremath{\mathcal{B}}_{12}=-\frac{691}{2730}.$$
We now decompose $Q_1$ into a perturbation series in powers of $\mu$,
\begin{equation}
Q_1=\sum_{n=0}^\infty R_n\mu^n,
\label{e57}
\end{equation}
and obtain a sequence of commutator equations for the coefficients $R_n$:
\begin{equation}
\left[R_n,H_0\right]=Z_n\quad(n=0,\,1,\,2,\,\ldots).
\label{e58}
\end{equation}
In the next section we show how to solve these commutator equations for the
special simple case in which $H_1=q$.

\section{Solution of (\ref{e58}) for the shifted harmonic oscillator
$H=\mbox{$\textstyle{\frac{1}{2}}$} p^2+\mbox{$\textstyle{\frac{1}{2}}$} q^2+i\epsilon q$}
\label{s5}

Let us consider the shifted harmonic oscillator for which $H_0=\mbox{$\textstyle{\frac{1}{2}}$} p^2+\mbox{$\textstyle{\frac{1}{2}}$}
q^2$ and $H_1=iq$. This Hamiltonian has an unbroken $\ensuremath{\mathcal{PT}}$ symmetry for all real
$\epsilon$. Its eigenvalues $E_n=n+\mbox{$\textstyle{\frac{1}{2}}$}+\mbox{$\textstyle{\frac{1}{2}}$}\epsilon^2$ ($n=0,\,1,\,2,\,
\ldots$) are all real. One $\ensuremath{\mathcal{C}}$ operator for this theory is given exactly by
\cite{R11,R12}
\begin{equation}
\ensuremath{\mathcal{C}}=e^{-2\epsilon p}\ensuremath{\mathcal{P}}.
\label{e59}
\end{equation}
In the limit $\epsilon\to0$ the Hamiltonian becomes Hermitian and $\ensuremath{\mathcal{C}}$ in
(\ref{e59}) becomes identical with $\ensuremath{\mathcal{P}}$. However, the solution for $\ensuremath{\mathcal{C}}$ in
(\ref{e59}) is not unique, and by taking any or all of the $Q_0$ in (\ref{e14}),
we obtain an infinite number of operators $\ensuremath{\mathcal{C}}$. To find $Q_1$ we must calculate
$Z_0$ (which is $2iq$), $Z_2$, $Z_4$, and so on, and from these we must solve
(\ref{e58}) to obtain $R_0$, $R_2$, $R_4$, and so on.

For the case $n=0$ in (\ref{e58}) we have a simple exact solution to the
commutator equation for $R_0$:
\begin{equation}
R_0=-2p.
\label{e60}
\end{equation}
We emphasize that this solution is not unique.

The equation for $Z_2$ is
\begin{equation}
Z_2=-\frac{i}{6}\sum_{j=0}^\infty \sum_{k=1}^\infty a_j^{(\gamma)}a_k^{(\gamma)}
(1-2k-2\gamma) F_{j,k}^{(\gamma)},
\label{e61}
\end{equation}
where $F_{j,k}^{(\gamma)}= \left[ T_{-2j-2\gamma+1,2j},T_{-2k-2\gamma,2k}
\right]$, whose explicit form is obtained by using the algebra in
Ref.~\cite{R33} of the basis elements $T_{m,n}$:
\begin{equation}
F_{j,k}^{(\gamma)}=\sum_{\alpha=0}^{j+k-1} \sum_{\beta=0}^{2\alpha+1}
C_{j,k,\alpha,\beta}^{(\gamma)}T_{-2(k+j)-4\gamma - 2 \alpha,2(k+j)-2\alpha-1}
\label{e62}
\end{equation}
with coefficients $C_{j,k,\alpha,\beta}^{(\gamma)}$ given by
\begin{equation}
C_{j,k,\alpha,\beta}^{(\gamma)}=\frac{i^\alpha(-1)^{\alpha+\beta}(2j)!(2k)!(2j+
\beta-2)!(2k+2\alpha-\beta)!}{4^\alpha(2j-2)!(2k-1)!(2j+\beta-2\alpha-1)!(2k-
\beta)!(2\alpha+1-\beta)!\beta!}.
\label{e63}
\end{equation}
A more compact form for $Z_2$ in (\ref{e61}) is
\begin{equation}
Z_2=-\frac{i}{6}\sum_{k=1}^\infty\sum_{\alpha=0}^\infty A_{k,\alpha}^{(\gamma)}
T_{-2k-4\gamma-4\alpha,2k-1},
\label{e64}
\end{equation}
where
\begin{equation}
A_{k,\alpha}^{(\gamma)}=2\frac{(-1)^{k-\alpha+1}\Gamma(k+2\alpha+2\gamma)
\Gamma^2(\alpha+\gamma+1/2)}{\Gamma(\gamma-1/2)\Gamma(k)\Gamma(\alpha+1)\Gamma
(\alpha+2\gamma)(\alpha+\gamma)}
\label{e65}
\end{equation}
and for simplicity we have set $a_0=1$.

For the special case $\gamma=0$ we have the following results for $R_2^{(0)}$,
where from now on we omit the superscript $(0)$. The commutator equation is
\begin{equation}
\left[R_2,\,H_0\right]=\frac{i}{6}\sum_{k=1}^\infty\sum_{\alpha=0}^\infty
A_{k,\alpha}T_{-2k-4\alpha,2k-1}.
\label{e66}
\end{equation}
This is a linear equation, so we solve it for each $\alpha$ separately and
express the solution as a sum over $\alpha$: $R_2=\sum_{\alpha=0}^\infty\rho_{
k,\alpha}R_{2,\alpha}$. For $\alpha=0$ we seek a solution of the form
\begin{equation}
R_{2,0}=\sum_{k=0}^\infty\rho_{k,0}T_{-2k-1,2k}
\label{e67}
\end{equation}
whose coefficients $\rho_{k,0}$ satisfy the recursion relation
\begin{equation}
(2k-1)\rho_{k-1,0}+2k\rho_{k,0}=A_{k,0}.
\label{e68}
\end{equation}
(The techniques used here are described in detail in Ref.~\cite{R33}.) The
simplest solution to this recursion relation is $\rho_{k,0}=\pi(-1)^k$.

For $\alpha=1$ we set
\begin{equation}
R_{2,1}=\sum_{k=0}^\infty\rho_{k,1}T_{-2k-5,2k}
\label{e69}
\end{equation}
and so the recursion relation for the coefficients $\rho_{k,1}$ is
\begin{equation}
(2k+3)\rho_{k-1,1}+2k\rho_{k,0}=A_{k,1},
\label{e70}
\end{equation}
whose solution is $\rho_{k,0}=-\frac{\pi}{4}(-1)^k(k+2)!/k!$.

For general $\alpha$ we have
\begin{equation}
R_{2,\alpha}=\sum_{k=0}^\infty\rho_{k,\alpha}T_{-2k-4\alpha-1,2k}
\label{e71}
\end{equation}
and the recursion relation for the coefficients $\rho_{k,\alpha}$ is
\begin{equation}
(2k+4\alpha-1)\rho_{k-1,\alpha}+2k\rho_{k,\alpha}=A_{k,\alpha},
\label{e72}
\end{equation}
whose solution is
\begin{equation}
\rho_{k,\alpha}=(-1)^k\frac{(k+2\alpha)!}{k!}\left[\frac{\Gamma(\alpha+1/2)}
{\alpha!}\right]^2.
\label{e73}
\end{equation}

\subsection{Complete evaluation of the first-order expansion $Q_1$}
\label{ss5a}

We now derive the general form of the first-order expansion in $\epsilon$ of the
operator $Q=Q_0+\epsilon Q_1+\epsilon^2 Q_2+\ldots$, which takes the form of a
series in even powers of the parameter $\mu$:
\begin{equation}
Q_1=\sum_{n=0}^\infty\mu^{2n}R_{2n},
\label{e74}
\end{equation}
whose coefficients $R_{2n}$ satisfy the commutator equation
\begin{equation}
\left[R_{2n},H_0\right]=Z_{2n}\quad(n=1,2,3,\ldots).
\label{e75}
\end{equation}
Recall that the operator $Z_{2n}$ is proportional to the recursive evaluation of
the double commutator $[Q_0,[Q_0,\,.\,]]$ acting on the operator $Z_{2n-2}$,
starting from $Z_0=2H_1$. (As established earlier, $Z_2=[Q_0,[Q_0,H_1]]/6$, $Z_4
=-[Q_0,[Q_0,[Q_0,[Q_0,H_1]]]]/360$, and so on.) The operator $Z_{2n}$ can be
written as a double series over the basis elements $T_{m,n}$ once their algebra
has been repeatedly applied and its closed-form expression is
\begin{equation}
Z_{2n}=2i\frac{\ensuremath{\mathcal{B}}_n}{n!}\sum_{k=n}^\infty\sum_{\alpha=0}^\infty A^{(2n)}_{k,
\alpha}T_{-2k-4\alpha,2k-2n+1}\quad(n=1,2,3,\ldots),
\label{e76}
\end{equation}
where the coefficients $A^{(2n)}_{k,\alpha}$ are given by
\begin{equation}
A^{(2n)}_{k,\alpha}=W_\alpha^{(2n)}(-1)^k\Gamma(k+2\alpha+n-1)/\Gamma(k).
\label{e77}
\end{equation}

It is extremely laborious to evaluate explicitly the function $W_\alpha^{(2n)}$,
even for the first few values of $n$. For example, for $n=1$ we have $W_\alpha^{
(1)}=[\Gamma(\alpha+1/2)/\alpha!]^2$; the evaluation of the next commutator with
$Q_0$ gives
$$W_\alpha^{(2)}=\sum_{\beta=0}^\alpha\frac{\Gamma(2\alpha-\beta+1)\Gamma(\beta+
1/2)\Gamma^2(\alpha-\beta+1/2)}{(4\alpha+1)\beta!\Gamma(2\alpha-\beta+1/2)
\Gamma^2(\alpha-\beta+1)}.$$
Moreover, the existence of solutions $R_{2n}$ to (\ref{e75}) is not affected by
the explicit form of the functions $W_\alpha^{(2n)}$ because (\ref{e75}) is a
linear equation.

Substituting (\ref{e76}) into (\ref{e75}) and noting the last two commutator
equations in (\ref{e8}), we argue that the operator $R_{2n}$ has the form
\begin{equation}
R_{2n}=\sum_{k=n-1}^\infty\,\sum_{\alpha=0}^\infty\rho^{(2n)}_{k,\alpha}
T_{-2k-4\alpha-1,2k-2n+2},
\label{e78}
\end{equation}
where the coefficients $\rho_{k,\alpha}^{(2n)}$ satisfy the recursion relation
\begin{equation}
(2k+4\alpha-1)\rho_{k-1,\alpha}^{(2n)}+2(k-n+1)\rho_{k,\alpha}^{(2n)}=A_{k,
\alpha}^{(2n)}.
\label{e79}
\end{equation}
Equation (\ref{e79}) is a first-order linear inhomogeneous difference equation
that we can rewrite as
\begin{equation}
\rho_{k+1,\alpha}^{(2n)}+\frac{2k+4\alpha+1}{2(k-n+2)}\rho_{k,\alpha}^{(n)}=\frac{A_{k+1,\alpha}^{(2n)}}{2(k-n+2)}.
\label{e80}
\end{equation}

To find solutions to (\ref{e80}) we divide both sides of the equation by the
{\it summing factor} $Y_k$, where
\begin{equation}
Y_k=(-1)^k\prod_{j=n}^k\frac{(2j+4\alpha+1)}{2(j-n+2)}=\frac{(-1)^k\Gamma(k+2
\alpha+3/2)}{\Gamma(2\alpha+n+1/2)\Gamma(k-n+3)}.
\label{e81}
\end{equation}
Equation (\ref{e80}) then becomes
\begin{equation}
\frac{\rho_{k+1,\alpha}^{(2n)}}{Y_k}-\frac{\rho_{k,\alpha}^{(2n)}}{Y_{k-1}}=
\frac{A_{k+1,\alpha}^{(2n)}}{2(k-n+2)Y_k}.
\label{e82}
\end{equation}
Note that (\ref{e82}) has taken the form of an exact discrete difference of the
function $\rho_{k,\alpha}^{(2n)}/P_{k-1}$. Summing both sides of (\ref{e82})
from $1$ to $k-1$ gives the solution
\begin{equation}
\rho_{k,\alpha}^{(2n)}=\frac{(-1)^{k+1}\Gamma(k+2\alpha+1/2)}{\Gamma(n+2\alpha+
1/2)(k-n+1)!}\left[G_\alpha^{(2n)}+W_\alpha^{(2n)}\sum_{j=1}^{k-1}\frac{\Gamma
(n+2\alpha+1/2)(j+2\alpha+n-2)!(j-n+1)!}{2\,\Gamma(j+2\alpha+3/2)(j-1)!}\right],
\label{e83}
\end{equation}
where $G_\alpha^{(2n)}$ is an arbitrary constant.

\subsection{Semiclassical approximation to the $\ensuremath{\mathcal{C}}$ operator}
\label{ss5b}

In this subsection we attempt a semiclassical calculation of the $\ensuremath{\mathcal{C}}$ operator.
In such a calculation we expand $Q$ as a series in powers of $\hbar$:
\begin{equation}
Q=Q_0+\hbar\,Q_1+\hbar^2 Q_2+\ldots.
\label{e84}
\end{equation}
The semiclassical approximation terminates after the $\hbar$ term in this
expansion. The ordering of powers of $p$ and $q$ in this expansion becomes
unimportant because commuting $p$ with $q$ introduces additional powers of
$\hbar$. Furthermore, every factor of $pq$ has dimensions of $\hbar$, and thus
only the first term in a sum needs to be kept.

A recursive determination of the first-order solution $Q_1$ in (\ref{e84})
arises as a natural simplification of the difficult problem that we formally
solved in Subsec.~(\ref{ss5a}). To proceed, a dimensional analysis of the
operators is required. Because of the commutator equation $[q,p]=i\hbar$, we can
assign the dimensions of $q$ and $p$ to be $\hbar^{1/2}$. With this convention
the Hamiltonian for the shifted harmonic oscillator becomes
\begin{equation}
H=\mbox{$\textstyle{\frac{1}{2}}$} p^2+\mbox{$\textstyle{\frac{1}{2}}$} q^2+i\hbar^{1/2}q.
\end{equation}
\label{e85}

For $Q_0$ in (\ref{e19}) to be the zeroth-order solution in $\hbar$, we must
introduce its explicit dependence on $\hbar$; that is
\begin{equation}
Q_0=\hbar^{-1/2}\sum_{k=0}^\infty a_k T_{1-2k,2k}.
\end{equation}
\label{e86}
Following the procedure illustrated in Sec.~(\ref{s4}), we make the replacement
$Q_0\to\mu\,Q_0$, where $\mu$ is a small {\it dimensionless} parameter. The
operator $Q_1$ in (\ref{e74}) admits dimensionless solutions in terms of the
operators $R_{2n}$ only for $\alpha=0$ in (\ref{e78}). In fact, for $\alpha=0$
the series representation of both $Z_{2n}$ in (\ref{e76}) and $R_{2n}$ in
(\ref{e78}) can be drastically simplified. Introducing the explicit dependence
on $\hbar$, the operator $Z_{2n}$ becomes
\begin{equation}
Z_{2n}=\hbar^{n+1/2}\sum_{k=n}^\infty\frac{(-1)^{n+k-1}(k-1)!}{(n-1)!(k-n)!}
T_{-2k,2k-2n+1},
\label{e87}
\end{equation}
while for the operators $R_{2n}$ we get
\begin{equation}
R_{2n}=\hbar^{n-1/2}\sum_{k=n-1}^\infty\rho_k^{(2n)}T_{-2k-1, 2k-2n+2},
\label{e88}
\end{equation}
where the coefficients $\rho_k^{(2n)}$ satisfy the recursion relation
\begin{equation}
\rho_{k+1}^{(2n)}+\frac{2k+1}{2(k-n+2)}\rho_k^{(2n)}=\frac{(-1)^{n+k}k!}{2(n-1)!(k-n+2)!}.
\label{e89}
\end{equation}

The general solution to (\ref{e89}) contains an arbitrary constant $C_n$:
\begin{equation}
\rho_k^{(2n)}=C_n+\frac{(-1)^n\Gamma(n+1/2)[\sqrt{\pi}\,k!-\Gamma(k-1/2)]}{
\sqrt{\pi}(n-1)!\Gamma(k+1/2)}.
\label{e90}
\end{equation}
With the choice $C_{n}=(-1)^n\pi^{-1/2}\Gamma(n+1/2)/(n-1)!$ the result in
(\ref{e90}) is considerably simplified. The simplest first-order solution $Q_1$
in the series (\ref{e84}) for the operator $Q$ is
\begin{equation}
Q_1=\hbar^{-1/2}\sum_{n=0}^\infty\frac{(-\mu^2\hbar)^n}{(n-1)!}
\sum_{k=n-1}^\infty\frac{k!}{\Gamma(k+1/2)}\,T_{-2k-1,2k-2n+2}.
\end{equation}
\label{e91}
This illustrates the nature of a semiclassical expansion for the operator $Q$.

\section{Conclusions}
\label{s6}

The principal result in this paper is that while there is a unique bounded
metric and $\ensuremath{\mathcal{C}}$ operator for the quantum harmonic oscillator, which is the
simplest $\ensuremath{\mathcal{PT}}$-symmetric quantum theory, there is an infinite number of
unbounded metric and $\ensuremath{\mathcal{C}}$ operators, and we have calculated them exactly.
To produce these unbounded operators we have had to sum infinite series of
singular operators (involving powers of $1/p$) and have observed that the
resulting sums are no longer singular. Of course, our summation procedure is at
best only formal. However, we have verified our results by using dimensional
summation procedures and have shown that the $\ensuremath{\mathcal{C}}$ operators that we have
constructed satisfy exactly their defining equations.

As anticipated in Ref.~\cite{R31}, the properties of nonuniqueness and
unboundedness of the $\ensuremath{\mathcal{C}}$ operators are connected. There is a unique bounded
$\ensuremath{\mathcal{C}}$ operator for the harmonic oscillator, namely $\ensuremath{\mathcal{P}}$, and an infinite class
of unbounded $\ensuremath{\mathcal{C}}$ operators. Interestingly, the unbounded metrics grow for
large $q$ like $e^q$. This does not pose a serious problem if we want to
calculate matrix elements of eigenstates of $H_0$ because in $q$ space these
states vanish like $e^{-q^2}$. Thus, any finite linear combination of
eigenstates is an acceptable state in the Hilbert space associated with the
unbounded metric.

Finally, while we have performed formal summations of operators in this paper,
we have justified our results by doing careful summation calculations that rely
on analytic continuation. Our calculation are modeled on the dimensional
continuation evaluations that are used to regulate divergent Feynman integrals.
We conjecture that such techniques might be applied to generalize the notions of
Cauchy sequences and completeness sums for Hilbert-space vectors.

\acknowledgments
We thank S.~Kuzhel for many discussions regarding Hilbert-space theory and
Q.~Wang for useful comments regarding Sec.~\ref{s3}. MG is grateful for the
hospitality of the Department of Physics at Washington University. CMB thanks
the U.S.~Department of Energy and the U.K.~Leverhulme Foundation and MG thanks
the INFN (Lecce) for financial support.

redpajama_set_name: RedPajamaArXiv


Review Request #106304 - Created Sept. 1, 2012 and submitted Sept. 24, 2012, 9:42 p.m.
Hide empty additional information lines in icons view mode.  Previously they were "-", which took up a lot of space while providing no additional information.
I would write ... - (additionalRolesCount - emptyRolesCount) here. That makes it more obvious what happens IMHO. Frank Reininghaus Sept. 2, 2012, 10:31 a.m.
Hm, at first sight this looks like the loop does essentially nothing - if it finds an empty string in ... Frank Reininghaus Sept. 2, 2012, 10:31 a.m.
Please remove the '\'. Frank Reininghaus Sept. 3, 2012, 11:29 p.m.
Sept. 1, 2012, 12:17 p.m.
Sept. 1, 2012, 4:25 p.m.
Thanks for the patch, looks good! I have two small comments on the code.
I would write ... - (additionalRolesCount - emptyRolesCount) here. That makes it more obvious what happens IMHO.
Todd Jennings Sept. 1, 2012, 5:08 p.m.
Hm, at first sight this looks like the loop does essentially nothing - if it finds an empty string in the hash, it puts an empty string with the same key into the hash. Am I overlooking something, or could the entire loop be removed?
That depends on whether we can guarantee that the values will always only be strings.  I assumed that since it uses a qvariant that this cannot be guaranteed, in which case the loop overwrites non-string values with an empty string.  If we can guarantee it will always be a string, then does it really need to be a qvariant?
Thomas Lübking Sept. 2, 2012, 8:30 a.m.
So what this code eventually did was to override few (mostly custom) types with empty strings resp. a dash, ie. at best melt unknown information - what would likely have been a bug since toString() would still return an empty string when this is used. With the patch from "-" -> "" the loop is pointless unless you actively want to destroy information where you explicitly do not know what it is.
Todd Jennings Sept. 2, 2012, 10:42 a.m.
Sept. 2, 2012, 10:41 a.m.
Sept. 3, 2012, 11:46 a.m.
Thanks, looks good from my point of view! I can't test it at the moment, but you can commit this the the 4.9 branch. There are a couple of weeks to go before 4.9.2 is tagged anyway, and I (and hopefully, more users of the 4.9 branch) will test it until then.
Todd Jennings Sept. 3, 2012, 12:11 p.m.
Are you sure backporting to 4.9 is safe?  This has a impacts on a lot of things besides just the icon view, it also impacts the other views, group headers, sorting, and perhaps other things as well.  I cannot personally guarantee there are no regressions or unexpected behavior. My personal inclination would be to hold off on this until 4.10, but of course you know the code much better than I do so I will backport if you think it is safe.
Sept. 3, 2012, 11:29 p.m.
Remove trailing slash per request.
I don't have git access.  Should I go ahead and request that?
Sept. 6, 2012, 5:53 a.m.
This review has been submitted with commit 20b0cb68bf5cc1099fd6e61982817d9e2ae0130c by Frank Reininghaus to branch master.
Sept. 6, 2012, 5:58 a.m.
This review has been submitted with commit f64a82bd69bcc0f35c9024a8d43e486997b860dd by Frank Reininghaus to branch KDE/4.9.
Sept. 24, 2012, 9:42 p.m.
Unfortunately, there were really some problems due to this fix in master (icons jumping around unexpectedly in the view while info was retrieved asynchronously because the number of rows required by each item could change, see comments in https://bugs.kde.org/show_bug.cgi?id=304752 ). Nontheless, the idea was very nice and looked good at first sight, and we still have the part of the patch that removes the annoying "-" in 4.9 and master, so thanks again for looking into this, Todd!


text:
package commands

import (
	"errors"
	"fmt"
	"io"
	"io/ioutil"
	"os"
	"strings"

	"github.com/codegangsta/cli"
)

// NewRunCommand returns the CLI command for "run".
func NewRunCommand() cli.Command {
	return cli.Command{
		Name:        "run",
		ShortName:   "r",
		Usage:       "Runs a task for a given service and version",
		Description: "run [service] [version] [command]",
		Action: func(ctx *cli.Context) {
			service, version, command, err := extractRunParameters(ctx)
			if err != nil {
				errorAndBail(err)
			}

			c := getClientFromContext(ctx)
			resp, err := c.CreateTask(service, version, command)
			if err != nil {
				errorAndBail(err)
			}

			defer resp.Body.Close()

			if resp.StatusCode == 200 {
				io.Copy(os.Stdout, resp.Body)
			} else {
				fmt.Println("Task failed to launch :(")
				fmt.Println("-- Response Status:", resp.Status)
				body, _ := ioutil.ReadAll(resp.Body)
				fmt.Println("-- Response Body:", string(body))

				os.Exit(1)
			}
		},
	}
}

func extractRunParameters(ctx *cli.Context) (service string, version string, command string, err error) {
	if len(ctx.Args()) == 0 {
		return "", "", "", errors.New("Service required")
	}
	service = ctx.Args()[0]

	if len(ctx.Args()) == 1 {
		return service, "", "", errors.New("Version required")
	}
	version = ctx.Args()[1]

	if len(ctx.Args()) == 2 {
		return service, version, "", errors.New("Command required")
	}
	command = strings.Join(ctx.Args()[2:], " ")

	return service, version, command, nil
}


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solution(n = noun.substance) - a homogeneous mixture of two or more substances; frequently (but not necessarily) a liquid solution; "he used a solution of peroxide and water"
solution(n = noun.communication) answer, resolution, result, solvent - a statement that solves a problem or explains how to solve the problem; "they were trying to find a peaceful solution"; "the answers were in the back of the book"; "he computed the result to four decimal places"
solution(n = noun.cognition) - a method for solving a problem; "the easy solution is to look it up in the handbook"
solution(n = noun.act) - the successful action of solving a problem; "the solution took three hours"
solution, n. [OE. solucion, OF. solucion, F. solution, fr. L. solutio, fr. solvere, solutum, to loosen, dissolve. See Solve.].
(Chem.), a solution which is used as a reagent, and is of a known and standard strength; specifically, a normal solution, containing in each cubic centimeter as many milligrams of the element in question as the number representing its atomic weight; thus, a normal solution of silver nitrate would contain 107.7 mgr. of silver in each cubic centimeter.
1 the act or a means of solving a problem or difficulty.
2 a the conversion of a solid or gas into a liquid by mixture with a liquid solvent. b the state resulting from this (held in solution).
3 the act of dissolving or the state of being dissolved.
4 the act of separating or breaking.
5 = rubber solution (see RUBBER(1)).
solution set Math. the set of all the solutions of an equation or condition.
N liquefaction, liquescence, liquescency, melting, colliquation, colliquefaction, thaw, liquation, deliquation, deliquescence, lixiviation, dissolution, solution, apozem, lixivium, infusion, flux, solvent, menstruum, alkahest, liquefied, liquescent, liquefiable, deliquescent, soluble, colliquative.
N answer, response, reply, replication, riposte, rejoinder, surrejoinder, rebutter, surrebutter, retort, repartee, rescript, rescription, antiphon, antiphony, acknowledgment, password, echo, counter statement, discovery, solution, rationale, clue, Oedipus, oracle, return, answering, responsive, respondent, conclusive, because, on the scent, on the right scent, Int, eureka!.


April 10 - Fairstar Heavy Transport's semi-submersible newbuilding Forte has started its sea trials off the coast of China, under the supervision of Det Norske Veritas (DNV), the ship's Classification Society.
Willem Out, COO of Fairstar, commented from the bridge of the Forte: "I have been in the shipping business for almost 40 years, out of which I have been at sea for over 20 years on many different types of vessels. This is one of my proudest moments at Fairstar and one of the most satisfying for me personally. Our crew onboard the Forte are all full time Fairstar professionals; a combination of veterans from the Fjord and Fjell, as well as a number of newly-hired Dutch mariners and engineers.
"Our partners at Guangzhou Shipyard International (GSI) have done an excellent job. They have maintained a disciplined schedule and we are totally satisfied with the design of the ship and the quality of her construction.
We look forward to testing her limits over the next two weeks. After she returns to port, we will begin to prepare for the formal naming ceremony on May 23.
"In July, Forte will be on contract for the next two years in the Gorgon LNG Project. She is scheduled to transport some of the biggest and most valuable modules from Korea to Barrow Island, Australia over the course of our time charter. The Forte will be delivered on time and on budget."
Forte's sister ship, Finesse was launched from the drydock in Qidong on Thursday, April 5. She is scheduled for delivery to Fairstar in late October 2012.