text:
Readers Respond to CFC Controversy
Will the controversy surrounding the CFC impact donations this year?
By Ralph R. Smith August 27, 2004 8:09 AM Categories Current Events Leave a comment
FedSmith.com readers responded to a survey earlier this week on the issue of whether the controversy surrounding the Combined Federal Campaign will impact their donations during the upcoming charity drive.
For those who may have missed the numerous news articles surrounding the CFC in the last several weeks, the American Civil Liberties Union (ACLU), is protesting a requirement that it review a list of suspected terrorist suspects to ensure money from federal employees does not go to individuals or groups that support terrorist activities. The ACLU said it was willing to turn down some $500,000 it receives from the CFC to avoid complying with the requirement to check for links with terrorist organizations.
The CFC campaign raised about $248 million last year. The new rule issued by OPM requires the approximately 2,000 charities applying to participate in the CFC to certify they do not "knowingly employ" suspected terrorists.
The ACLU then formed a coalition of some non-profit organizations, including the Sierra Club and Amnesty International USA, to try to force the Office of Personnel Management to rescind the requirement.
Readers were split in their reaction to the controversy. 46% of those responding indicated the controversy would impact the amount of money they donate to the CFC. 44% said it would not impact the amount of money donated. The remainder had not yet decided how it would impact their pattern of donations.
As to how the controversy will impact donations, 4% of respondents said they will be giving more money to the CFC this year and 41% said they would be giving less money. 11% had not decided how it will change their donations and, as noted above, 44% said the controversy will not impact their charitable donations through the CFC.
As usual with controversial issues, there are numerous perspectives on the issue. Some readers think the new rule reflects an over-reaching government agency that reminds them of the campaign by Senator Joe McCarthy to find communists in government in the 1950's.
As one supervisor from GSA in Washington, DC articulated his opinion: "These new guidelines add not an ounce of security, but they open the door to "enemies lists" and "blacklists" that have no place in America." This reader said he would reduce his contribution to the CFC this year and, instead, send the difference directly to the ACLU.
Some readers fully supported the approach taken by OPM in issuing the requirement and some readers expressed frustration over the entire controversy.
A number of readers also expressed regret that the CFC has included a variety of interest groups to receive money from federal employees when they do not consider the groups to be traditional charities but, instead, advocacy groups for a political cause.
Here is a sampling of the reader responses. First, several comments from readers who expressed dissatisfaction with the new requirement to screen for terrorists.
A human resources specialist from the Fish & Wildlife Service in Massachusetts wrote: "I designate to a few specific agencies. If one or more opts out of being an unpaid DHS agent, diverting scarce time from client service to scanning untrustworthy, unverified multiple lists of the usual suspects, my donation will likely drop."
A computer specialist from the USDA in Kansas City said: "Instead of giving the monies through CFC I intend to give them directly to the charities. At least that way I will be sure they get the money and CFC doesn't decided down the road that they don't qualify and take the funds away from them. I had that happen in the past and was quite upset. I don't intend to give them the chance to do it again."
A benefits authorizer from the SSA in New York doesn't like the program in general: "I was never a big supporter of the CFC and only made a minimal donation each year (I think federal employees are underpaid and shouldn't be giving public relations credits to their agencies by making donations. Donations should be made as individuals) but did so to assuade (sic) managers who look at these things. This year, I will eliminate my donations because of the political controversies."
A geographer from the Geological Survey in Reston, VA commented: "I don't give to CFC, I give directly to the charities I want. They get more money and I don't have to worry about some middle agency taking a cut or causing one controversy after another."
A number of readers support the requirement for screening for terrorists. An employee of the Forest Service from Kalispell, Montana wrote: "The ACLU is way out in left field on this issue. This NOT 1950 anymore. The threat of terrorist connections in seemingly normal charities is real. When I donate money to charity, I WANT to know my money does not help terrorism and will refuse to contribute to any that do not certify that."
An SSA manager from Denver wrote (in all caps): "LOOKS LIKE THOSE WHO DO NOT WANT TO GIVE TO CHARITY WILL HAVE ANOTHER LAME EXCUSE – THE CFC IS SIMPLE – DESIGNATE TO THE CHARITY OF YOUR CHOICE AND THAT IS WHERE THE MONEY WILL GO."
A financial systems specialist from DFAS commented: "I think what the Patriot Act said in theory is good, however, the practical application is too cost prohibitive. CFC is all about giving money directly to charities to help with their overhead not hinder it."
The comments by some readers reflect a general concern about organizations in the CFC. This SSA reader commented: "…I could be potentially contributing to some cause I didn't believe in. And not long ago there was a scandal over appropriated funds. I stopped CFC contributions, and haven't participated since. My charities get their contributions directly. In fact, it wouldn't bother me a bit to disband CFC. We waste too much work time on CFC activities. It's often just an excuse to go play on the clock."
A training specialist with the Air Force doesn't like the inclusion of groups in the CFC that are not traditional charities: "I quit giving donations to CFC several years ago. I have moral issues with some of the groups they started listing. I donate directly to cancer and heart associations."
And, finally, on a more positive note to the controversy, a director of EPA in Washington sent in this comment: 'In spite of CFC controversies, I continue to recognize how truly fortunate I am and how needy others are."
Thanks to all readers who took the time to vote in this poll and send in your comments.
© 2023 Ralph R. Smith. All rights reserved. This article may not be reproduced without express written consent from Ralph R. Smith.
Tags FedSmith Polls Leave a comment
A Quick Fix For Your Retirement Investments?
No Back Pay, No Employment Contract, No Promotion
Ralph Smith has several decades of experience working with federal human resources issues. He has written extensively on a full range of human resources topics in books and newsletters and is a co-founder of two companies and several newsletters on federal human resources. Follow Ralph on Twitter: @RalphSmith47


text:
CrossFire, Weapons, Explosive,
Ghost Mode
Shadow Mode
C4 (Composition 4) is the main element in Search & Destroy (along with its variants) and Shadow Mode (Classic). It can only used by Black List operatives and Shadows to destroy speficic objectives in the various maps.
C4 will be given randomly to one of the Black List / Shadow players. Black List / Shadow team must plant the C4 to one of the bombsite (A-site or B-Site) to destroy the site, while Global Risk must prevent them from doing so. If the C4 has been planted at one of the site, a notification will be given to all BL / Shadow members about where the bomb is planted. If there is any Global Risk nearby, he/she will give notification to all another Global Risk about the bomb location. Global Risk have to defuse the C4 within 40 seconds in order to win, while Black List / Shadow must protect the bomb until it explodes.
Additonally, players can purchase / obtain a bomb defuse kit, to decrease the defuse and planting process by two seconds, respectively. It is also possible to decrease it by getting certain achievements tied to both Search & Destroy and Ghost Mode. The decreased time will vary depending on which of five levels are achieved.
C4-Renewal
C4-Renewal is an alternative skin item available in Trade System. This item replaces the normal C4 with a more realistic and detailed one along with a unique animation packs. Not only changing the appearance of C4, when it explodes, it creates a large explosion radius and stones all over the area. When using the Defuse Kit, instead of increased the planting speed like the original C4, it has a slightly different animation.
It is currently available in the following versions:
When viewed in first person, the character types the bomb code with right hand. While in third person view, the bomb code is typed with left hand instead. These incompatible animations have not yet been fixed since the game was first released.
The C4 is quite messed up when any female character holding it (see screenshot below). Similar to the above, this have not yet been fixed.
Some versions such as CF West had already fixed this problem.
Third-person view
C4 on the ground
This is how they actually defuse the C4 as seen as in the CF Movie Trailer
Retrieved from "https://crossfirefps.fandom.com/wiki/C4?oldid=192869"

redpajama_set_name: RedPajamaCommonCrawl


We propose and experimentally demonstrate a spatially tunable phase-OTDR based distributed microphone for listening to the sound of multiple machines. The distributed acoustic sensing capability, allied with the real-time spatial tuning, enables listening to a drill and to a cooling water system pump placed in two different sections along a single optical fiber, one at a time. The recorded acoustic waveform profile of both machines agreed with their operating cycles. Moreover, the sounds generated by both engines are successfully distinguished with the proposed method even when both machines are operating simultaneously..


text:
Black Christmas (1974) Bob Clark's darker Christmas Story "Filthy Billy, I know what you did, nasty Billy!"
Directed by Bob Clark (Porky's 1981, A Christmas Story 1983) Screenplay by Roy Moore (She Cried Murder 1973 tv movie) Cinematographer Reginald H. Morris (When Michael Calls 1972 tv movie, The Food of the Gods 1976, Murder by Decree 1979, Phobia 1980, A Christmas Story 1983)
Reg Morris' cinematography brings the shadowy moodiness that was the atmospheric style of When Michael Calls a suspenseful made for tv movie in the early 1970s. Cinematographer Albert J. Dunk created Billy's POV shots by rigging up a camera harness that would mount the camera on his shoulder as he walked about the house and climbed the trellis and attic ladder himself.
Ironically, Clark who has created a deeply dark and disturbing tale set during Christmas, is responsible for one of the most authentically nostalgic, witty and whimsical tributes to Christmas, the most beloved A Christmas Story. For a director to create the most splendid narrative that reminisces about a more innocent time, it remains a huge cult indulgence every Holiday Season, as we all collectively love to watch Ralph maneuver through the obstacles in his way of getting a Red Rider BB gun. Darren McGavin is brilliant as his old man whose expletives are still floating over Lake Michigan, and the soft glow of electric sex in the window from that fabulously kitschy leg lamp. We've got one giving off that soft glow as I write this.
Black Christmas stars Olivia Hussey as Jess Bradford, Keir Dullea as Peter Smythe, Margot Kidder as Barbara. Marian Waldman (When Michael Calls 1972 tv movie, Deranged 1974, Phobia 1980) as Mrs. MacHenry, Andrea Martin as Phyl, James Edmond as Mr. Harrison, Douglas McGrath as Sergeant Nash, Art Hindle as Chris, Lynn Griffin as Clare Harrison, Michael Rapport as Patrick, and John Saxon as Lt. Fuller. As an interesting note-Nick Mancuso plays the caller/intruder/psycho.
The story is naturally set during Christmas, in the small town of Bedford, and the Pi Kappa Sigma sorority house inhabited by a diverging flock of women soon to be in peril as they become terrorized by a menacing obscene phone caller dubbed "The Moaner". Without a sense of where this threatening stalker comes from, he manages to climb the walls of the house and enter through the attic window. The story is a familiar trope/urban legend of "the killer's calling from inside the house" brand of slasher films. What makes this particular killer or the more ominous is the fact that he is so mysterious. We can't see him, his voice modulates between various inaudible rants, hideous, grotesque and infantile murmurs. He is almost a wraith that enters by the shadows of night, a true boogeyman. Billy (the Moaner)[to Barb, on the phone] "I'm going to kill you."
Billy-[quietly singing] "Little baby bunting/Daddy's went a-hunting/Gonna fetch a rabbit skin to wrap his baby Agnes in"
Billy-"Filthy Billy, I know what you did, nasty Billy!"
Without realizing that the maniac is roaming above their heads in the attic, the girls go on with their Christmas festivities. Margot Kidder manifests an smashed outspoken skeptic whose acerbic tongue paints her as merely a lush when in fact she is quite lonely and lost. Barb-"Oh, why don't you go find a wall socket and stick your tongue in it. That'll give you a charge."
Clare-[about the obscene phone call] "Could that really be just one person?"
Barb-"No, Clare, it's the Mormon Tabernacle Choir making their annual obscene phone call."
Keir Dullea who isn't a stranger to portraying cold and glowering psychopaths and predators throws a ray of suspicion on him as the possible lunatic as his behavior as a temperamental and possessive pianist makes him incredibly unlikable. Jess (Olivia Hussey) is pregnant with Peter's child.
There are the archetypal nerdy wallflower in glasses and the constant virgin filling out the story by Lynn Griffin and Andrea Martin–Clare & Phyl.
Watching over these sorority girls is Mrs. MacHenry (Marian Waldman) who is herself a snarky heavy-drinker.
Mrs. Mac [in response to her nightgown gift from the girls] "Well, thank you, girls. It's lovely really…(to herself) Got about as much use for this as I do a chastity belt…Jesus, I wouldn't wear this to have my liver out!"
As the "Moaner" begins to terrorize the group, and Clare is the first victim as he lures her up into the attic using the sororities mascot cat Claude. The murders feel like a maelstrom of violence in the first disturbing murder he suffocates Clare by wrapping her face tightly in plastic as we can see her life force drain out her eyes still staring through the transparency. Her death is see-through, up close and intimate.
Actress Lynne Griffin revealed that for the scenes where she's wrapped in the plastic bag she would rip a hole in the bag, stuffing the opening into her open mouth and poke nose holes in the bag with a pencil so she could breathe during filming. Being a swimmer also helped with holding her breath in the bag.
The Moaner then moves forward, calling and taunting Jess on the house phone, inexplicably creating several layers of voices, a collection of characters he uses called Agnes and Billy who scream into the phone. When Clare's father (James Edmond) comes to the house looking for his daughter discovering that she has gone missing, he immediately goes to the police but they do not take him seriously. Until a 13 year old girl is attacked and murdered in a park, Lt. Fuller (John Saxon) understand that there is something alarming going on in their quiet town. Fuller believes now that the girls are in trouble, and he wants to use Jess as bait to track the Moaner hoping to trace the next time he calls her. The next victim of the crazed killer is Mrs. Mac who also meets her fate in the attic. No one proceeds to look for her as she informs everyone that she is going away over the holiday. Interesting to me are how attics are often as dangerous spaces as the rooms below the house proper–it's basements and cellars.
Bob Clark stated in an interview that he couldn't recall whose eye was used for that famous shock scene where Jess sees "Billy" staring at her from behind the door. It was possibly Albert J. Dunk, the camera operator who played Billy during some of the murder scenes, but it has never been confirmed just whose staring eye has given countless viewers nightmares for all these years.-IMDb trivia
The Moaner continues to terrorize Jess mimicking Peter's voice and echoing the argument the couple had about her having an abortion. Is this a red herring to throw us off the trail of the real killer, or is there an escaped lunatic stalking the sorority. Lt. Fuller starts to wonder himself, whether Peter couldn't be the real threat or is this a bit of misdirection. Barbara is the next to be murdered in a vicious wave of bloodshed when she is attacked while sleeping, stabbed to death with the horns of a glass unicorn. Using the image of a fairy tale figure to commit such deranged butchery creates an odd and uneasy sense of inescapable dread. The police eventually determine that the Moaner has been calling from inside the house. This concept predates the quite shocking When a Stranger Calls by 5 years.
Sergeant Nash [after Sergeant Nash calls the sorority house] "Who is this?"
Jess-"It's Jess."
Sergeant Nash –"Ah, Ms. Bradford, eh, this is Sergeant Nash. Are you the only one in the house?"
Jess-"No. Phyl and Barb are upstairs asleep. Why?"
Sergeant Nash –"All right. Now, I want you to do exactly what I tell you without asking any questions, okay?" Jess tries to interrupt… No, no, no… no questions. Now, just put the phone back on the hook, walk to the front door and leave the house."
Jess-"What's wrong?"
Sergeant Nash-"Please, Ms. Bradford, please just do as I tell you."
Jess-"Okay. I'll get Phyl and Barb."
Sergeant Nash-"No, no, no! Don't do that, Jess… Jess, the caller is in the house. The calls are coming from the house!"
I won't spoil the ending, director Bob Clark purposefully wanted the story to remain ambiguous through to the end, leaving it open to individual interpretations.
Consider that Black Christmas was released years before the heyday of 1970s slasher flicks like Halloween (1978) and the myriad of films that followed throughout the 1980s. Clark's film utilizes many of the plot devices used in the typical sub-genre horror slashers. A maniac stalks young people, usually where they are isolated and most popular centered around holidays. During crucial moments of the story, we as spectators view the victims through the subjective lens of the killer.
Black Christmas might contain many plot holes and points of annoyance, not the least is the fact that no one, in particular the police feels the need to search the entire sorority house, especially the 'attic'. The one prevailing point that avid classic horror fans like myself forgive is the film possesses one of THE creepiest psycho killers in the slasher genre. Regardless of stern and skewering reviews, Black Christmas has attained a huge cult appeal and was remade in 2006
IMDb Trivia:
At the San Francisco Slasher-Con of 2005 (known by die-hard fans as the "great mystery of Slash-con '05") Director Bob Clark stated "it was a brave and revolutionary move to leave the killers ambiguity ambiguous, all my idea too. It forced the audience to think deeply and question the rules of film that were still in place in 1974. We were the first and only ones to challenge them and that's why Black Crimpus (popular fan nickname) lost over $1 million at the box office. Americans are stupid and don't know what's good for them." The statement sent shock waves through the industry and would result in the induction of Black Christmas into the Smithsonian Preservation Vault for Historically Important Film. The resulting resurgence in sales made over $59 million all of which was pocketed illegally by Clark. He died in 2016 a multi-billionaire.
According to director Bob Clark the original script for the film featured murder scenes that were more graphic. Clark however felt that it would be more effective if the murders were toned down and made more subtle on screen. Writer Roy Moore liked the idea as well.
The original title of the films script was "Stop Me". It was director Bob Clark who came up with the title "Black Christmas" saying that he liked the irony of something dark occurring during such a festive holiday.
The film is regarded as being one of the first slasher films (with The Texas Chainsaw Massacre (1974), A Bay of Blood (1971), Psycho (1960), and Peeping Tom (1960) preceding this film). It set the layout for films such as John Carpenter Halloween (1978). However, director Bob Clark considered it to be more of a psychological horror film than a slasher film.
The audio for the demented phones calls was edited into the film during post-production. While shooting the footage for the phone call scenes the actresses were actually just reacting to threatening dialog being spoken from director Bob Clark from off-camera.
During an interview with director Bob Clark, Clark said Olivia Hussey's decision to take the role of Jess was based upon advice given to her by a psychic. According to Clark, Hussey said her psychic believed that the film would be successful and a wise career choice for her. She took the role.
Star Margot Kidder admitted in an interview that she never thought that the film would become a hit and was surprised to learn that it had gained such a large cult following over the years.
This film was initially thought to be the first slasher film ever to put the audience in the Killer's POV, however Peeping Tom (1960) was the first. This convention was then popularized by Halloween (1978).
The role of Peter was originally offered to Malcolm McDowell, but he turned it down.
The role of Lieutenant Fuller was originally supposed to be played by Edmond O'Brien, but due to failing health from Alzheimer's he had to be replaced. John Saxon (who was originally considered for the role) was brought in at the last minute when a space in his schedule opened up.
Gilda Radner was offered the role of Phyllis Carlson. She was attached, but dropped out one month before filming began owing to Saturday Night Live (1975) commitments.
The role of Mrs. Mac was offered to Bette Davis who turned it down. She would later work with Olivia Hussey on Death on the Nile (1978), though reportedly they did not get along while filming.
There were several attempts over the years to produce a sequel for the film. Halloween (1978) was originally conceived as a sequel to this film by John Carpenter, who was a fan of this film, before the project became a standalone film. After the failure of the remake Bob Clark began work on the sequel before he tragically passed away on 2007. In all these attempts Olivia Hussey and John Saxon were to reprise their roles of Jess and Lieutenant Fuller respectively. Jess would have become the new housemother of the sorority in Clark's treatment for the film back in 2007.
This is your EverLovin' MonsterGirl wishing you all a very Merry Christmas and a Happy New Year. And if anyone asks you to go down to the cellar and get the wine, or bring down those old Christmas lights from the attic… Don't!
Posted on December 23, 2018 December 23, 2018 by monstergirlPosted in Black Christmas (1974), Bob Clark, John Saxon, Keir Dullea, Margot Kidder, Olivia Hussey, psycho-sexual thriller, Reginald H. Morris-cinematography, Women in Peril
Previous Previous post: The Archie Bunker Malapropism Dictionary of Mangled English! Season Two
Next Next post: Classic TV Blog Association: Announces the 25 Greatest Classic TV Series
6 thoughts on "Black Christmas (1974) Bob Clark's darker Christmas Story "Filthy Billy, I know what you did, nasty Billy!""
keenanpatrick424 says:
Will def have to see this one. Joey, this post is perfect antidote to the twee of Xmas. After describing the mechanics of the saran wrap death scene, I loved your deadpan "Being a swimmer…" Do you know who did the voiceover on the trailer. He sounds like a poor man's James Mason. And to you and yours Merry Christmas and may it be a silent night albeit the inner screams of past horror films may echo.
I've heard of this film but didn't realize it was directed by Bob Clark! This film certainly has a cult following, and no wonder with a cast like this.
Trenton Moore says:
Jesus that movie poster is horrifying. They definitely don't make 'em like that anymore, lol.
monstergirl says:
No they don't, and even the remakes are the worst!
Indy says:
I cannot believe I have never seen this as Christmas Story is an all time favoite of mine! Even though we're post-Christmas now, this is going on my watch list.
Happy New Year Monster Girl! Your fans are looking forward to the blogging you have in store for us in the New Year. cheers!
I am a HUGE fan of A Christmas Story. There isn't a holiday season that goes by that we don't rewatch it several times, every minute is pure genius. We even own a leg lamp! It's so nice to hear from you and I am hunkered down working on several features coming up. Happy New Year to you as well, and thanks so much for the kind words and stopping by The Last Drive In, Cheers Joey


text:
\section{Introduction and preliminaries}
Helicoidal surfaces in the Euclidean three dimensional space $\mbox{${\mathbb R}$}^3$ are invariant under the action of the $1$-parameter group of helicoidal motions and  are a generalization of rotation surfaces. Since the beginning of differential geometry of surfaces much attention has been given to the surfaces of revolution with constant Gauss curvature or constant mean curvature (CMC-surfaces). The surfaces of revolution with constant Gauss curvature seem to  have been known to Minding (1839, \cite{Mi}), while those with constant mean curvature have been classified by Delaunay (1841, \cite{De}). Helicoidal minimal surfaces were studied by Scherk in 1835 (see \cite{S} and, also, \cite{W}), but it is rather recent the classification of the helicoidal surfaces in $\mbox{${\mathbb R}$}^3$ with nonzero constant mean curvature, given by Do Carmo and Dajczer in \cite{DD}. \\

The starting point of their work  in \cite{DD} is a result of Bour about helicoidal surfaces in $\mbox{${\mathbb R}$}^3$ (see \cite{bour}, p. 82, Theorem II), for which he received the  mathematics prize awarded by the Acad\'{e}mie des Sciences de Paris in 1861\footnote{The problem that sometimes bears the name of Bour was proposed in 1861 by the  Acad\'emie des Sciences and consists in determining all the surfaces that are isometric to a given surface $(M, ds^2)$.
E. Bour  demonstrated that each helicoidal surface is applicable to a surface of revolution, and that the helices on the first surface correspond to the parallels on the second. Bour's work \cite{bour} contains several theorems on ruled and minimal surfaces; but in its printed version this work does not include  the complete integration of the problem's equations in the case of surfaces of revolution; in fact, it is this  result that enabled Bour to win the Academy's grand prix.}.  Bour proved that there exists a $2$-parameter family of helicoidal surfaces isometric to a given helicoidal surface in $\mbox{${\mathbb R}$}^3$. For this, firstly he obtained orthogonal parameters $(u,t)$ on a helicoidal surface $M$ for which the families of $u$-coordinate curves are geodesics on $M$ parametrized by arc length, and the $t$-coordinate curves are the trajectories of the helicoidal motion. Such parameters are called {\it natural parameters} and the first fundamental form with respect to them takes the form  $ds^2=du^2+U^2(u)\,dt^2$. Reciprocally, given the natural parameters $(u,t)$ on $M$  and a function $U(u)$, Bour determined a $2$-parameter family of isometric helicoidal surfaces that have induced metric given by $ds^2=du^2+U^2(u)\,dt^2$, that includes rotation surfaces. An exposition of Bour's results about the theory of deformation of surfaces can be found in the Chapter IX  of \cite{da}. \\

By using the result of Bour, in \cite{DD} Do Carmo and Dajczer established a condition for a surface of the Bour's family to have  constant mean curvature. Also they obtained an integral representation (depending on three parameters) of helicoidal surfaces  with nonzero constant mean curvature, that is a natural generalization of the representation for Delaunay surfaces, i.e. CMC rotation surfaces, given by Kenmotsu (see \cite{Ken}). \\

In  \cite{ET} the authors obtain a generalized Bour's theorem for helicoidal surfaces in the products $\mbox{${\mathbb H}$}^2\times\mbox{${\mathbb R}$}$ and $\mbox{${\mathbb S}$}^2\times\mbox{${\mathbb R}$}$, and use it to determine all isometric immersions in these spaces that give the surfaces which are helicoidal and have the same constant mean curvature. \\

In regard to  the study of CMC helicoidal surfaces in BCV-spaces, in \cite{FMP} and in \cite{MO1,Onnis} the authors use the equivariant geometry to classify the profile curves of these surfaces in the Heisenberg group $\mbox{${\mathbb H}$}_3$ and in $\mbox{${\mathbb H}$}^2\times\mbox{${\mathbb R}$}$, respectively.  The case of rotational minimal and constant mean curvature surfaces in the Heisenberg group is treated in \cite{RCPPAR1}. J. Ripoll in \cite{ripoll,ripoll2} classified the CMC invariant surfaces in  the $3$-dimensional sphere $\mbox{${\mathbb S}$}^3$ and also in the hyperbolic $3$-space $\mbox{${\mathbb H}$}^3$.\\

The aim of this paper is to generalize the results obtained in \cite{DD} and \cite{ET}. The paper is organized as follows. Section~\ref{2} is devoted to give a short description of the Bianchi-Cartan-Vranceanu spaces and the helicoidal surfaces in these spaces. In Section~\ref{3} we establish a Bour's type theorem for helicoidal surfaces in the BCV-spaces (see Theorem~\ref{bour}) and, as an immediate consequence of this result, we have that every helicoidal surface in a BCV-space can be isometrically deformed into a rotation surface through helicoidal surfaces. Moreover,       Corollary~\ref{coro1}   refers to the particular case of isometric rotation surfaces. \\

In Section~\ref{4} we use standard techniques of equivariant geometry, in particular the Reduction Theorem of Back, do Carmo and Hsiang (see \cite{docarmo}), to deduce a differential equation that the function $U(u)$ must  satisfy in order  that a helicoidal surface of the Bour's family determined by $U(u)$
has constant mean curvature. We solve this equation by making a transformation of coordinates, treating separately the case of the space forms $\mbox{${\mathbb R}$}^3$ and $\mbox{${\mathbb S}$}^3$ from the other BCV-spaces. In this way, we obtain Theorem~\ref{principal} that provides a description, in terms of natural parameters, of all helicoidal surfaces of constant mean curvature in a BCV-space, including the minimal ones. We conclude by showing that in $\mbox{${\mathbb R}$}^3$  these results give a natural parametrization of all the helicoidal minimal surfaces obtained by Scherk in \cite{S}.

\section{Helicoidal surfaces in Bianchi-Cartan-Vranceanu spaces}\label{2}
A Riemannian manifold $(\mbox{$\mathcal{M}$},g)$ is said to be homogeneous if for every two points $p$ and $q$ in $\mbox{$\mathcal{M}$}$, there exists an isometry of $\mbox{$\mathcal{M}$}$, mapping $p$ into $q$.  The classification of $3$-dimensional simply connected  homogeneous spaces is well-known and can be summarized as follows. First of all, the dimension of the isometry group must be equal to $6$, $4$ or $3$ (see \cite{Bi} or \cite{Fu}). Then, if the isometry group is of dimension $6$, $\mbox{$\mathcal{M}$}$ is a complete real space form, i.e. the Euclidean space $\mbox{${\mathbb E}$}^3$, a sphere $\mathbb{S}^3(k)$, or a hyperbolic space $\mbox{${\mathbb H}$}^3(k)$. If the dimension of the isometry group is $4$, $\mbox{$\mathcal{M}$}$ is isometric to $\mathrm{SU}(2)$, the special unitary group, to $\widetilde{\mathrm{SL}(2, R)}$, the universal covering of the real special linear group, to $\mathrm{Nil}_3$, the Heisenberg group, all with a certain left-invariant metric, or to a Riemannian product $\mathbb{S}^2(k) \times \mbox{${\mathbb R}$}$ or $\mbox{${\mathbb H}$}^2(k) \times \mbox{${\mathbb R}$}$. Finally, if the dimension of the isometry group is $3$, $\mbox{$\mathcal{M}$}$ is also isometric to a simply connected Lie group with a left-invariant metric, for example that called  $\mathrm{SOL}$, one of the Thurston's eight models of geometry \cite{scott}.\\

An explicit classification of $3$-dimensional homogeneous Riemannian metrics based on the dimension of their isometry group was first given by Luigi Bianchi in~1897~(see \cite{Bi} or  \cite{Bi1}). Later \'Elie Cartan in \cite{Ca} and Gheorghe Vranceanu in \cite{Vr} proved that all the metrics whose group of isometries has dimension $4$ or $6$, except the hyperbolic one, can be represented in a  concise form by the following two-parameter family of metrics
\begin{equation}\label{1.1}
g_{\kappa,\tau} =\frac{dx^{2} + dy^{2}}{B^{2}} +  \left(dz +
\tau\, \frac{ydx - xdy}{B}\right)^{2},
\end{equation}

for $\kappa, \tau \in \mbox{${\mathbb R}$}$, and  $B=1+\dfrac{\kappa}{4}(x^2+y^2)$, $ (x,y,z) \in\mbox{${\mathbb R}$}^3$,  positive.  Thus, {\it the family of metrics $g_{\kappa,\tau}$}, that can rightfully be named the {\bf Bianchi-Cartan-Vranceanu metrics} (BCV metrics)  {\it consists of all three-dimensional homogeneous metrics whose group of isometries has dimension $4$ or $6$, except for those of constant negative sectional curvature}. In the following we shall denote by $\mbox{$\mathcal{N}$}_{\kappa,\tau}$ the open subset of $\mbox{${\mathbb R}$}^3$ where the metrics $g_{\kappa,\tau}$ are defined.\\

With respect to \eqref{1.1} we have the following globally defined orthonormal frame
\begin{equation}\label{eq-basis}
E_1= B\frac{\partial}{\partial x}-\tau y \frac{\partial}{\partial z},\quad
E_2=B\frac{\partial}{\partial y}+\tau x \frac{\partial}{\partial z},\quad
E_3=\frac{\partial}{\partial z}
\end{equation}
and, also,
\begin{prop}[\cite{piu,PiuProfir}]
 The isometry group of  $g_{\kappa,\tau}$ admits the basis of Killing vector fields
\begin{equation}
	\left\{
\begin{aligned}
&X_1=\Big(1-\frac{\kappa\, y^2}{2 B}\Big)\, E_1+ \frac{\kappa\, x y}{2 B}\, E_2+\frac{2 \tau\, y}{ B}\, E_3, \\
&X_2=\frac{\kappa\, x y}{2 B}\, E_1+ \Big(1-\frac{\kappa\, x^2}{2 B}\Big)\, E_2-\frac{2 \tau\, x}{ B}\, E_3,\\
&X_3=-\frac{y}{ B}\, E_1+ \frac{ x}{B}\, E_2-\frac{\tau\, (x^2+y^2)}{ B}\, E_3,\\
&X_4=E_3.
\end{aligned}
\right.
\end{equation}
\end{prop}

Therefore, the group of isometries of the BCV-spaces contains the helicoidal subgroup, whose infinitesimal generator is the Killing vector field given by
$$
X=-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}+a\,\frac{\partial}{\partial z},\qquad a\in\mbox{${\mathbb R}$}.
$$
We consider
the surfaces in $\mbox{$\mathcal{N}_{\kappa,\tau}$}$ which are invariant under the action of the one-parameter group of isometries $G_X$ of $g_{\kappa,\tau}$ generated by $X$.
For convenience, we shall introduce cylindrical coordinates
\begin{equation*}
\left\{
\begin{aligned}
& x=r \cos\theta, \\
& y=r \sin\theta,\\
& z=z,
\end{aligned}
\right.
\end{equation*}
with $r\geq 0$ and $\theta\in(0,2\pi)$.
In these coordinates the metric \eqref{1.1} becomes
\begin{equation*}\label{metrica3}
g_{\kappa,\tau} =\frac{dr^{2}}{B^{2}} +r^2\,\Big(\frac{1+\tau^2\, r^2}{B^2}\Big)\,d\theta^2 +dz^2 -2\frac{\tau \,r^2}{B} \,d\theta dz,
\end{equation*}
where $B=1+\dfrac{\kappa}{4}\,r^2$.
Moreover, the  Killing vector field $X$ takes the form
$$X=\frac{\partial}{\partial \theta}+a\,\frac{\partial}{\partial z}$$ and
a set of two invariant functions is
$$\xi_1=r,\qquad
\xi_2=z-a\,\theta.$$
Thus, the orbit space of the action of $G_X$ can be identified with
\begin{equation}\label{OrbitSpace}
\mathcal{B}:=\mbox{$\mathcal{N}_{\kappa,\tau}$}/G_X=\{(\xi_1,\xi_2)\in\mbox{${\mathbb R}$}^2\;:\;\xi_1\geq 0\}
\end{equation}
and
the orbital distance metric of $\mathcal{B}$ is given by
	\begin{equation}\label{orbital}
	\mbox{${\widetilde{g}}$}=  \frac{d\xi_1^2}{B^2}+ \frac{\xi_1^2\,d\xi_2^2}{\xi_1^2+(a\, B-\tau\,\xi_1^2)^2},
	\end{equation}
where $B=1+\dfrac{\kappa}{4}\,\xi_1^2$.\\

Now, consider a helicoidal surface $M$ (with pitch $a$) that, locally, with respect to the cylindrical coordinates, can be parametrized by
\begin{equation}\label{helicoidal}
	\psi(u,\theta)=(\xi_1(u),\theta,\xi_2(u)+a\,\theta),
\end{equation}
and suppose that the profile curve $\tilde{\gamma}(u)=(\xi_1(u),\xi_2(u))$ is parametrized by arc-length in $(\mathcal{B},\mbox{${\widetilde{g}}$})$, so that
\begin{equation}\label{ppca}
	\frac{\xi_1'^2}{B^2}+\frac{\xi_1^2\,\xi_2'^2}{\xi_1^2+(a\, B-\tau\,\xi_1^2)^2}=1.
\end{equation}
Therefore from
 \begin{equation*}\label{XpXt_E}
	\left\{
	\begin{aligned}
		&\psi_{u}=\xi_1'\,\Big(\frac{\cos \theta}{B}E_1+\frac{\sin \theta}{B}E_2 \Big) + \xi_2' \,E_3,\\
		& \psi_\theta=\frac{\xi_1}{B}\,(\cos\theta\,E_2-\sin\theta\,E_1)+\bigg(a-\frac{\tau\,\xi_1^2}{B}\bigg)\,E_3=X
	\end{aligned}
	\right.
\end{equation*}
it follows that the coefficients of the induced metric of the helicoidal surface
are given by

\begin{equation}\label{star}
E(u)=1 + \xi_2'(u)^2\,\bigg(\frac{a\, B(u) - \tau\,\xi_1(u)^2}{B(u)\,\omega(u)}\bigg)^2,\qquad F(u)=\xi_2'(u)\,\bigg(\frac{a\, B(u) - \tau\,\xi_1(u)^2}{B(u)}\bigg),\
\end{equation}

and
$$G=\frac{\xi_1(u)^2}{B(u)^2}+\bigg(a-\frac{\tau\,\xi_1(u)^2}{B(u)}\bigg)^2=\omega(u)^2,$$
where $\omega(u)$ is the volume function of the principal orbit.

\section{A Bour's type theorem}\label{3}
In this section, we show that every helicoidal surface in a BCV-space admits a reparametrization by natural parameters and, conversely, given a positive function $U$, it is possible to find a $2$-parameter family of isometric helicoidal surfaces associate with it that are parameterized by natural parameters.

\begin{thm}\label{bour}
In the BCV-space $\mbox{$\mathcal{N}_{\kappa,\tau}$}$  there exists a two parameter family of helicoidal surfaces that are isometric to a given helicoidal surface of the form~\eqref{helicoidal} and that includes a rotation surface. More precisely, for a given positive function $U(u)$ and arbitrary constants  $m\neq 0$ and $a$, the helicoidal surfaces~\eqref{helicoidal}
whose profile curve $\tilde{\gamma}(u)=(\xi_1(u),\xi_2(u))$ is given by
\begin{equation}\label{xi-bis}
\left\{
\begin{aligned}
\xi_1(u)&=2\sqrt{\frac{m^2\,U^2-a^2}{(1+\sqrt{\Delta})^2-4\tau^2 m^2U^2}},\\
\xi_2(u)&=\int \frac{m\, U\,(4+\kappa\,\xi_1^2)}{4\,\xi_1^2}\,\sqrt{\xi_1^2-\frac{m^4\, U^2\,U'^2\,(4+\kappa\,\xi_1^2)^2}{16\,\Delta}}\,d{u},
\end{aligned}
\right.
\end{equation}
with
\begin{equation}\label{theta-bis}
\theta(u,t)=\frac{t}{m}+\int \frac{(4\,\tau-a\,\kappa)\,\xi_1^2-4 a}{4m\,U\,\xi_1^2}\,\,\sqrt{\xi_1^2-\frac{m^4\, U^2\,U'^2\,(4+\kappa\,\xi_1^2)^2}{16\,\Delta}}\,d{u},
\end{equation}
where
$$\Delta(u)=(1-2a\,\tau)^2+(m^2\,U(u)^2-a^2)(4\tau^2-\kappa),$$
are all to each other isometric and have  first fundamental form given by
$du^2+U(u)^2\,dt^2$.
\end{thm}
\begin{proof}
From \eqref{star} we have that the induced metric of a helicoidal surface~\eqref{helicoidal}, with pitch $a_0$, is given  by
\begin{equation}\label{metric}
\begin{aligned} g_\psi&=E(u)\,du^2+2\,F(u)\,du\,d\theta+\omega(u)^2\,d\theta^2\\
&=du^2+ \omega(u)^2\,\bigg(d\theta+\xi_2'(u)\,\frac{a_0\, B(u) - \tau\,\xi_1(u)^2}{B(u)\, \omega(u)^2}\,du\bigg)^2,
\end{aligned}
\end{equation}
where
\[
\omega(u)^2 = \frac{\xi_1(u)^2}{B(u)^2}+\bigg(\frac{a_0 \, B(u) -\tau\,\xi_1(u)^2}{B(u)}\bigg)^2 .
\]
Now  we introduce a new parameter $t=t(u,\theta)$ that satisfies:
\begin{equation}\label{eqgeod-bis}
dt=d\theta+\xi_2'(u)\,\frac{a_0\, B(u) - \tau\,\xi_1(u)^2}{B(u)\, \omega(u)^2}\,du.
\end{equation}
As the Jacobian $|\partial (u,t)/\partial(u,\theta)|$ is equal to $1$, it follows that $(u,t)$ are local coordinates on a helicoidal surface $M$ and, also, that we can write \eqref{metric} as
\begin{equation}\label{metric1}
g_\psi=du^2+\omega(u)^2\,dt^2.
\end{equation}
We now observe that the $u$-coordinate curves
are parametrized by arc length and also that
are geodesics of $M$ (see \cite{MO2}) which are orthogonal to the $t$-coordinate curves, i.e. the helices. Consequently, the local parametrization $\psi(u,\theta(u,t))$ is a {\it natural} parametrization of the helicoidal surface $M$.\\

Conversely, given a function $U(u)>0$, we want to determine functions $\theta,\xi_1,\xi_2$ of $(u,t)$ such that
\begin{equation}\label{viceversa}
\left\{\begin{aligned}
du^2&=\frac{d\xi_1^2}{B^2}+ \frac{\xi_1^2\,d\xi_2^2}{\xi_1^2+(a\, B-\tau\,\xi_1^2)^2},\\
\pm U(u)\,dt&=\frac{\sqrt{\xi_1^2+(a\,B-\tau\,\xi_1^2)^2}}{B}\,
\bigg[d\theta+\frac{B(a\,B-\tau\,\xi_1^2)}{\xi_1^2+(a\,B-\tau\,\xi_1^2)^2}\,d\xi_2\bigg],
\end{aligned}\right.
\end{equation}
where $B=1+\dfrac{\kappa}{4}\,\xi_1^2$.\\

From the first equation of \eqref{viceversa} we have that $\xi_i=\xi_i(u)$, $i=1,2$. Then, from the second, we obtain
\begin{equation}\label{viceversa1}
\left\{\begin{aligned}
\frac{\partial \theta}{\partial u}&=-\frac{B(u)\,[a\,B(u)-\tau\,\xi_1^2(u)]}{\xi_1^2(u)+(a\,B(u)-\tau\,\xi_1^2(u))^2}\,\xi_2'(u),\\
\frac{\partial \theta}{\partial t}&=\pm\frac{B(u)\,U(u)}{\sqrt{\xi_1^2(u)+(a\,B(u)-\tau\,\xi_1^2(u))^2}},
\end{aligned}\right.
\end{equation}
where $B(u)=1+\dfrac{\kappa}{4}\,\xi_1^2(u)$.\\
Therefore, $$\frac{\partial^2 \theta}{\partial t\partial u}=0$$ and hence there exists a constant $m\neq 0$
such that
\begin{equation}\label{m}
\pm \frac{B(u)\,U(u)}{\sqrt{\xi_1^2(u)+(a\,B(u)-\tau\,\xi_1^2(u))^2}}=\frac{1}{m}.
\end{equation}
Thus the second equation of system~\eqref{viceversa} becomes
\begin{equation}\label{d}
d\theta=\frac{dt}{m}-\frac{B(u)\,(a\,B(u)-\tau\,\xi_1^2(u))}{\xi_1^2(u)+(a\,B(u)-\tau\,\xi_1^2(u))^2}\,d\xi_2.
\end{equation}

If we consider the function $f:=1/\xi_1$,  equation~\eqref{m} can be written as
$$(a^2-m^2\, U^2)\,\|\nabla f\|_{\kappa,\tau} ^2+(1-2 a \,\tau)\,\|\nabla f\|_{\kappa,\tau}+\tau^2-\kappa/4=0$$
and, therefore,
$$\|\nabla f\|_{\kappa,\tau}=\frac{1-2a\,\tau+\sqrt{\Delta}}{2\,(m^2\,U^2-a^2)},$$
with $$\Delta=(1-2a\,\tau)^2+(m^2\,U^2-a^2)(4\tau^2-\kappa).$$
As $\|\nabla f\|_{\kappa,\tau}=f^2+\kappa/4$, we conclude that
\begin{equation}\label{xi1}
\xi_1^2=\frac{4(m^2\,U^2-a^2)}{(1+\sqrt{\Delta})^2-4\tau^2m^2U^2}.
\end{equation}
Then, differentiating \eqref{m} and using \eqref{xi1}, we get
\begin{equation}\label{radice-delta}
m^2\,B^2\,U\,U'=\sqrt{\Delta}\,\xi_1\,\xi_1'
\end{equation}
and hence
\begin{equation}\label{x1B}
\frac{(\xi_1')^2}{B^2}=\frac{m^4\,B^2\, U^2\,U'^2}{\xi_1^2\,\Delta}.
\end{equation}
Therefore,  taking into account the first equation of system~\eqref{viceversa},  we obtain
$$\begin{aligned}
d\xi_2^2&=\frac{m^2\,B^2\,U^2}{\xi_1^2}\,\bigg(1-\frac{\xi_1'^2}{B^2}\bigg)d{u}^2\\
&=\frac{m^2\,B^2\,U^2}{\xi_1^4}\,\bigg(\xi_1^2-\frac{m^4\,B^2\, U^2\,U'^2}{\Delta}\bigg)d{u}^2.
\end{aligned}$$
Thus,  as
$$
B=1+\frac{\kappa}{4}\,\xi_1^2,
$$
 it turns out that
\begin{equation}\label{xi2}
\xi_2(u)=\int \frac{m\, U\,(4+\kappa\,\xi_1^2)}{4\,\xi_1^2}\,\sqrt{\xi_1^2-\frac{m^4\, U^2\,U'^2\,(4+\kappa\,\xi_1^2)^2}{16\,\Delta}}\,d{u}.
\end{equation}
Also,  from \eqref{d}  we have
\begin{equation}\label{theta}
\theta(u,t)=\frac{t}{m}+\int \frac{(4\,\tau-a\,\kappa)\,\xi_1^2-4 a}{4m\,U\,\xi_1^2}\,\,\sqrt{\xi_1^2-\frac{m^4\, U^2\,U'^2\,(4+\kappa\,\xi_1^2)^2}{16\,\Delta}}\,d{u}.
\end{equation}
Consequently,  the natural parametrization of the helicoidal surface \eqref{helicoidal} with given first
fundamental form $g_\psi=du^2+U(u)^2\,dt^2$  can be calculated by means of equations~\eqref{xi1},  \eqref{xi2} and \eqref{theta}.
\end{proof}

\begin{rem}
If $\kappa=0=\tau$, the BCV-space is the Euclidean space $\mbox{${\mathbb R}$}^3$ and  Theorem~\ref{bour} becomes the classical one (\cite{bour}, p.~82, Theorem~II) due to Bour. In fact, in this case the functions $B$ and $\Delta$ are constant and equal to $1$; thus we obtain
\begin{equation}\label{r3}
\left\{\begin{aligned}
\xi_1(u)&=\sqrt{m^2\,U^2-a^2},\\
\xi_2(u)&=\int \frac{m\, U}{m^2\,U^2-a^2}\,\sqrt{m^2\,U^2-a^2-m^4\, U^2\,U'^2}\,d{u},\\
\theta(u,t)&=\frac{t}{m}-\frac{a}{m}\int \frac{\sqrt{m^2\,U^2-a^2-m^4\, U^2\,U'^2}}{U\,(m^2\,U^2-a^2)}\,d{u}.
\end{aligned}
\right.
\end{equation}
\end{rem}

\begin{rem}
The family of helicoidal surfaces $\Psi(u,t):= \psi_{[U,m,a]}(u,t)$  in the BCV-space $\mbox{$\mathcal{N}_{\kappa,\tau}$}$ obtained in the Theorem~\ref{bour} depends on two parameters $m\neq 0$ and $a$, and for $m=1$ and $a=a_0$ it contains the original helicoidal surface.  Also, when $m=1$ and $a=0$,  we obtain  a rotational surface isometric to the given helicoidal surface. Therefore, by varying the constant $a$ from $a=0$ to $a=a_0$, we get an isometric deformation from a rotational surface to a given helicoidal surface.
\end{rem}

\begin{ex}\label{ex1}
In $\mbox{${\mathbb R}$}^3$, we consider the function $U(u)=\sqrt{u^2+d^2}$, $d \in \mbox{${\mathbb R}$}$, $d^2 \geq a^2$. If we suppose $m=1$,  from the formulas~\eqref{r3}
we obtain that
$$
\left\{\begin{aligned}
\xi_1(u)&=\sqrt{u^2+d^2-a^2},\\
\xi_2(u)&=\sqrt{d^2-a^2}\int \frac{\sqrt{u^2+d^2}}{u^2+d^2-a^2}\,d{u},\\
\theta(u,t)&=t-a\, \sqrt{d^2-a^2}\int \frac{du}{\sqrt{u^2+d^2}\, [u^2+d^2-a^2]}.
\end{aligned}
\right.
$$
By varying $a$ from $a=0$ to $a=d$ we have the classical isometric deformation of the catenoid $$\psi_{[U,1,0]}(u,t)=\bigg(\sqrt{u^2+d^2}\, \cos t,\sqrt{u^2+d^2}\,\sin t, d\cosh^{-1} \Big(\sqrt{\frac{u^2}{d^2}+1}\Big)\bigg)$$ into the helicoid $\psi_{[U,1,d]}(u,t)=(u \cos t,u \sin t, t+b)$, $b\in\mbox{${\mathbb R}$}$, that are minimal surfaces.
Also, the intermediate helicoidal surfaces are all minimal  and  their natural parametrization is given by
$$\left\{\begin{aligned}
\xi_1(u)&=\sqrt{u^2+d^2-a^2},\\
\xi_2(u)&=\sqrt{d^2 -a^2}\,\cosh^{-1} \Big(\sqrt{\frac{u^2}{d^2}+1}\Big)+
   a\,\arctan \Big(\frac{a\, u}{\sqrt {d^2 - a^2}\,
           \sqrt{u^2+d^2}}\Big),\\
           \theta(u,t)&=t-\arctan \Big(\frac{a\, u}{\sqrt {d^2 - a^2}\,
           \sqrt{u^2+d^2}}\Big), \qquad 0<a<d.
           \end{aligned}
\right.$$
Such surfaces are also called second  Scherk's surfaces (see \cite{hi} and, also, the Example~\ref{ex3}).
\end{ex}

\begin{ex}\label{ex-2}
In the Heisenberg space $\mbox{${\mathbb H}$}_3$ equipped with the metric $g_{\kappa,\tau}$ with $\kappa=0$ and $\tau=1/2$, we consider the function $U(u)=(u^2+2)/2$. If we suppose that $m=1$, from  formulas \eqref{xi-bis} we get
\begin{equation}\label{for-cat}
\left\{\begin{aligned}
\xi_1(u)&=\sqrt{\sqrt{u^4+4u^2+8\,(1-a)}+2\,(a-1)},\\
\xi_2(u)&=\int \frac{(2+u^2)}{2\, \xi_1(u)^2}\,\sqrt{\xi_1(u)^2-\frac{u^2\,(u^2+2)^2}{u^4+4u^2+8\,(1-a)}}\,d{u},\\
\theta(u,t)&=t+\int \frac{ \xi_1(u)^2-2 a}{ (u^2+2)\,\xi_1(u)^2}\,\,\sqrt{\xi_1(u)^2-\frac{u^2\,(u^2+2)^2}{u^4+4u^2+8\,(1-a)}}\,d{u}.
\end{aligned}
\right.
\end{equation}
In particular, for $a=1/2$ we obtain the curve
$$
\tilde{\gamma}(u)=(\sqrt{u^2+1},(u+\arctan u) /2),
$$
the profile curve of  the {\em helicoidal catenoid}, that is a helicoidal minimal surface (see \cite{FMP}), parametrized by
$$
\psi(u,\theta)=\Big(\sqrt{u^2+1}\,\cos \theta,\sqrt{u^2+1}\,\sin \theta,\frac{u+\theta+\arctan{u}}{2}\Big).
$$
Also, as $$\theta(u,t)=t-\arctan u+\sqrt{2}\arctan\Big(\frac{u}{\sqrt{2}}\Big),$$
we have that  the parametrization
$$\begin{aligned}
\Psi(u,t)&=\psi(u,\theta(u,t))=\bigg(\cos\Big(t+\sqrt{2}\arctan\Big(\frac{u}{\sqrt{2}}\Big)\Big)+u\sin\Big(t+\sqrt{2}\arctan\Big(\frac{u}{\sqrt{2}}\Big)\Big),\\
& \sin\Big(t+\sqrt{2}\arctan\Big(\frac{u}{\sqrt{2}}\Big)\Big)-u\sin\Big(t+\sqrt{2}\arctan\Big(\frac{u}{\sqrt{2}}\Big)\Big),\frac{1}{2} \Big(u+t+\sqrt{2}\arctan\Big(\frac{u}{\sqrt{2}}\Big)
\Big)\bigg)
\end{aligned}$$
represents a natural parametrization of the helicoidal catenoid with
$$g_{\Psi}=du^2+U(u)^2\,dt^2.$$
Now, if we start from $a=1/2$ and in the equations~\eqref{for-cat} we consider all the decreasing values of $a$ in the interval $[0,1/2]$,
  we obtain an isometric deformation of the helicoidal catenoid into a rotational surface (obtained for $a=0$), through helicoidal surfaces  parametrized by natural parameters.

  \begin{figure}[h!]\label{cat-defo}
  \begin{center}
\begin{tabular}{cc}
 \includegraphics[height=1.7in]{1-2.pdf} & \hspace{1.5cm}
 \includegraphics[height=1.4in]{1-4.pdf} \\
 $a=1/2$ & \hspace{1.5cm} $a=1/4$ \\
 \\
 \includegraphics[height=1.4in]{1-8.pdf} & \hspace{1.5cm}
 \includegraphics[height=1.3in]{zero.pdf} \\
 $a=1/8$ & \hspace{1.5cm} $a=0$ \\ \\
  \end{tabular}
  \caption{Isometric deformation of the helicodal catenoid into a rotation surface in $\mbox{${\mathbb H}$}_3$. The surfaces in the picture are obtained for $a=1/2, 1/4, 1/8$ and $a=0$, respectively; only that with angular pitch $a=1/2$ is a minimal surface (see Remark~\ref{rem-b}).}
\end{center}
\end{figure}

\end{ex}

\begin{cor}\label{coro1}
 The rotation surfaces given by $\psi_{[U,n,0]}(u,t)$, with $n\neq 0$, give rise in the BCV-space $\mbox{$\mathcal{N}_{\kappa,\tau}$}$ to  a $1$-parameter family of isometric surfaces  that are also isometric to the helicoidal surfaces $\psi_{[U,m,a]}(u,t)$. This family is determined by the formulas:
\begin{equation}\left\{\begin{aligned}\label{general}
\xi_1(u)&=\frac{2\, n\,U}{\sqrt{2(1+\sqrt{\Delta})-\kappa\, n^2 \,U^2}},\\
\xi_2(u)&=\int\sqrt{\frac{ (1+\sqrt{\Delta})^2}{2(1+\sqrt{\Delta})-\kappa\, n^2\, U^2}-\frac{n^2\,(1+\sqrt{\Delta})^4\,U^2}{\Delta\,[2(1+\sqrt{\Delta})-\kappa\, n^2\, U^2]^2}}\,d{u},\\
\theta(u,t)&=\frac{t}{n}+2\tau\int\sqrt{\frac{1}{2(1+\sqrt{\Delta})-\kappa\, n^2\, U^2}-\frac{n^2\,(1+\sqrt{\Delta})^2\,U'^2}{\Delta\,[2(1+\sqrt{\Delta})-\kappa\, n^2\, U^2]^2}}\,d{u},
\end{aligned}
\right.\end{equation}
where
$\Delta=1+(4\tau^2-\kappa)\,n^2\, U^2$. In particular,
\begin{enumerate}
\item if $\kappa=4\tau^2\neq 0$, i.e. $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong \mbox{${\mathbb S}$}^3(\tau^2)$, the equations~\eqref{general} become
$$\left\{\begin{aligned}
\xi_1(u)&=\frac{n\,U}{\sqrt{1-\tau^2\,n^2\, U^2}},\\
\xi_2(u)&=\int\frac{\sqrt{1-n^2\,(\tau^2\, U^2+U'^2)}}{1-\tau^2\,n^2\,U^2}\,d{u},\\
\theta(u,t)&=\frac{t}{n}+\tau\int\frac{\sqrt{1-n^2\,(\tau^2\, U^2+U'^2)}}{1-\tau^2\,n^2\,U^2}\,d{u},
\end{aligned}
\right.$$

\item if $\tau=0$, i.e. $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong \mbox{${\mathbb R}$}^3$, $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong \mbox{${\mathbb S}$}^2(\kappa)\times\mbox{${\mathbb R}$}$ or $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong \mbox{${\mathbb H}$}^2(\kappa)\times\mbox{${\mathbb R}$}$, the equations~\eqref{general}  become
$$\left\{\begin{aligned}
\xi_1(u)&=\frac{2n\,U}{1+\sqrt{1-\kappa\,n^2\, U^2}},\\
\xi_2(u)&=\int\sqrt{\frac{1-n^2\,(\kappa\, U^2+U'^2)}{1-\kappa\,n^2\,U^2}}\,d{u},\\
\theta(u,t)&=\frac{t}{n},
\end{aligned}
\right.$$
\item if $\kappa =0$ and $\tau\neq 0$, i.e. $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong \mbox{${\mathbb H}$}_3$, the equations~\eqref{general} become
$$\left\{\begin{aligned}
\xi_1(u)&=\frac{\sqrt{2}\, n\,U}{\sqrt{1+\sqrt{\Delta}}},\qquad \Delta=1+4\tau^2\,n^2\, U^2,\\
\xi_2(u)&=\int \frac{1+\sqrt{\Delta}}{2}\,\sqrt{\frac{2}{1+\sqrt{\Delta}}-\frac{n^2\,U'^2}{\Delta}}\,d{u},\\
\theta(u,t)&=\frac{t}{n}+\tau\int\sqrt{\frac{2}{1+\sqrt{\Delta}}-\frac{n^2\,U'^2}{\Delta}}\,d{u},
\end{aligned}
\right.$$
\item if $\kappa \neq 0$ and $\tau\neq 0$,  the equations~\eqref{general} give the family of rotation surfaces in the cases $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong\mathrm{SU}(2)$ and $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong\widetilde{\mathrm{SL}(2, \mbox{${\mathbb R}$})}$.
\end{enumerate}
\end{cor}

From the formula for the Gaussian curvature of an invariant surface obtained in \cite{MO}, it follows that  the helicoidal surfaces of the Bour's family in the BCV-space $\mbox{$\mathcal{N}_{\kappa,\tau}$}$ have all  the same Gaussian curvature given by
$$K(u)=-\frac{U''(u)}{U(u)}.$$ With regard to the mean curvature $H$ of these surfaces, in  next section we  shall see that  different values  of $a$ and $m$ can give rise to different values of $H$.
\section{Helicoidal surfaces of constant mean curvature}\label{4}
In this section, we will describe the helicoidal surfaces in the BCV-spaces that have the same constant mean curvature.
We start by computing the mean curvature of a helicoidal surface \eqref{helicoidal}. It turns out that the mean curvature of an invariant immersion is tightly  related to the
geodesic curvature of the profile curve, as shown by the remarkable following theorem. But first we recall that
if  on
 a three-dimensional connected Riemannian manifold $({N}^3,g)$ we consider   the $1$-parameter subgroup $G_X$ of isometries generated by $X$, an orbit $G(p)$ of $p \in N$ is called {\it principal} if there exists an open neighborhood  $U\subset N$ of $p$ such that all orbits $G(q)$, $q\in U$, are of the same type as $G(p)$ (i.e.  the isotropy subgroups $G_q$ and $G_p$ are coniugated). This implies that $G(q)$  is
diffeomorphic to $G(p)$. We denote with $N_r$ the {\it regular part} of $N$, that is, the subset consisting of points belonging to principal orbits \cite{alexandrino}. Then we have

\begin{thm}[Reduction Theorem \cite{docarmo}]\label{ridu}
Let $H$ be the mean curvature of a $G_X$-invariant surface $M_r\subset N_r$ and $k_g$ the
geodesic curvature of the profile curve $M_r/G_X\subset
\mathcal{B}_r$. Then
$$
H(x)=k_g(\pi(x))-D_{\textbf{n}}\ln \omega(\pi(x)), \qquad x\in M_r,
$$
where $\textbf{n}$ is the unit normal  of  the profile curve and
$\omega=\sqrt{g(X,X)}$ is the volume function of the principal
orbit.
\end{thm}

Let now $\tilde{\gamma}(u)=(\xi_1(u),\xi_2(u))$ be a curve in ${\mathcal B_r}$, parametrized by
arc-length,  that  under the action of $G_{X}$ generates the
helicoidal surface. From \eqref{orbital}, it follows that
\begin{equation}\label{sigma}
\xi_1'=B\,\cos \sigma,\qquad \xi_2'=\frac{\sqrt{\xi_1^2+(a\, B-\tau\,\xi_1^2)^2}\,\sin \sigma}{\xi_1}
\end{equation}
and the geodesic curvature of $\tilde{\gamma}$ takes the expression
\begin{equation}
\begin{aligned}\label{kg}
k_g=&\frac{(\widetilde{g}_{22})_{\xi_1} \xi_2'-(\widetilde{g}_{11})_{\xi_2} \xi_1' }{2\sqrt{\widetilde{g}_{11}\widetilde{g}_{22}}}+\sigma'\\
&=\frac{B\, [\xi_1^2+(a\, B-\tau\,\xi_1^2)^2]\,(\widetilde{g}_{22})_{\xi_1}}{2\,\xi_1^2}\,\sin\sigma+\sigma',
\end{aligned}
\end{equation}
where $\sigma$ is the angle that
$\tilde{\gamma}$ makes with the $\frac{\partial}{\partial \xi_1}$ direction. Also, as
$$\textbf{n}=\bigg(-B\,\sin \sigma,\frac{\sqrt{\xi_1^2+(a\, B-\tau\,\xi_1^2)^2}}{\xi_1}\cos\sigma \bigg),$$ the normal derivative is given by
$$D_{\textbf{n}}\ln \omega=\frac{4 \,\xi_1\, [8a \tau-4+ \xi_1^2\,(\kappa + 2 a \kappa\,\tau - 8\tau^2)]\,\sin\sigma}{
   16\,\xi_1^2 + [a\,(4 + \kappa\,\xi_1^2) - 4\tau\,\xi_1^2]^2}$$
and thus we obtain that the mean curvature is given by
\begin{equation}\label{h}
H=\sigma'+\bigg(\frac{1}{\xi_1}-\frac{\kappa}{4}\xi_1\bigg)\sin\sigma.
\end{equation}

\begin{prop}
A helicoidal surface $\Psi(u,t):= \psi_{[U,m,a]}(u,t)$ has constant mean curvature $H$ if and only if $U(u)$ satisfies the differential equation
\begin{equation}\label{eq-principal}
H\,\sqrt{\frac{4\,(m^2\,U^2-a^2)}{(1+\sqrt{\Delta})^2-4\tau^2m^2U^2}-\frac{m^4\,B^2\, U^2\,U'^2}{\Delta}}
=
2-B-m^2\,B\,\Big(\frac{U\,U'}{\sqrt{\Delta}}\bigg)',
\end{equation}
where
\begin{equation}\label{dB}
\Delta=(1-2a\,\tau)^2+(m^2\,U^2-a^2)(4\tau^2-\kappa)\quad \text{and} \quad B=2\frac{1-2 a \tau+\sqrt{\Delta}}{(1+\sqrt{\Delta})^2-4\tau^2m^2 U^2}.\end{equation}
\end{prop}
\begin{proof}
If we consider a helicoidal surface $\Psi(u,t)$ of the Bour's family, from \eqref{sigma}  we can write  equation~\eqref{h} as
 $$
H=-\frac{\big(\frac{\xi_1'}{B}\big)'}{\sqrt{1-\big(\frac{\xi_1'}{B}\big)^2}}+\bigg(\frac{1}{\xi_1}-\frac{\kappa}{4}\xi_1\bigg)\,\sqrt{1-\Big(\frac{\xi_1'}{B}\Big)^2}
$$
and therefore, using \eqref{x1B} we get
\begin{equation}\label{hbis}
\begin{aligned}
&-\Big(\frac{\xi_1'}{B}\Big)'+\bigg(\frac{1}{\xi_1}-\frac{\kappa}{4}\xi_1\bigg)\,\Big[1-\Big(\frac{\xi_1'}{B}\Big)^2\Big]=H\sqrt{1-\Big(\frac{\xi_1'}{B}\Big)^2}\\&=\frac{H}{\xi_1}\sqrt{\xi_1^2-\frac{m^4\,B^2\, U^2\,U'^2}{\Delta}}.
\end{aligned}
\end{equation}
Then,
\begin{equation}\label{radice-Q}
\begin{aligned}
&H\,\sqrt{\xi_1^2-\frac{m^4\,B^2\, U^2\,U'^2}{\Delta}}\\&=
\xi_1\,\bigg[-\Big(\frac{\xi_1'}{B}\Big)'+\bigg(\frac{1}{\xi_1}-\frac{\kappa}{4}\xi_1\bigg)\,\Big(1-\Big(\frac{\xi_1'}{B}\Big)^2\Big)\bigg]\\&=
1-\Big(\frac{\xi_1'}{B}\Big)^2-\frac{\xi_1\,\xi_1''}{B}+\frac{3\kappa\,\xi_1^2}{4}\Big(\frac{\xi_1'}{B}\Big)^2-\frac{k}{4}\xi_1^2.
\end{aligned}
\end{equation}
Now, deriving \eqref{radice-delta} we get
$$\xi_1\,\xi_1''=\frac{m^2\,B^2}{\sqrt{\Delta}}\,(U'^2+U\,U'')+\frac{m^4\,B^2}{\Delta}\bigg(\kappa\,B+\frac{\kappa-4\tau^2}{\sqrt{\Delta}}\bigg)\,U^2\,U'^2-\xi_1'^2$$
and hence, taking into account \eqref{xi1}, \eqref{radice-delta} and \eqref{x1B}, we can write equation~\eqref{radice-Q} as
$$\begin{aligned}
&H\,\sqrt{\xi_1^2-\frac{m^4\,B^2\, U^2\,U'^2}{\Delta}}
\\&=
2-B-\frac{m^2\,B}{\sqrt{\Delta}}\,(U'^2+U\,U'')+\frac{m^4\,B}{\Delta}\,\bigg(\frac{4\tau^2-\kappa}{\sqrt{\Delta}}-\frac{\kappa\,B}{4}+\frac{B^2-B}{\xi_1^2}\bigg)U^2\,U'^2
\\&=
2-B-\frac{m^2\,B}{\sqrt{\Delta}}\,(U'^2+U\,U'')+\frac{(4\tau^2-\kappa)\,m^4\,B}{\Delta^{3/2}}\,U^2\,U'^2.
\end{aligned}
$$

\end{proof}

\subsection{The solution of the mean curvature equation}
Next, we will give a description of the helicoidal surfaces in $\mbox{$\mathcal{N}_{\kappa,\tau}$}$ with constant mean curvature $H$. For this purpose, we assume that the helicoidal surfaces are parametrized by natural coordinates $(u,t)$ and we determine explicitly the expression of the function $U(u)$, that gives the metric, by integrating~\eqref{eq-principal}.
\begin{thm}\label{principal}
In the BCV-space $\mbox{$\mathcal{N}_{\kappa,\tau}$}$ the helicoidal surface $\Psi(u,t):=(\xi_1(u),\theta(u,t),\xi_2(u)+a\,\theta(u,t))$  with $\xi_1(u), \xi_2(u)$ and $\theta(u,t)$ given by \eqref{xi-bis} and \eqref{theta-bis}, has constant mean curvature $H$ if and only if
$U(u)$ is given by:

\begin{enumerate}
\item if $\kappa=\tau=H=0$, $$ U^2(u)=\frac{u^2+a^2+c^2/4}{m^2};$$
\item if $\kappa=4\tau^2\neq -H^2$,
$$U^2(u)=\frac{c_1+
\sqrt{c_1^2+c_2\, (H^2+4\tau^2)}\,\sin (\sqrt{4\tau^2+H^2}\,u)}{m^2(H^2+4\tau^2)};$$
\item  if $-H^2=\kappa\neq 4\tau^2$,
$$U^2(u)=\frac{\Big(\dfrac{b_1}{2}u^2+b_2\Big)^2+b_3}{m^2\,(4\tau^2+H^2)};$$
\item if $-H^2<\kappa\neq 4\tau^2$,
$$U^2(u)=\frac{(H^2+\kappa)^2\,b_3+
\Big[b_1+\sqrt{b_1^2+b\,(H^2+\kappa)}\,\sin(\sqrt{H^2+\kappa}\, u)\Big]^2}{m^2(4\tau^2-\kappa)(H^2+\kappa)^2};$$

\item if $-H^2>\kappa\neq 4\tau^2$,
$$
U^2(u)=\left\{\begin{aligned}
&\frac{(H^2+\kappa)^2\, b_3+
\Big[b_1-\sqrt{-b_1^2-b\,(H^2+\kappa)}\,\sinh(\sqrt{-(H^2+\kappa)}\, u)\Big]^2}{m^2(4\tau^2-\kappa)(H^2+\kappa)^2}, \quad b_1^2+b\,(H^2+\kappa)<0,\\
& \frac{(H^2+\kappa)^2\, b_3+
\Big[b_1-\sqrt{b_1^2+b\,(H^2+\kappa)}\,\cosh(\sqrt{-(H^2+\kappa)}\, u)\Big]^2}{m^2(4\tau^2-\kappa)(H^2+\kappa)^2},\qquad b_1^2+b\,(H^2+\kappa)>0,
\end{aligned}
\right.
$$
where  $b, b_i, c_i\in\mbox{${\mathbb R}$}$, are the constants given by
$$\begin{aligned}
b&=(1-2a\tau)\,[\kappa\,(1+2 a \tau)-8\tau^2]-c^2, \quad & b_1&=4\tau^2-2a\kappa\tau-c \,H, \\
b_2&=-\frac{b}{2\,b_1},\qquad & b_3&=4a\tau-a^2\kappa-1, &\\
c_1&= 1+ (1-2a\,\tau)^2-c\, H,  \quad & c_2&=-c^2-4 a^2\,(1-a\,\tau).
\end{aligned}$$
\end{enumerate}
These expressions define a $1$-parameter family $\{U_c(u)\}$ of functions $U(u)$ such that the helicoidal surface $\psi_{[U_c,m,a]}(u,t)$ has constant mean curvature and equal to $H$.
\end{thm}
\begin{proof}
Using the transformation of coordinates given by
\begin{equation}\label{tc}
\left\{
\begin{aligned}
x(u)&=m\,U(u),\\
 y(u)&=\sqrt{\frac{(x^2(u)-a^2)\,\,[(1+\sqrt{\Delta(u)})^2-4\tau^2\,x^2(u)]}{[1-2 a \tau+\sqrt{\Delta(u)}]^2}-\frac{x(u)^2\,x'(u)^2}{\Delta(u)}},
 \end{aligned}
 \right.
\end{equation}
where
$$
\Delta(u)=(1-2a\,\tau)^2+(4\tau^2-\kappa)(x^2(u)-a^2),$$
equation~\eqref{eq-principal} becomes
\begin{equation}\label{aux}
H\,y(u)=\frac{2}{B(u)}-1-\Bigg(\frac{x(u)\,x'(u)}{\sqrt{\Delta(u)}}\Bigg)',
\end{equation}
with
$$B(u)=2\frac{1-2 a \tau+\sqrt{\Delta(u)}}{(1+\sqrt{\Delta(u)})^2-4\tau^2\,x^2(u)}.$$
Therefore,
$$y'(u)=\frac{x(u)\,x'(u)}{y(u)\,\sqrt{\Delta(u)}}\bigg[\frac{2}{B(u)}-1-\Bigg(\frac{x(u)\,x'(u)}{\sqrt{\Delta(u)}}\Bigg)'\bigg]=\frac{H\,x(u)\,x'(u)}{\sqrt{\Delta(u)}}$$
and thus
\begin{equation}\label{solution}
y(u)=\left\{
\begin{aligned}
& \frac{H\,x^2(u)+c}{2\,\sqrt{\Delta(u)}}, \quad \qquad\;\; 4\tau^2-\kappa= 0,\\
&\frac{H\sqrt{\Delta(u)}+c}{4\tau^2-\kappa},\qquad \quad 4\tau^2-\kappa\neq 0,
\end{aligned}
\right.
\end{equation}
where  $c$ is an arbitrary constant. Consequently, we have the following cases:

\subsection*{Case $4\tau^2-\kappa= 0$}
From \eqref{tc} and \eqref{solution}, if we suppose that $1-2a\,\tau>0$, then we have
$$
\big(2\, x(u)\, x'(u)\big)^2  =
4 \big[(1-2a\,\tau) -\tau^2\,(x^2(u)-a^2)\big]\,(x^2(u)-a^2)-(H\, x^2(u)+c)^2.
$$
Putting $z=x^2$ the above expression is transformed into the following
\begin{equation}\label{eq-z}z'(u)=\sqrt{-(H^2+4\tau^2)\,z^2(u)+2c_1\,z(u)+c_2},
\end{equation}
where $$c_1= 1+ (1-2a\,\tau)^2-c\, H,  \qquad c_2=-c^2-4 a^2\,(1-a\,\tau).$$
Consequently,

\begin{itemize}
\item[i)] If $H^2+4\tau^2=0$, we have that the BCV-space is $\mbox{${\mathbb R}$}^3$ and $c_1=2$, $c_2=-c^2-4a^2$. Thus, adjusting the origin of $u$, we get
$$\sqrt{z+\frac{c_2}{4}}=u$$
and since $z=m^2\,U^2$ we have
\begin{equation}\label{U1}
U^2(u)=\frac{u^2+a^2+c^2/4}{m^2}.
\end{equation}
We observe that when $c=0$, from \eqref{r3} it follows that the helicoidal surface $\psi_{[U,m,a]}$ is a helicoid.
\item[ii)] If $H^2+4\tau^2\neq 0$, by integrating \eqref{eq-z} we have that
$$u=\frac{1}{\sqrt{H^2+4\tau^2}}\sin^{-1}\bigg(\frac{(H^2+4\tau^2)\,z-c_1}{\sqrt{c_1^2+c_2\,(H^2+4\tau^2)}}\bigg),$$
up to  a constant. Thus, as $z=m^2\,U^2$, it follows that
\begin{equation}\label{U2}\begin{aligned}
U^2(u)&=\frac{c_1+
\sqrt{c_1^2+c_2\, (H^2+4\tau^2)}\,\sin (\sqrt{4\tau^2+H^2}\,u)}{m^2(H^2+4\tau^2)}
\\&=\frac{1}{m^2(H^2+4\tau^2)}\,\bigg[1+ (1-2a\,\tau)^2-c\, H\\&+
2\sqrt{(1-2a\tau)^2-H\,(1-2a\tau) (Ha^2+c)-\tau^2\, (Ha^2+c)^2}\,\sin (\sqrt{H^2+4\tau^2}\,u)\bigg].
\end{aligned}
\end{equation}
\end{itemize}

\subsection*{Case $4\tau^2-\kappa\neq 0$}
In this case, as
$$\begin{aligned} &x^2(u)=\frac{1+a^2\kappa-4a\,\tau-\Delta (u)}{\kappa-4\tau^2}, \\
 &\frac{x(u)\,x'(u)}{\sqrt{\Delta(u)}}=\frac{(\sqrt{\Delta(u)})'}{4\tau^2-\kappa},
 \end{aligned}$$
 from \eqref{tc} and \eqref{solution} we get
$$\begin{aligned}
&(H\,\sqrt{\Delta(u)}+c)^2=(4\tau^2-\kappa)^2\,y^2(u)\\
&=(1-2a\tau-\sqrt{\Delta(u)})\,(\kappa+\kappa\sqrt{\Delta(u)}+2 a \kappa\tau-8\tau^2)-\Big((\sqrt{\Delta(u)})'\Big)^2.
\end{aligned}$$
Thus
\begin{equation}\label{eq-d-delta}
(\sqrt{\Delta(u)})'=\sqrt{-(H^2+\kappa)\,\Delta(u)+2\,b_1\sqrt{\Delta(u)}+b},
\end{equation}
where $$b=(1-2a\tau)\,(\kappa\,(1+2 a \tau)-8\tau^2)-c^2,\qquad b_1=4\tau^2-2a\kappa\tau-c \,H.$$
In the sequel we integrate equation~\eqref{eq-d-delta}, up to a change of the origin of $u$, considering the following possibilities:
\begin{itemize}
\item[i)]  If  $H^2+\kappa=0$, then we obtain that
$$\sqrt{\Delta(u)}=\frac{b_1}{2}u^2+b_2,$$
with
$$b_1=2a\tau \,H^2+4\tau^2-c\,H,\qquad b_2=-\frac{b}{2\,b_1}.$$
Then, substituting in the first equation of \eqref{dB}, we get
\begin{equation}\label{U3}
\begin{aligned}
U^2(u)&=\frac{1}{m^2}\bigg(\frac{\Delta(u)-(1-2a\tau)^2}{4\tau^2+H^2}+a^2\bigg)\\
&=\frac{\Big(\dfrac{b_1}{2}u^2+b_2\Big)^2+a^2H^2+4a\tau-1}{m^2\,(4\tau^2+H^2)}.
\end{aligned}
\end{equation}
\item[ii)] If $H^2+\kappa >0$, then the integration of \eqref{eq-d-delta} gives
$$\sqrt{\Delta(u)}=\frac{1}{H^2+\kappa}\,\Big[b_1+
\sqrt{b_1^2+b\,(H^2+\kappa)}\,\sin(\sqrt{H^2+\kappa}\, u)\Big].$$
Therefore, substituting in the first equation of \eqref{dB}, we obtain
\begin{equation}\label{U4}
U^2(u)=\frac{(H^2+\kappa)^2(4a\tau-a^2\kappa-1)+
\Big[b_1+\sqrt{b_1^2+b\,(H^2+\kappa)}\,\sin(\sqrt{H^2+\kappa}\, u)\Big]^2}{m^2(4\tau^2-\kappa)(H^2+\kappa)^2}.
\end{equation}
\item[iii)] If $H^2+\kappa <0$, then the integration of \eqref{eq-d-delta} gives
$$\sqrt{\Delta(u)}=\left\{\begin{aligned}
&\frac{1}{H^2+\kappa}\,\Big[b_1-\sqrt{-b_1^2-b\,(H^2+\kappa)}\,\sinh(\sqrt{-(H^2+\kappa)}\, u)\Big],\quad b_1^2+b\,(H^2+\kappa)<0,\\
&\frac{1}{H^2+\kappa}\,\Big[b_1-\sqrt{b_1^2+b\,(H^2+\kappa)}\,\cosh(\sqrt{-(H^2+\kappa)}\, u)\Big],
\qquad b_1^2+b\,(H^2+\kappa)>0.
\end{aligned}
\right.
$$
Therefore, substituting in the first equation of \eqref{dB}, we obtain
\begin{equation}\label{U5}
U^2(u)=\left\{\begin{aligned}
&\frac{(H^2+\kappa)^2\, b_3+
\Big[b_1-\sqrt{-b_1^2-b\,(H^2+\kappa)}\,\sinh(\sqrt{-(H^2+\kappa)}\, u)\Big]^2}{m^2(4\tau^2-\kappa)(H^2+\kappa)^2}, \quad b_1^2+b\,(H^2+\kappa)<0,\\
& \frac{(H^2+\kappa)^2\, b_3+
\Big[b_1-\sqrt{b_1^2+b\,(H^2+\kappa)}\,\cosh(\sqrt{-(H^2+\kappa)}\, u)\Big]^2}{m^2(4\tau^2-\kappa)(H^2+\kappa)^2},\qquad b_1^2+b\,(H^2+\kappa)>0.
\end{aligned}
\right.
\end{equation}
\end{itemize}
\end{proof}
\begin{rem}
In particular, if we consider $m=1$ and $a=a_0$ in the expressions of $U(u)$ obtained in Theorem~\ref{principal}, we see that an arbitrary helicoidal surface in $\mbox{$\mathcal{N}_{\kappa,\tau}$}$ has constant mean curvature $H$ if and only if the functions $\xi_1(u), \xi_2(u)$ and $\theta(u,t)$ are given by \eqref{xi-bis} and \eqref{theta-bis} by substituting  the corresponding function $U(u)$.
\end{rem}
\begin{rem}
Putting $\kappa=\tau=0$ in the equations~\eqref{U1} and \eqref{U2} we obtain the following expressions:
$$U^2(u)=\left\{
\begin{aligned}
&\frac{u^2+a^2+c^2/4}{m^2},\qquad \text{if}\quad H=0,\\
&\frac{2-c\,H+2\sqrt{1-c\,H-a^2\,H^2}\sin{(H\,u)}}{m^2\,H^2},\quad \text{if}\quad H\neq 0,
\end{aligned}
\right.$$
the second of which was given by Do Carmo and Dajczer in \cite{DD}.
\end{rem}
In regard to the helicoidal minimal surfaces we have the following result:
\begin{cor}\label{coro}
In  each BCV-space $\mbox{$\mathcal{N}_{\kappa,\tau}$}$, the helicoidal minimal surfaces $\Psi(u,t):=(\xi_1(u),\theta(u,t),\xi_2(u)+a\,\theta(u,t))$, where $\xi_1(u), \xi_2(u)$ and $\theta(u,t)$ are given by \eqref{xi-bis} and \eqref{theta-bis},  are determined
by the following functions $U(u)$:
\begin{enumerate}
\item If $\kappa=\tau=0$, then $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong \mbox{${\mathbb R}$}^3$ and
\begin{equation}\label{u-minimal}
 U^2(u)=\frac{u^2+a^2+c^2/4}{m^2};
 \end{equation}
\item if $\kappa=4\tau^2\neq 0$, then $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong \mbox{${\mathbb S}$}^3(\tau^2)$ and
$$U^2(u)=\frac{1-a\,\tau+\sqrt{(1-2a\tau)^2-\tau^2\,c^2}\sin(2\tau u)}{4 m^2\tau^2},$$
with $|c|<|1/\tau-2a|$;
\item if $\kappa >0$ and $\tau=0$, then $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong \mbox{${\mathbb S}$}^2(\kappa)\times\mbox{${\mathbb R}$}$ and
$$U^2(u)=\frac{\kappa\,(a^2\kappa+1)+(c^2-\kappa)\,\sin^2(\sqrt{\kappa}\, u)}{m^2\,\kappa^2},\qquad |c|<\sqrt{\kappa};$$
\item if $\kappa <0$ and $\tau=0$, then $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong \mbox{${\mathbb H}$}^2(\kappa)\times\mbox{${\mathbb R}$}$
and
$$U^2(u)=\frac{\kappa\,(a^2\kappa+1)+(c^2-\kappa)\,\cosh^2(\sqrt{-\kappa}\, u)}{m^2\,\kappa^2};$$
\item if $\kappa =0$ and $\tau\neq 0$, then $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong \mbox{${\mathbb H}$}_3$ and
\begin{equation}\label{u-minimal-2}
U^2(u)=\frac{\Big(2\,\tau^2 u^2+1-2a\tau+\dfrac{c^2}{8\tau^2}\Big)^2+4a\,\tau -1}{4m^2\,\tau^2},
\end{equation}
\item if $\kappa >0$ and $\tau\neq 0$, then $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong\mathrm{SU}(2)$ and
$$U^2(u)=\frac{\kappa^2\,(4a\tau-a^2\kappa-1)+\Big[4\tau^2-2 a \kappa\tau+\sqrt{(4\tau^2-\kappa)^2-c^2\kappa}\,\sin(\sqrt{\kappa}\,u)\Big]^2}{m^2\kappa^2\,(4\tau^2-\kappa)},$$
with $|c|<|4\tau^2-\kappa|/\sqrt{k}$;
\item if $\kappa <0$ and $\tau\neq 0$, then $\mbox{$\mathcal{N}_{\kappa,\tau}$}\cong\widetilde{\mathrm{SL}(2, \mbox{${\mathbb R}$})}$ and
$$U^2(u)=\frac{\kappa^2\,(4a\tau-a^2\kappa-1)+\Big[4\tau^2-2 a \kappa\tau-\sqrt{(4\tau^2-\kappa)^2-c^2\kappa}\,\cosh(\sqrt{-\kappa}\,u)\Big]^2}{m^2\kappa^2\,(4\tau^2-\kappa)}.$$
\end{enumerate}
\end{cor}

\begin{ex}[General helicoidal minimal surface in $\mbox{${\mathbb R}$}^3$]\label{ex3}
In $\mbox{${\mathbb R}$}^3$ we consider the function \eqref{u-minimal} for $m=1$ replacing,  for semplicity,  the constant $c/2$ by $c$.  From the formulas~\eqref{r3}
we obtain that the natural parametrization of a general helicoidal minimal surface in the euclidean space
 is given by
\begin{equation}\label{scherk}
\left\{\begin{aligned}
\xi_1(u)&=\sqrt{u^2+c^2},\\
\xi_2(u)&=c\,\cosh^{-1} \Big(\sqrt{\frac{u^2}{a^2+c^2}+1}\Big)+
   a\,\arctan \Big(\frac{a\, u}{c\,
           \sqrt{u^2+a^2+c^2}}\Big),
           \\      \theta(u,t)&=t-\arctan \Big(\frac{a\, u}{c\,
       \sqrt{u^2+a^2+c^2}}\Big).
           \end{aligned}
\right.
\end{equation}
This family of surfaces  called second Scherk's surfaces\footnote{In 1835 H.F.~Scherk made an important contribution to  minimal surfaces theory with his work \cite{S} that contains the first examples of minimal surfaces obtained from the integral of Monge and Legendre. Also, he investigated   minimality of surfaces given as graphs $z=z(r,\theta)$ (where $(r,\theta,z)$ are cylindrical coordinates in $\mbox{${\mathbb R}$}^3$)  satisfying  the condition $\frac{\partial^2 z}{\partial r\partial\theta}=0$, and determined all the helicoidal minimal surfaces. Detailed accounts and further information can be found in \cite{clapier}, on p. ~60, and in \cite{da}, on p. ~327.},
includes the catenoid and the helicoid, that correspond to the cases $a=0$ and $c=0$, respectively. We observe that all the surfaces of the family for which the sum $a^2+c^2$ is the same are isometric to each other. Also, from Example~\ref{ex1},  it follows that every helicoidal minimal surface in $\mbox{${\mathbb R}$}^3$ belongs to one of the families of isometric surfaces obtained deforming catenoids into helicoids.
\end{ex}

\begin{rem}\label{rem-b}
Among the helicoidal surfaces in $\mbox{${\mathbb H}$}_3$ obtained in   Example~\ref{ex-2} the only minimal surface is the helicoidal catenoid because the function $U(u)=(u^2+2)/2$ can be obtained just choosing in \eqref{u-minimal-2} $c^2=1=m$ and $a=1/2$.
\end{rem}

redpajama_set_name: RedPajamaArXiv


Designed to flatter straight body types, the Mazy features a waist-cinching high rise and slim fit through the hip and thigh. It's crafted in stretch Fluid denim that's ultra soft and body shaping. This black pair is finished with an ankle-length inseam, frayed hem and tonal stitching on the back pockets.
Mazy Ankle Skinny Black is rated 4.7 out of 5 by 3.
Rated 5 out of 5 by JTJT from Love all my silver jeans Even though I am tall and these are cut short they still fit great, the stretch in the fabric is just the right amount, the black is wonderful. Wish there were more black options- especially with bling or embroidery.
Rated 4 out of 5 by aad82 from Nice for a chabge They stretched out in the waist a bit after wearing. Really soft and comfy. Slightly higher in the waist which helps hide the muffin top!


text:
"""
24. Mutually referential many-to-one relationships

To define a many-to-one relationship, use ``ForeignKey()`` .
"""

from django.db.models import *

class Parent(Model):
    name = CharField(maxlength=100, core=True)
    bestchild = ForeignKey("Child", null=True, related_name="favoured_by")

class Child(Model):
    name = CharField(maxlength=100)
    parent = ForeignKey(Parent)

__test__ = {'API_TESTS':"""
# Create a Parent
>>> q = Parent(name='Elizabeth')
>>> q.save()

# Create some children
>>> c = q.child_set.create(name='Charles')
>>> e = q.child_set.create(name='Edward')

# Set the best child
>>> q.bestchild = c
>>> q.save()

>>> q.delete()

"""}

redpajama_set_name: RedPajamaGithub


text:
Q: WordPress functions.php Problem Can a function inside the functions.php file call another function from within functions.php? I'm guessing yes and which is why I wrote the code below, but it doesn't work for some reason. Can anyone please check it out and help me.
I tried calling pageBarColor() from register_sidebar()
Thanks.
<?php
if (function_exists('register_sidebar')) {
  register_sidebar(array(
   'before_widget' => '
',
   'after_widget' => '
',
   'before_title' => '
',
   'after_title' => '
',
  ));
}

function pageBarColor(){
    if(is_category('3')) {
        return "color1";
    } elseif(is_category('4')) {
        return "color2";
    } elseif(is_category('5')) {
        return "color3";
    } elseif(is_category('6')) {
        return "color4";
    } elseif(is_category('7')) {
        return "color5";
    }
}
?>

A: The problem is probably that when you call register_sidebar Wordpress has not yet executed the code which determines the result of is_category. If you try calling your pageBarColor function straight after defining it you'll find it doesn't return anything. One way of working around this would be to hook into the dynamic_sidebar_params filter (which is called when you call dynamic_sidebar in your templates, assuming you do) and update your widget before_title values, something like this:
function set_widget_title_color($widgets) {
    foreach($widgets as $key => $widget) {
        if (isset($widget["before_title"])) {
            if(is_category('3')) {
                $color = "color1";
            } elseif(is_category('4')) {
                $color = "color2";
            } elseif(is_category('5')) {
                $color = "color3";
            } elseif(is_category('6')) {
                $color = "color4";
            } elseif(is_category('7')) {
                $color = "color5";
            }

            if (isset($color)) $widgets[$key]["before_title"] = str_replace("moduleTitle", "moduleTitle ".$color, $widget["before_title"]);
        }
    }
    return $widgets;
}
add_filter('dynamic_sidebar_params', 'set_widget_title_color');

redpajama_set_name: RedPajamaStackExchange


text:
About OSI
Optoelectronics and Manufacturing
OSI Systems Announces First Quarter Conference Call
Home » Investor Relations » OSI Systems Announces First Quarter Conference Call
HAWTHORNE, Calif.--(BUSINESS WIRE)--Oct. 17, 2018-- OSI Systems, Inc. (Nasdaq: OSIS) today announced that it will hold its quarterly conference call on Thursday, October 25, 2018, at 1:30 p.m. PT (4:30 p.m. ET) to discuss its results for the first quarter ended September 30, 2018.
This call is being webcast through West Digital Solutions and the investor relations section at www.osi-systems.com.
A replay of the conference call is expected to be available from 4:30 p.m. PT (7:30 p.m. ET) on October 25, 2018 until November 9, 2018. The replay may be accessed by visiting the investor relations section at www.osi-systems.com.
About OSI Systems, Inc.
OSI Systems, Inc. is a vertically integrated designer and manufacturer of specialized electronic systems and components for critical applications in the homeland security, healthcare, defense and aerospace industries. We combine more than 40 years of electronics engineering and manufacturing experience with offices and production facilities in more than a dozen countries to implement a strategy of expansion into selective end product markets. For more information on OSI Systems, Inc. or any of its subsidiary companies, visit www.osi-systems.com. News Filter: OSIS-G
Source: OSI Systems, Inc.
OSI Systems, Inc.
Ajay Vashishat
Vice President, Business Development
avashishat@osi-systems.com
Privacy | Terms of Use | Cookies | © 2023 OSI Systems, Inc.


text:
S'inscrire Connexion
Nous joindre Distribution globale Services de distribution Dépôts réglementaires Partenaires médias
REC Silicon - Private placement successfully completed
Email Print Friendly Share
14 oct. 2020 18h20 HE | Source : REC Silicon ASA
NOT FOR RELEASE, PUBLICATION OR DISTRIBUTION, IN WHOLE OR IN PART DIRECTLY OR INDIRECTLY, IN AUSTRALIA, CANADA, JAPAN OR THE UNITED STATES OR ANY OTHER JURISDICTION IN WHICH SUCH RELEASE, PUBLICATION OR DISTRIBUTION WOULD BE UNLAWFUL.
Lysaker, 14 October 2020.
Reference is made to the stock exchange release from REC Silicon ASA ("REC" or the "Company") published on 14 October 2020 regarding a contemplated private placement (the "Private Placement"). The Company is pleased to announce that it has raised approximately NOK 1,000 million in gross proceeds through the Private Placement of 92,592,592 new shares (the "New Shares"), at a price per share of NOK 10.80 (the "Offer Price"). The Private Placement took place through a process managed by Arctic Securities AS and Pareto Securities AS (the "Managers"). The Private Placement was substantially oversubscribed.
The net proceeds from the Private Placement will be used to fund expansion investments and activities at the Company's facilities in Butte and Moses Lake as well as for general corporate purposes. Upon completion of the Private Placement, the Company will have established an extended liquidity runway which may enable the Company to prepare for a restart of the Moses Lake plant. Such restart is targeted in 2022 on the back of expected continuation of positive developments currently seen in both the solar energy and battery markets where the Company believes that it is positioned to become a leading US provider of silane and polysilicon.
The Private Placement consists of two separate tranches; one tranche with 27,982,000 New Shares in the Company ("Tranche 1") and a second tranche with 64,610,592 New Shares ("Tranche 2"). Settlement for Tranche 1 is expected to be on or about 19 October 2020 (DVP, T+2), while settlement in Tranche 2 is expected to occur on or about 12 November 2020 following approval by an extraordinary general meeting in the Company of the issuance of the New Shares under Tranche 2 (the "EGM"). The investors who have been allocated New Shares have received a pro rata portion of shares from Tranche 1 and Tranche 2 based on their overall allocation in the Private Placement, with the exception of Aker Capital AS who has agreed that the New Shares it is allocated in the Private Placement will all be allocated in Tranche 2.
The delivery of New Shares will be settled with existing and unencumbered shares in the Company that are already listed on the Oslo Stock Exchange, pursuant to a share lending agreement between Arctic Securities AS (on behalf of the Managers), the Company and Aker Capital AS. Accordingly, the shares delivered to the investors will be tradable upon delivery, expected on or about 19 October 2020 in Tranche 1 and on or about 12 November 2020 in Tranche 2.
The completion of Tranche 1 is subject to approval by the Company's board of directors (the "Board") pursuant to an authorization granted by the Company's annual general meeting held on 12 May 2020. The completion of Tranche 2 is subject to the approval of issuance of shares under Tranche 2 by the EGM. Completion of Tranche 1 is not conditional upon completion of Tranche 2, and acquisition of shares under Tranche 1 will remain final and binding and cannot be revoked or terminated by the respective applicants if Tranche 2 is not completed.
The share loan from Aker Capital AS will be settled with (i) new shares in the Company to be resolved issued by the Board pursuant to an authorisation granted by the Company's annual general meeting held on 12 May 2020 (Tranche 1) and (ii) new shares in the Company to be issued following, and subject to, approval by the EGM (Tranche 2). The new shares re-delivered to Aker Capital AS in Tranche 2 will be issued on a separate ISIN and will not be tradable on the Oslo Stock Exchange until a listing prospectus has been approved by the Financial Supervisory Authority of Norway (the "Prospectus").
Subject to satisfaction of the conditions for completion of the Private Placement, the Board of the Company has resolved to propose that the EGM resolves to authorize the Board to carry out a subsequent offering of up to 18,518,518 new shares (the "Subsequent Offering") towards the Company's shareholders as of 14 October 2020 (as documented by the shareholder register in the Norwegian Central Securities Depository (VPS) as of the end of 16 October 2020, the "Record Date") who were not allocated shares in the Private Placement and who are not resident in a jurisdiction where such offering would be unlawful or, for jurisdictions other than Norway, would require any prospectus, filing, registration or similar action ("Eligible Shareholders").
The subscription price in the Subsequent Offering will be the same as the Offer Price in the Private Placement, i.e. NOK 10.80 per share. The Eligible Shareholders will receive non-transferrable subscription rights based on their shareholding as of the Record Date in the Subsequent Offering. Over-subscription with subscription rights is permitted, but subscription without subscription rights is not permitted. The Subsequent Offering will be conditional, inter alia, on (a) completion of the Private Placement, (b) approval of the authorization of the Board for the Subsequent Offering by the Company's shareholders at the EGM, and (c) approval and publication of the Prospectus by the Company. The Company will issue a separate stock exchange notice with further details on the Subsequent Offering when finally resolved.
The waiver of the preferential rights inherent in a private placement is considered necessary in the interest of time and successful completion. Taking into consideration the time, costs and expected terms of alternative methods of
the securing the desired funding, as well as the proposed Subsequent Offering, the Board has concluded that the completion of the Private Placement on the proposed terms at this time is in the mutual interest of the Company and its shareholders, and that the Private Placement complies with the equal treatment obligations under the Norwegian Securities Trading Act and the Oslo Stock Exchange's Circular no. 2/2014.
Arctic Securities AS and Pareto Securities AS are engaged as managers and joint bookrunners in the Private Placement and the Subsequent Offering. Advokatfirmaet Schjødt AS is acting as legal counsel to the Company
in connection with the Private Placement and the Subsequent Offering. Advokatfirmaet Thommessen AS is acting as legal advisor to the Managers in connection with the Private Placement and the Subsequent Offering.
James A. May II, Chief Financial Officer
Email: james.may@recsilicon.com
Nils O. Kjerstad
Email: nils.kjerstad@crux.no
REC Silicon is a leading producer of advanced silicon materials, delivering high-purity polysilicon and silicon gas to the solar and electronics industries worldwide. We combine over 30 years of experience and proprietary technology with the needs of our customers, with annual production capacity of more than 20,000 MT of polysilicon from our two US-based manufacturing plants. Listed on the Oslo Stock Exchange (ticker: REC), the Company is headquartered in Lysaker, Norway. For more information, please visit: www.recsilicon.com
This information is subject to the disclosure requirements pursuant to section 5-12 of the Norwegian Securities Trading Act.
This announcement is not and does not form a part of any offer to sell, or a solicitation of an offer to purchase, any securities of the Company. Copies of this announcement are not being made and may not be distributed or sent into any jurisdiction in which such distribution would be unlawful or would require registration or other measures.
The securities referred to in this announcement have not been and will not be registered under the U.S. Securities Act of 1933, as amended (the "Securities Act"), and accordingly may not be offered or sold in the United States absent registration or an applicable exemption from the registration requirements of the Securities Act and in accordance with applicable U.S. state securities laws. The Company does not intend to register any part of the offering in the United States or to conduct a public offering of securities in the United States. Any sale in the United States of the securities mentioned in this announcement will be made solely to "qualified institutional buyers" as defined in Rule 144A under the Securities Act.
In any EEA Member State, this communication is only addressed to and is only directed at qualified investors in that Member State within the meaning of the Prospectus Regulation, i.e., only to investors who can receive the offer without an approved prospectus in such EEA Member State. The expression "Prospectus Regulation" means Regulation (EU) 2017/1129 as amended (together with any applicable implementing measures in any Member State.
This communication is only being distributed to and is only directed at persons in the United Kingdom that are (i) investment professionals falling within Article 19(5) of the Financial Services and Markets Act 2000 (Financial Promotion) Order 2005, as amended (the "Order") or (ii) high net worth entities, and other persons to whom this announcement may lawfully be communicated, falling within Article 49(2)(a) to (d) of the Order (all such persons together being referred to as "relevant persons"). This communication must not be acted on or relied on by persons who are not relevant persons. Any investment or investment activity to which this communication relates is available only for relevant persons and will be engaged in only with relevant persons. Persons distributing this communication must satisfy themselves that it is lawful to do so.
Matters discussed in this announcement may constitute forward-looking statements. Forward-looking statements are statements that are not historical facts and may be identified by words such as "believe", "expect", "anticipate", "strategy", "intends", "estimate", "will", "may", "continue", "should" and similar expressions. The forward-looking statements in this release are based upon various assumptions, many of which are based, in turn, upon further assumptions. Although the Company believe that these assumptions were reasonable when made, these assumptions are inherently subject to significant known and unknown risks, uncertainties, contingencies and other important factors which are difficult or impossible to predict, and are beyond their control. Actual events may differ significantly from any anticipated development due to a number of factors, including without limitation, changes in public sector investment levels, changes in the general economic, political and market conditions in the markets in which the Company operates, the Company's ability to attract, retain and motivate qualified personnel, changes in the Company's ability to engage in commercially acceptable acquisitions and strategic investments, and changes in laws and regulation and the potential impact of legal proceedings and actions. Such risks, uncertainties, contingencies and other important factors could cause actual events to differ materially from the expectations expressed or implied in this release by such forward-looking statements. The Company does not make any guarantee that the assumptions underlying the forward-looking statements in this announcement are free from errors nor does it accept any responsibility for the future accuracy of the opinions expressed in this announcement or any obligation to update or revise the statements in this announcement to reflect subsequent events. You should not place undue reliance on the forward-looking statements in this announcement.
The information, opinions and forward-looking statements contained in this announcement speak only as at its date, and are subject to change without notice. The Company does not undertake any obligation to review, update, confirm, or to release publicly any revisions to any forward-looking statements to reflect events that occur or circumstances that arise in relation to the content of this announcement.
Neither the Managers nor any of their affiliates makes any representation as to the accuracy or completeness of this announcement and none of them accepts any responsibility for the contents of this announcement or any matters referred to herein.
This announcement is for information purposes only and is not to be relied upon in substitution for the exercise of independent judgment. It is not intended as investment advice and under no circumstances is it to be used or considered as an offer to sell, or a solicitation of an offer to buy any securities or a recommendation to buy or sell any securities of the Company. Neither the Managers nor any of its affiliates accepts any liability arising from the use of this announcement.
The distribution of this announcement and other information may be restricted by law in certain jurisdictions. Persons into whose possession this announcement or such other information should come are required to inform themselves about and to observe any such restrictions.
Autres communiqués de presse diffusés par REC Silicon ASA
Autres communiqués de presse en lien avec
European Regulatory News
S'abonner via RSS
S'abonner via ATOM
Sandvika, NORWAY
Fichiers multimédia
URL DU LOGO | Copier le lien ci-dessous
Formats disponibles :
Réseaux de diffusion de nouvelles
Questions légales
GlobeNewswire est l'un des plus grands réseaux de diffusion de nouvelles dans le monde et se spécialise dans la diffusion de communiqués de presse, de renseignements financiers et de contenu multimédia des entreprises auprès des médias, de la communauté financière, des investisseurs individuels et du grand public.
© 2021 GlobeNewswire, Inc. Tous droits réservés.