I'm sure this has been discussed but....you know...as the search is broken etc. etc.
I'm on the list for the second batch of Cirklon and would really appreciate some user feedback, especially if anyone who bought one got the CV/IO expansion too (who wouldn't?).
I'm trying to decide whether to drop the cash on that or try (well, plan) building something modular.
i was hoping you'd grab one so i could check it out and see if its for me. im thinking of reducing my euro sequencing so maybe see how you get on with it and we could talk swapsies/sales.
That was my idea. Get your own ideas.
Do the right thing, get them both!
I use my Cirklon with my OT, and between them, there isn't a whole lot that can't be done. I have been having the rhythmic manipulated sampled stuff going on on the OT synched to the Cirlklon, sequencing external sources with the Cirklon and running them into the OT for both mixing and effects. It rocks!
The aux events on the Cirklon are worth the asking price alone. If you're curious about it, check out the P3 (the Cirklon's predecessor) manual. The aux-events are basically unchanged, as of now.
It's the Aux functions that are really whetting my appetite for the Cirklon.
I already have an Octatrack and while it's MIDI/sequencing capabilities get better with every release, I think my needs are somewhere between that and the Cirklon.
That's why I'm wondering if I just built something modular, something a bit quirky (yes, I'm looking at you 'Rene'...) then I might have my needs sated without dropping a grand on a Cirklon.
Is it easy to do triplets on one of those?
come on halvard- more general details needed here. whats so great about it, how does it compare to other seq's i may have used. how fluid is it? im still not sure i like analogue style sequencing akll that much, though the cirklon obviously builds on that in a huge way. will it take pitch from an external keyboard by the way- i mean can i input data using a keyboard via midi in?
I have my cvio box (banana version, made by Colin) finally arriving in a few days.
I have not been using the cirklon much until now, and hope that this will inspire me to get started on it again.
My problems with it thus far are as follows. I don't write 'songs' - what I really want in a sequencer is something that is very similar to an analog sequencer, but with lots more of it, and then some uniquely midi features built in.
I love the aux events, but they are embedded in a structure that just drives me batty. It takes me a few seconds to set up something on the schrittmacher where I have notes on track one, with the velocity values (from a sequence of another length with different timings) on track 2, with the note timings on track 3, and with the pitches being modulated by track 4. I can then mess with the specific values, but setting this up is very, very fast. I find it somewhat of a pain to do on the cirklon, which at this point (for my use) is simply a p3 with a nicer display.
The cirklon also does not have nearly the granularity of individual event times that the schrittmacher does, and it is not anywhere near as easy to set up a track where the individual notes have different timings.
I've spoken to Colin about my issues, which admittedly are not the usual ones, and he seems to have some empathy for someone who is using the sequencer in this way. As a starting point, I'd really like to have him allow the different event types within a track to have their own # of steps, and he agreed in theory to do this....but I'm just not sure that any of this will happen. The design of the instrument is just so, so heavily influenced by song writers, especially Paul Nagle, who just pull him in another direction.
Colin is great - I can understand where he's coming from with all of this. I've really never had this level of support - I asked him for nrpn support (to more completely control the gotharman anamono) and he implemented it in a couple of days.
I tend to not 'write songs' so it's interesting to read your struggles with the Cirklon.
be interested to know how you get on nelson. the schritmacher really appeals too, though id need a midi to cv with that which puts me off. its a shame the only way you can experience stuff like this is by buying it but there you go. the cirklon seems to tie everything together into one box which appeals.
btw, that cvi/o box, is it something easily diy-able?
neil as i said if you do decide against the cirklon let me know and ill likely take it off your hands... maybe try it first and then decide.
The CVIO is easily DIY-able. If you can hack a printer cable apart and solder jacks to the wires, then you can do it. If you can drill holes in a panel and mount the jacks, then it's essentially what Colin is offering soon.
I don't write "songs", but listening to a pattern loop for a hour gets a bit much. so the ability to chain patterns together into songs is what allows you to take your pattern, copy paste it, change it, copy paste it etc.. build something that evolves.
The aux events can be used to construct something that evolves too of course. So can simply using patterns of different lengths.
Setting up tracks and patterns isn't instant but it is by no means cumbersome or a pita. Like anything with massive functionality and depth you have to get used to using it. For something that run your entire studio, it's pretty slick.
I do hope the feature creep ends soon and UI optimization begins. The problem with Colin's unrivalled "customer support" is that there exists the possibility of adding too many features that might clutter up the usability of the UI's first "layer".. if you know what I mean. But watching all the requests come in, I can't imagine anything else that caters to so many different modes of operation.
Even Nelson's way of looking at it as a big rack of analog sequencers is right there, albeit not as instantly as Schrittmacher. Although CV outs arguably puts it ahead of Schrittmacher in that aspect.
Think Techno tracks - That's Cirklon's roots.
I have a cirklon (without the cvio on order..shock horror, but I have no cv gear and don't want to go down that road just yet).
I haven't used a schrittmacher but can understand where Nelson is coming from with regards setting up tracks where velocity, note gates and note values are on different lengths..i used to do this on my doepfer maq and it was a great source of evolving patterns. However, I think the cirklons way of doing it is, whilst slightly slower, much more advanced (than what I understand of other seqs). Being able to grab, swap or push values between other events but only on certain steps (and modulating between different tracks each time). And then assigning accumulators (essentially similar to LFOs) or randomness generators to change when things happen. Ie. you can do everything the schrittmacher can do but with more control.
As mentioned Colin has said multiphasic patterns (tracks with different lengths) are on his list, along with a number of other things.
I'm confused what you mean about control over individual note times? You can set each notes duration and delay to 48ths of 1/16th. That certainly enough flexibility for me. You can also use an aux to change timebase which is a great feature for interesting rhythms.
However, where the cirklon is by far the best thing I've ever used is in terms of it's UI. The gang feature and encoder slope feature are absolutely amazing.
It's also the little things. I sold my motu midi express since getting it since it basically serves as my USB midi router now. It also works as a mungo sync, allowing you to to stop say an elektron mid-live set, load a new snapshot or project, and then re-start without stopping the cirklon. It's having 5 midi ins, 5 midi outs, a dedicated sync port, USB hub and an analogue drum-trigger feature buillt into the sync port. Everything has been thought about deeply.
It just gets better everytime i use it and the mind boggles at the things it can do.
PS. I don't really write songs either, but the song feature is the only song feature I have started using on all the sequencers i've had..it's really quick and intuitive.
^ are tracks with different lengths not implemented yet then? good to know the ui is there.
cheers for the info revtor, i could just about do that.
every track can have a different length time and timebase. Every track can also have different playback directions (including random, and not-so-random type modes). Every track can also be modulated to change direction, timebase and length.
What is key to remember though is that a track on the cirklon contains a note value row, a velocity row, a duration rown, a delay row and 4 aux rows. Therefore all of those rows share the same settings. So....to have a set-up where a 4 step note value row is played using a 5 step velocity row (for example, to create a dynamically changing pattern) you have to either use inter-track events. ie. Track 1 holds all your note values (and also velocities, but these are irrelevant). Track 2 you can use as a kind of dummy track if you want although it doesn't have to be dummy (where the note values you enter are irrelevant). You can then assign an aux on track 1 to grab the velocity values of track 2 on every step of track 1. The ultimate conclusion is the same as having a 4 step note value row and a 5 step velocity row but it's a bit long winded to set up. However, where it wins is that you can set when you want to grab values. So track 1 doesnt have to always grab track 2, just sometimes. This itself can be modulated by use of other auxs.
Of course you could just set up a random velocity aux to give a similar dynamic pattern.
I'd really suggest reading the p3 manual to give you an idea of what aux's can do. So imagine their power but with a nice big clear screen and fantastic UI.
Thanks for the info and long posts chaps, much appreciated.
yes - the cirklon is far more advanced than the schrittmacher. but it's way, way faster to set up stuff on the schrittmacher (for me) unless I want to really do more complex modulations and aux events.
As far as individual step times - well, if you can simply set every step to a different length with that level of granularity, it shows you that I haven't spent enough time on this. My recollection is that you can set a note length, but that the choices are very limited. But I'll have to take a look at this.
Perhaps you're taking about the 'gate time' (god, these terms are used so inconsistently - on the schrittmacher, gate time means what kind of note it is, 1/16, 1/2, etc....)? On the schrittmacher I can quickly set up a track where the first note is a couple of ticks, the second is 4 beats, etc, etc, and then I can also set it up so that another track of a different length controls the direction from step to step, etc. You can pretty much do anything on the cirklon, but for me, for this kind of direct modular type patching, I find the Cirklon so, so much faster. Yeah - you're using menus, but it doesn't feel like it. It feels like it on the Cirklon.
I suppose I prefer using a track for one type of event. On the schrittmacher I'll use track one for notes, track 2 for velocity, track 3 for what they call 'gate time', etc - and it is just so fast to set this up.
Of course, the default is that the note track has a consistent note/gate length, velocity, etc - but the first thing I do is to break this stuff out.
At one point a few years ago I had a schrittmacher and a p3, and I sold the schrittmacher simply because it was so easy and fast to set this stuff up that I'd never use the p3. I sold it so that I would.
thanks, i shall do that once ive either had a go on neils or geard back from colin... whichever comes first. if the UI is there i can use it, cheers for the info peoples.
I've now set up my elektron MD (OT next!) as an instrument on the cirklon. You can save track values (CCs) per track per scene. So...I've put the mute channel cc's for the elektron in as instrument level definitions. When I want to store the MD's mute channel settings with a scene (without resorting to using the MD song mode) I simply hit the 'save track values with scene' button and next time that scene loads the correct mute status for the MD follows. Combined with pattern change (via either note or prog change message) the cirklon is now master to everything in my studio...I'm doing the same with my Ensoniq DP4 to save settings for each scene..kind of like just storing prog change messages but allows you to capture up to 96 CC's to transmit too (which you can name and save with the instrument for easy pick-out from the dropdown list).
Nelson, I'm not sure what the correct term is but i take gate time and note length to mean the same thing. ie. the amount of time a note is played before release. I'm not sure you can stretch a note to play longer than 16 steps but I might be wrong. You can very easily though so set step one to play a few ticks, step two to play 4 steps, step three to play a quarter note etc. Where notes overlaps you get either polyphony or slide depending on how your synth is set up.
I'm curious about playing with schrittmacher..it was what i was considering pre-maq16/3 and pre-cirklon..but one sequencer is enough for me and I don't see me losing the cirklon.
sounds good, unless neil doesnt like his ill be a fair while i think- the ot has taught me that im happier waiting until things are stable os wise.
you decide to keep that dp4 in the end then?


To make a donation to Maple View Lodge, make cheques payable to Maple View Lodge Auxiliary and mail to 746 County Road 42, Box 100, Athens, ON K0E 1B0.
Please ensure your return address is enclosed. Tax receipts will be mailed out for donations over $10.00.


text:
Variety Plus Icon Read Next: 'Moon Knight' Trailer Reveals Oscar Isaac as Marvel's Newest Superhero
Sep 28, 2015 8:33am PT
CBS to Air 2016 Tony Awards on June 5
By Whitney Friedlander
Whitney Friedlander
@loislane79
Jodie Comer, Penelope Cruz and Ruth Negga Talk Portraying Women at Turning Points 1 month ago
Female Stunt Performers From 'Hawkeye' to 'Kung Fu' Talk Kicking Butt and Trouncing Stereotypes on TV 1 month ago
How the 2021 Reality Host Emmy Race Reflects Major Strides in Diversity and Inclusion 5 months ago
The 2016 Tony Awards will air June 5 on CBS, the broadcaster announced Monday.
The ceremony, which honors achievement in Broadway theater, will air from 8 p.m. to 11 p.m. that Sunday. It is exec produced by Ricky Kirshner and Glenn Weiss of White Cherry Entertainment. This also makes the 17th consecutive year that Weiss will also serve as director. A host has yet to be named.
The Tony Awards have been broadcast on CBS since 1978. The 2015 ceremony, which was hosted by Kristin Chenoweth and Alan Cumming, flirted with near-record ratings lows as it went against the NBA Finals on ABC.
The upcoming ceremony also marks a milestone for the Tony Awards, which are presented by Tony Award Productions — a joint venture of the Broadway League and the American Theatre Wing. It is the 70th anniversary of the event, which was first held on April 6, 1947 at the Waldorf Astoria's Grand Ballroom.
The nominees for the 2016 Tony Awards will be announced on April 29.

redpajama_set_name: RedPajamaCommonCrawl


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    case 1:
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text:
Oreophryne inornata est une espèce d'amphibiens de la famille des Microhylidae.

Répartition
Cette espèce est endémique de l'île Goodenough en Papouasie-Nouvelle-Guinée.

Description
Oreophryne inornata mesure entre 36 et .

Publication originale
 Zweifel, 1956 : Microhylid frogs from New Guinea, with descriptions of new species. American Museum novitates, ,  (texte intégral).

Liens externes

Notes et références

Anoure (nom scientifique)
Asterophryinae
Faune endémique de l'archipel d'Entrecasteaux

redpajama_set_name: RedPajamaWikipedia


text:
Smart sensing
Mark England, CEO of Sentec, considers the possibility of a 'smart grid' and the implications that it will have.
A report by ZPryme Research that was conducted in 2009 predicted that the global smart grid market would reach $171 billion by 2014. Perhaps surprisingly, smart metering was forecast as the smallest segment, with only 11% of the total. Communications was next at 16%.
In fact, the largest category was smart sensors and devices, with a full 50% of the market, predicted to hit $85.5 billion. But when you consider the new challenges the grid will face in the coming years, this high market share is easily explained.
As we move towards a smart grid, equipment such as substations will face stresses from new applications on top of the growing challenges in monitoring performance. With many major car manufacturers now showcasing eco-friendly vehicles, it seems that the electric car is on the verge of achieving a long-awaited breakthrough in the UK market. But the grid in its current form would be unable to cope with the pressure from thousands of new electric vehicles, the electrification of residential heating and the input of micro-generated power back into a grid which was designed to transport centrally generated power in one direction; to the end user. Most of the UK high and medium voltage sections of the existing grid are well understood and monitored efficiently, whereas in the low voltage distribution network there is almost no monitoring or control and this area needs to be addressed.
Some distribution network operators (DNOs) in the UK use expensive mobile units to measure low voltage networks, but this equipment has to be moved between transformer units. Currently less than 1% of units are monitored continuously and on each feeder. With Smart Grid Analysis predicting that by 2015 more than $7.5 billion in smart sensors will be sold for applications in electricity grids around of the world, there is a clear opportunity for these products to be developed.
Bringing intelligence to the low voltage grid with a sensing solution is vital to achieving the objectives of an overall 'smart' grid. It's crucial to know the health of major assets in near real time and identify in detail the load those assets are serving so that problems can be pinpointed before they cause trouble. This simultaneously minimises maintenance costs and maximises uptime. 'Smart' distribution grids can also help companies scientifically value and prioritise projects and capital investments and co-ordinate and integrate capacity from distributed generation and storage solutions. The low voltage section of the distribution network requires a solution that can be retrofitted to existing infrastructure if it is to be cost-effective and implemented quickly enough. The typical UK substation also faces several specific issues that make a custom designed product an obvious choice.
Space in substations is tight, so product size must be considered. In addition, it's crucial that there is no disruption to supply when the sensors are being fitted. Ease of fitting should also be a priority in a low voltage sensor, and avoiding the need for calibration on site can make it much quicker and easier to train the fitters to install a new product.
Robustness and reliability once fitted is also vital. Due to the specific substation requirements, the sensor system must be impervious to weather, damp or physical location. To ensure longevity, it's also important that the sensor can continue to operate under challenging conditions; for example following large fault currents or when installed in external locations over extended periods of time.
When it comes to electricity distribution, accuracy saves time and money. Maintaining accuracy over the full lifetime of the sensor without any need for in situ recalibration is a key improvement over existing technology.
Finally, the cost point of the system must be low enough to enable continuous monitoring on all phases in all substations. The full cost of ownership must also be taken into account including the cost of the sensor system itself, transmitting and processing data, training staff, installation time and maintenance visit time. The business case can also include the savings as a result of the reduction in asset damage from overloads, and the costs of managing unexpected outages.
If predictions hold true, the sensor market is set to boom, but DNOs need to consider the custom product requirements of a low voltage substation. Bringing intelligence to the low voltage grid is vital, and by ensuring that sensors are customised to prioritise ease of installation, durability and accuracy, it will be possible to truly deliver on the promise of a 'smart' grid.
Control Stations available in stores
The Renewables Event reports record visitor registrations
SenseLogix set for growth following £1 million investment
Highland onshore wind farm gets extended
Smart meter projects awarded by Opus Energy


Internazionale coach Roberto Mancini says his side can still improve after last night's 2-0 win against Chievo.
The Nerazzurri were not at their very best but registered a much-needed win as they move up to 11th in the table.
The win was Mancini's first in the Serie A since taking over from Walter Mazzarri.
"Yes, [happy with the performance], it wasn't easy and the pitch wasn't the best we've ever played on," Mancini said.
"We kept the pressure and the moment we will improve when on the ball, we will do even better.
"They [Chievo] were coming from five consecutive positive results. But we deserved to win and their chances all came from our mistakes.
"The full backs did well, Dodo is quality, Danilo [D'Ambrosio] is improving a lot and will be important for us, just like all other players.
"Mateo [Kovacic] still has a lot of growing to do and he will not be used only in one particular role. He can play anywhere and the goal is important for him.
"[Fredy] Guarin did a lot of positive things but made mistakes as well. Sometimes he makes mistakes he shouldn't be making but I'm sure he will improve just like the others."


I have to say that I'm shocked that it's been over a year since my last blog post, where has the time go & what have I done in this past year?
Firstly, it's almost a year to the day that my mum came out of hospital after having spent a good 8 weeks being looked after by some wonderful doctors and nurses. My mum is a schizophrenic & an alcoholic and was free from the booze a year ago for a good 4 months until someone bought her a bottle of wine. Since that day she's been on a very slippery slope and is drinking plenty again. I'm sharing this because my mum is part of the reason I haven't blogged (in a year! I still can't believe it) Below (second of the two) is a very rare moment (selfie) taken in June this year of my mum and myself. It's rare because my mum never likes having her photo taken and the fact that she's smiling and has agreed to the photo is a small miracle. I think the last photo of me and my mum together where she smiles was when I was in my late teens (I'm in my 40's now) which says something.
I'm not sure how best to sum up a year's worth of posts in one post so I though maybe a selection of photos (in no particular order) would be a great way to summarise what I've got up to this past year.
I was lucky enough to get onto a pilot photoshop course run by a lovely lady called Sherry London www.prancingpixel.com which was a real eye opener for me as my photoshop skills aren't what they should be but I'm getting better all the time. Below is my 'Dudes' pattern after using the Kaleidoscope template.
Back in November I had a stall at The Spice Island Inn, Old Portsmouth. The venue is beautiful and the staff were wonderful along with the other stall holders and customers alike.
In January I took up going to the gym for the first time ever and thank my sister-in-law for coming with me the first few times as I don't think I would have done it with out her. I'm now a regular and love it. Below is a pic of me on the scales at the gym, you have to bear in mind that 3 or 4 years ago my weight was at least 105kg. I'm in the research process of coming up with some gym related designs which I'm excited about.
In July I had my first attempt at playing with henna, it's a lot harder that it looks bit was great fun.
I've been very fortunate to have been invited to some events called 'Poise', organised by some wonderful people. Each event is a mixture of amazing poetry, awesome music and incredible art which in turn creates a fantastic and inspiring evening. One such event motivated me to pen this poem.
And read a poem I wrote about the fear of reading aloud . . . at my age.
But not from the light that would glare so bright.
Rather than stand there in a paralyzed stance.
Right into the very pit of my stomach.
Instead they drift off into their 'wonders and what ifs'.
And pretend they were listening all along.
Would that make the experience any less painful?
I hope you've enjoyed the quick skip through my past year, I'm glad life isn't as edited as this. Here's to more regular posts.
A little while ago I made a little video featuring many of my doodle patterns and posted it to Youtube, I was really pleased about as I'd never made one before. I'm not technically minded when it comes to video editing so when the idea came to me to submit my work to my local Big Screen Portsmouth I thought I'd have some problems . . . how wrong was I, it was so much easier than I thought and I'm really please I did it.
I had a lovely time at the Omega Art fair yesterday, it was so lovely to see such amazingly talented & friendly people gathered together for the first Omega Art Fair. Here's hoping they will organise another in the near future.
I'm currently in the process of finding my creative mojo, it's playing hide & seek with me.
Betty's adventures before she left for a new home . . .
Betty loved posing with this great road sign.
Farewell Betty, enjoy your new adventures.
As Spring is coming I thought I would put together some samples to promote my work. I have had printed around 20 designs/artworks & have put 4 random prints into each envelope. I hope to place the envelopes in various coffee shops & libraries in and around Southsea for unsuspecting passers-by to enjoy.
I've (finally) decided to give my blog the attention it deserves with a new look just in time for 2015. The plan is to post more regularly than in the past & create posts worth viewing, but that's it as far as the plans go . . . I like the idea of being spontaneous, you never know what the day may bring.


Faceook's major deal of expending billion to buy Instagram followers company is at this instant the hot news basically Facebook is quite more self-assured to bring its high-class popularity in all terms and conditions of usage. Industrial show is also a fundamental point now as people may able to bring into play industrial animation in Myspace in the near next. This recent deal opens all cloud about added development of Facebook as a way to use in all how for users. Therefore, quite a few critics are hoping that would have better process identify of industrial animation additionally in Facebook soon. However, the deal has a number of other phases to be delivered out but the agreement is expected to strong by the end pertaining to June and also one of them more important point will be about hot tech upshots these days because with the lack of any kind of business model rarely dampens the excitement of deal.
Michael Pachter an important Wedbush analyst documented with regard that can this, "Facebook quickly this IPO can going to nevertheless be in a positioning to be aggressive. They can render sure no just one particular steps in your way and deal anyone who has in their concept. Buying Instagram people not only prevents a rival yet somehow gives Facebook your current technology that definitely is gaining crazy extender." Facebook said by having regard to this, "We plan within order to keep Instagram fanatics running independently. That a departure right from its tendency to assist you to buy small online companies and integrate the exact technology or enclosed them down entirely just so my wife and i can hire as qualified in cooking engineers and web designers.
comprar seguidores instagram is almost certainly an valuable milestone pertaining to us when it's all first second we've truly acquired a new product and as a result company while having so a large number of users. Almost all don't process on executing many a lot of of these, if virtually any at each." Melissa Parrish, an professional with Forrester Research said, "It's ones Web mythic that the only thing startups fancy of. These types of people took a definite simple attitudinal sharing footage with best friends and formulated it a trustworthy utility where it people love. Facebook is also getting anxious for their own substantial payday. It again is pointing for a good public which promise as eventually as adhering to month whom could estimate the concern around million.
That option it have the ability to easily have Instagram followers's price, any time only to be keep a definite rising movie star out connected the abs of others like The net." An analyzer at our own Altimeter Group, Rebecca Lieb said that includes regard so that it will this, "Buying Instagram admirers would make it possible for Facebook due to one within its virtually all urgent demands making its actual service a good deal more appealing on the smartphones. Really easier to actually update Myspace when your are on appropriate with an overview rather when compared with text. Myspace may likewise require Instagram practitioners to make a choice keep i'll carry on with the change regularly whims for the online listener.


Bloodless Coup: Good news from Iraq!
But the military has started censoring many [embedded reporting] arrangements. Before a journalist is allowed to go on an embed now, [the military] check[s] the work you have done previously. They want to know your slant on a story - they use the word slant - what you intend to write, and what you have written from embed trips before. If they don't like what you have done before, they refuse to take you. There are cases where individual reporters have been blacklisted because the military wasn't happy with the work they had done on embed.
I'd just love to see the blacklist.


text:
\section{Introduction}
The idea of associating graphs to algebraic structures for characterizing the algebraic structures with graphs and vice versa dates back to Bosak \cite{bosak}. Till then, a lot of research, e.g., \cite{graph-ideal,anderson-livingston,badawi,power2,mks-ideal,angsu-comm-alg-1,angsu-lin-mult-alg,angsu-comm-alg-2,angsu-jaa,survey2} has been done in connecting graph structures to various algebraic objects like groups, rings, vector spaces etc. However, the most prominent among them are the zero-divisor graphs \cite{anderson-livingston} and intersection graph of ideals of rings  \cite{mks-ideal}.  Recently, authors in \cite{weakly-perfect} proved that intersection graph of ideals of $\mathbb{Z}_n$ is weakly perfect for all $n>0$. In this paper, we characterize the values of $n$ for which the intersection graph of ideals of $\mathbb{Z}_n$ is perfect. In particular, we prove the following theorem.

\begin{theorem*}
The intersection graph of ideals of $\mathbb{Z}_n$ is perfect if and only if  $n={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$ where $p_i$'s are distinct primes and $\alpha_i \in \mathbb{N}\cup\{0\}$, i.e., the number of distinct prime factors of $n$ is less than or equal to $4$.
\end{theorem*}

\section{Definition, Preliminaries and Known Results}
In this section, for convenience of the reader and also for later use, we recall some definitions, notations and results concerning elementary graph theory and intersection graph of ideals of a ring. For undefined terms and concepts the reader is referred to \cite{west-graph-book}.

By a graph $G=(V,E)$, we mean a non-empty set $V$ and a symmetric binary relation (possibly empty) $E$ on $V$. The set $V$ is called the set of vertices and $E$ is called the set of edges of $G$. Two element $u$ and $v$ in $V$ are said to be adjacent if $(u,v) \in E$. $H=(W,F)$ is called an {\it induced subgraph} of $G$ if $\phi \neq W \subseteq V$ and $F$ consists of all the edges between the vertices in $W$ in $G$.  A complete subgraph of a graph $G$ is called a {\it clique}. A {\it maximal clique} is a clique which is maximal with respect to inclusion. The {\it clique number} of $G$, written as $\omega(G)$, is the maximum size of a clique in $G$. The {\it chromatic number} of $G$, denoted as $\chi(G)$, is the minimum number of colours needed to label the vertices so that the adjacent vertices receive different colours. It is easy to observe that $\omega(G)\leq \chi(G)$. A graph $G$ is said to be {\it weakly perfect} if $\omega(G)= \chi(G)$ and it is said to be {\it perfect} if $\omega(H)= \chi(H)$ for all induced subgraphs $H$ of $G$. Chudnovsky {\it et.al.} \cite{spgt} in 2004 settled a long standing conjecture regarding perfect graphs and provided a characterization of perfect graphs.

{\theorem[Strong Perfect Graph Theorem] \label{spgt} \cite{spgt} A graph $G$ is perfect if and only if neither $G$ nor its complement contains an odd cycle of length at least $5$ as an induced subgraph.}

Let $R$ be a ring. The intersection graph of ideals of $R$ (introduced in \cite{mks-ideal}), denoted by $G(R)$, consists of all non-trivial ideals as vertices and two ideals $I$ and $J$ are adjacent if and only if $I \cap J\neq \{0\}$. Throughout this paper, we take the ring $R$ to be $\mathbb{Z}_n$, the ring of integers modulo $n$. We know that $\mathbb{Z}_n$ is a principal ideal ring and each of its ideals is
generated by $\overline{m}\in \mathbb{Z}_n$ where $m$ is a factor of $n$. For convenience, we denote this ideal by $(m)$. Also without loss of generality, whenever we take an ideal $(m)$ of $\mathbb{Z}_n$, we assume that $m$ is a factor of $n$. It was proved in \cite{weakly-perfect} proved that intersection graph of ideals of $\mathbb{Z}_n$ is weakly perfect, i.e., $\omega(G( \mathbb{Z}_n))=\chi(G( \mathbb{Z}_n))$ for all $n>0$.

\section{Perfectness of Intersection Graph of Ideals of $\mathbb{Z}_n$}
In this section, we prove some preparatory results and subsequently use them to prove the main theorem of the paper.
{\proposition \label{lcm-lemma} Let $G(\mathbb{Z}_n)$ be the intersection graph of ideals of $\mathbb{Z}_n$ and $(a)$ and $(b)$ be two ideals in $\mathbb{Z}_n$ such that $a \mid n$ and $b\mid n$. Then $(a)$ and $(b)$ are adjacent in $G(\mathbb{Z}_n)$  if and only if $lcm(a,b)$ is a factor of $n$ and $1<lcm(a,b)<n$.}\\
\\
\noindent {\bf Proof: } Since $\mathbb{Z}_n$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$ as ring via the correspondence $\overline{a}\leftrightarrow a+n\mathbb{Z}$, the ideal $(a)$ in $\mathbb{Z}_n$ corresponds to the ideal $\langle a \rangle + n\mathbb{Z}$ in $\mathbb{Z}/n\mathbb{Z}$ where $\langle a \rangle$ denote the set of integer multiples of $a$. Now, let $(a)\sim (b)$ in $G(\mathbb{Z}_n)$, i.e., $(a)\cap(b)\neq \{\overline{0}\}$. Since $a \mid n$ and $b\mid n$, we have $lcm(a,b)\mid n$. On the other hand, using the correspondence described above, we have  $\langle a \rangle + n\mathbb{Z} \cap \langle b \rangle + n\mathbb{Z}\neq \{n\mathbb{Z}\}$. But, we know that $\langle a \rangle + n\mathbb{Z} \cap \langle b \rangle + n\mathbb{Z}=\langle lcm(a,b) \rangle + n\mathbb{Z}$. Hence, we have $\langle lcm(a,b) \rangle + n\mathbb{Z} \neq \{n\mathbb{Z}\}$. This, together with the fact that $lcm(a,b)\mid n$, implies that $1<lcm(a,b)<n$.

Conversely, let $lcm(a,b)$ is a factor of $n$ and $1<lcm(a,b)<n$. Clearly, $\overline{0} \neq \overline{lcm(a,b)} \in (a) \cap (b)$ in $\mathbb{Z}_n$ and hence $(a)\sim (b)$ in $G(\mathbb{Z}_n)$.  \hfill \rule{2mm}{2mm}

{\theorem \label{not-perfect} Let $n={p_1}^{\alpha_1}{p_2}^{\alpha_2}\cdots {p_k}^{\alpha_k}$. If $k\geq 5$, then $G(\mathbb{Z}_n)$ is not perfect.}\\
\\
\noindent {\bf Proof: } Let $n={p_1}^{\alpha_1}{p_2}^{\alpha_2}\cdots {p_5}^{\alpha_5}.s$ where $s=1$ if $k=5$ and $s={p_6}^{\alpha_6}\cdots {p_k}^{\alpha_k}$ if $k>5$. Consider the cycle $C$ given by $({p_1}^{\alpha_1}{p_2}^{\alpha_2} {p_3}^{\alpha_3}s)\sim({p_2}^{\alpha_2}{p_3}^{\alpha_3} {p_4}^{\alpha_4}s)\sim({p_3}^{\alpha_3}{p_4}^{\alpha_4} {p_5}^{\alpha_5}s)\sim({p_4}^{\alpha_4}{p_5}^{\alpha_5} {p_1}^{\alpha_1}s)\sim({p_5}^{\alpha_5}{p_1}^{\alpha_1}{p_2}^{\alpha_2} s)\sim ({p_1}^{\alpha_1}{p_2}^{\alpha_2} {p_3}^{\alpha_3}s)$. Simple calculation using Proposition \ref{lcm-lemma} shows that $C$ is an induced $5$-cycle in $G(\mathbb{Z}_n)$ and hence by Theorem \ref{spgt}, $G(\mathbb{Z}_n)$ is not perfect.\hfill \rule{2mm}{2mm}

{\theorem \label{no-odd-hole} Let $n={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$. Then $G(\mathbb{Z}_n)$ does not contain any induced cycle of length greater than $4$.}\\
\\
\noindent {\bf Proof: } Let, if possible $G(\mathbb{Z}_n)$  contains an induced cycle $C$ of length greater than $4$, say $(a_1)\sim(a_2)\sim(a_3)\sim(a_4)\sim(a_5)\sim \cdots \sim (a_1)$. By Proposition \ref{lcm-lemma}, we have $$lcm(a_1,a_3)=lcm(a_1,a_4)=lcm(a_2,a_4)=lcm(a_2,a_5)=lcm(a_3,a_5)=n.$$

{\it Claim: $gcd(a_1,a_3)>1$.} If possible, let $gcd(a_1,a_3)=1$. Since $gcd(a_1,a_3)\cdot lcm(a_1,a_3)=a_1a_3$, we have $lcm(a_1,a_3)=a_1a_3=n$. Note that as $lcm(a_3,a_5)=n$, we have $n=\frac{a_1a_3}{gcd(a_1,a_3)}=\frac{a_3a_5}{gcd(a_3,a_5)}$, i.e., $a_1\cdot gcd(a_3,a_5)=a_5\cdot gcd(a_1,a_3)$, i.e., $a_5=a_1\cdot gcd(a_3,a_5)$, i.e., $a_5$ is a multiple of $a_1$. Now as $a_1$ and $a_3$ are coprime and their lcm is $n$, without loss of generality, two cases may arise: either $a_1={p_1}^{\alpha_1}{p_2}^{\alpha_2}; a_3={p_3}^{\alpha_3}{p_4}^{\alpha_4}$ or $a_1={p_1}^{\alpha_1};a_3={p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$.

If $a_1={p_1}^{\alpha_1}{p_2}^{\alpha_2}; a_3={p_3}^{\alpha_3}{p_4}^{\alpha_4}$, we have $a_5={p_1}^{\alpha_1}{p_2}^{\alpha_2}\cdot s$ for some natural number $s$ such that $a_5 \mid n$. Also as $lcm(a_1,a_4)=n$, we have $a_4={p_3}^{\alpha_3}{p_4}^{\alpha_4}\cdot t$, for some natural number $t$ such that $a_4 \mid n$. Thus $lcm(a_4,a_5)=n$ contradicting Proposition \ref{lcm-lemma} and the fact that $a_4\sim a_5$ in $C$.

If $a_1={p_1}^{\alpha_1};a_3={p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$, similarly we have $a_5={p_1}^{\alpha_1}\cdot s$ and $a_4={p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4} \cdot t$ and hence $lcm(a_4,a_5)=n$ thereby leading to a contradiction. Thus by combining above two cases, we have $gcd(a_1,a_3)>1$.

Thus we have $lcm(a_1,a_3)=n$ and $gcd(a_1,a_3)>1$ with  $a_1 \mid n$ and $a_3\mid n$. Without loss of generality, let $p_1$ be a common factor of $a_1$ and $a_3$ and let $a_1={p_1}^x\cdot s$ and $a_3={p_1}^y \cdot t$ where $p_1$ is coprime with $s$ and $t$. Now, if $max\{x,y\}<\alpha_1$, then $lcm(a_1,a_3)<n$, a contradiction. Thus either $x=\alpha_1$ or $y=\alpha_1$, i.e., for any common prime divisor $p_i$ of $a_1$ and $a_3$, either ${p_i}^{\alpha_i} \mid a_1$ or ${p_i}^{\alpha_i} \mid a_3$ or both. Also as $lcm(a_1,a_3)=n$, all the ${p_i}^{\alpha_i}$ are factors of either $a_1$ or $a_3$ or both. Thus, without loss of generality, the forms of $a_1$ and $a_3$ are as follows: either $$\mathsf{Case ~ 1:}~ a_1={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\beta_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$$  or $$\mathsf{Case ~ 2:}~a_1={p_1}^{\alpha_1}{p_2}^{\beta_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$$ or $$\mathsf{Case ~ 3:}~a_1={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$$ or $$\mathsf{Case ~ 4:}~a_1={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$$ where $\beta_i < \alpha_i$. Note that in first two cases, $a_1$ and $a_3$ do not share any ${p_i}^{\alpha_i}$ as common factor. In the third case, they share only one ${p_i}^{\alpha_i}$ as common factor and in the fourth case, they share two ${p_i}^{\alpha_i}$'s as common factor.

{\it Case 1: ($a_1={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\beta_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$)} Since $lcm(a_1,a_4)=n$, we have $a_4={p_1}^{\gamma_1}{p_2}^{\gamma_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$ where $\gamma_1\leq \alpha_1,\gamma_2 \leq \alpha_2$ and $(\gamma_1,\gamma_2)\neq (\alpha_1,\alpha_2)$. Again, since $lcm(a_3,a_5)=n$, we have $a_5={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\delta_3}{p_4}^{\delta_4}$ where $\delta_3\leq \alpha_3,\delta_4 \leq \alpha_4$ and $(\delta_3,\delta_4)\neq (\alpha_3,\alpha_4)$. Hence, we have $lcm(a_4,a_5)=n$, a contradiction to the fact that $a_4 \sim a_5$. Thus Case 1 is an impossibility.

{\it Case 2: ($a_1={p_1}^{\alpha_1}{p_2}^{\beta_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$)} Since $lcm(a_1,a_4)=n$, we have $a_4={p_1}^{\gamma_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$ where $\gamma_1< \alpha_1$. Again, since $lcm(a_3,a_5)=n$, we have $a_5={p_1}^{\alpha_1}{p_2}^{\delta_2}{p_3}^{\delta_3}{p_4}^{\delta_4}$ where $\delta_i\leq \alpha_i$ and $(\delta_2,\delta_3,\delta_4)\neq (\alpha_2,\alpha_3,\alpha_4)$. Hence, we have $lcm(a_4,a_5)=n$, a contradiction to the fact that $a_4 \sim a_5$. Thus Case 2 is an impossibility.

{\it Case 3: ($a_1={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$)} Since $lcm(a_1,a_4)=n$, we have ${p_3}^{\alpha_3}{p_4}^{\alpha_4}\mid a_4$. Again, since $lcm(a_3,a_5)=n$, we have ${p_1}^{\alpha_1}\mid a_5$. Now, as $lcm(a_2,a_5)=n$, we have either ${p_2}^{\alpha_2}\mid a_2$ or ${p_2}^{\alpha_2}\mid a_5$. But if ${p_2}^{\alpha_2}\mid a_5$, then we have $lcm(a_4,a_5)=n$, a contradiction. Thus, we have ${p_2}^{\alpha_2}\mid a_2$. Again, as $lcm(a_2,a_4)=n$, we have either ${p_1}^{\alpha_1}\mid a_2$ or ${p_1}^{\alpha_1}\mid a_4$. If ${p_1}^{\alpha_1}\mid a_2$, then $lcm(a_2,a_3)=n$, a contradiction. On the other hand, if ${p_1}^{\alpha_1}\mid a_4$, then $lcm(a_3,a_4)=n$, a contradiction. Thus Case 3 is an impossibility.

{\it Case 4: ($a_1={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$)} Since $lcm(a_1,a_4)=n$, we have ${p_4}^{\alpha_4}\mid a_4$.  Now, as $lcm(a_2,a_4)=n$, we have either ${p_1}^{\alpha_1}\mid a_2$ or ${p_1}^{\alpha_1}\mid a_4$. If ${p_1}^{\alpha_1}\mid a_2$, then $lcm(a_2,a_3)=n$, a contradiction. On the other hand, if ${p_1}^{\alpha_1}\mid a_4$, then $lcm(a_3,a_4)=n$, a contradiction. Thus Case 4 is an impossibility.

Thus, combining all the cases we conclude that $G(\mathbb{Z}_n)$ does not contain any induced cycle of length greater than $4$.\hfill \rule{2mm}{2mm}

{\theorem \label{no-odd-antihole} Let $n={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$. Then $\overline{G(\mathbb{Z}_n)}$, the complement of $G(\mathbb{Z}_n)$, does not contain any induced cycle of length greater than $4$.}\\
\\ \noindent {\bf Proof: } Let, if possible $\overline{G(\mathbb{Z}_n)}$  contains an induced cycle $C$ of length greater than $4$, say $(a_1)\sim(a_2)\sim(a_3)\sim(a_4)\sim \cdots \sim(a_t)\sim (a_1)$ with $t\geq 5$. Then, by Proposition \ref{lcm-lemma}, $lcm(a_1,a_2)=lcm(a_2,a_3)=lcm(a_3,a_4)=\cdots =lcm(a_t,a_1)=n$.

[{\it Claim: $gcd(a_2,a_3)>1$}] If possible, let $gcd(a_2,a_3)=1$. Since $lcm(a_2,a_3)=n$, we have $n=a_2a_3$. Thus without loss of generality, either $$a_2={p_1}^{\alpha_1}{p_2}^{\alpha_2};a_3={p_3}^{\alpha_3}{p_4}^{\alpha_4} \mbox{ or } a_2={p_1}^{\alpha_1};a_3={p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$$

If $a_2={p_1}^{\alpha_1}{p_2}^{\alpha_2};a_3={p_3}^{\alpha_3}{p_4}^{\alpha_4}$, as $lcm(a_3,a_4)=lcm(a_1,a_2)=n$, we have $a_1={p_3}^{\alpha_3}{p_4}^{\alpha_4}\cdot s$ and $a_4={p_1}^{\alpha_1}{p_2}^{\alpha_2}\cdot t$ for some positive integer $s,t$. But this implies that $lcm(a_1,a_4)=n$, i.e., $a_1\sim a_4$ in $\overline{G(\mathbb{Z}_n)}$, a contradiction.

On the other hand, if $a_2={p_1}^{\alpha_1};a_3={p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$, as $lcm(a_3,a_4)=lcm(a_1,a_2)=n$, we have $a_1={p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}\cdot s$ and $a_4={p_1}^{\alpha_1}\cdot t$ for some positive integer $s,t$. But this implies that $lcm(a_1,a_4)=n$, i.e., $a_1\sim a_4$ in $\overline{G(\mathbb{Z}_n)}$, a contradiction. Hence the claim is true.

Now, we have $lcm(a_2,a_3)=n$ and $gcd(a_2,a_3)>1$ with  $a_2 \mid n$ and $a_3\mid n$. Without loss of generality, let $p_1$ be a common factor of $a_2$ and $a_3$ and let $a_2={p_1}^x\cdot s$ and $a_3={p_1}^y \cdot t$ where $p_1$ is coprime with $s$ and $t$. Now, if $max\{x,y\}<\alpha_1$, then $lcm(a_2,a_3)<n$, a contradiction. Thus either $x=\alpha_1$ or $y=\alpha_1$, i.e., for any common prime divisor $p_i$ of $a_2$ or $a_3$, either ${p_i}^{\alpha_i} \mid a_2$ or ${p_i}^{\alpha_i} \mid a_3$ or both. Also as $lcm(a_2,a_3)=n$, all the ${p_i}^{\alpha_i}$ are factors of either $a_2$ or $a_3$. Thus, without loss of generality, the forms of $a_2$ and $a_3$ are as follows: either $$\mathsf{Case ~ 1:}~ a_2={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\beta_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$$  or $$\mathsf{Case ~ 2:}~a_2={p_1}^{\alpha_1}{p_2}^{\beta_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$$ or $$\mathsf{Case ~ 3:}~a_2={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$$ or $$\mathsf{Case ~ 4:}~a_2={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$$ where $\beta_i < \alpha_i$. Note that in first two cases, $a_2$ and $a_3$ do not share any ${p_i}^{\alpha_i}$ as common factor. In the third case, they share only one ${p_i}^{\alpha_i}$ as common factor and in the fourth case, they share two ${p_i}^{\alpha_i}$'s as common factor.

{\it Case 1: ($a_2={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\beta_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$)} Since $lcm(a_1,a_2)=lcm(a_3,a_4)=n$, we have ${p_3}^{\alpha_3}{p_4}^{\alpha_4} \mid a_1$ and
 ${p_1}^{\alpha_1}{p_2}^{\alpha_2}\mid a_4$. But this implies $lcm(a_1,a_4)=n$, i.e., $a_1\sim a_4$ in $\overline{G(\mathbb{Z}_n)}$, a contradiction and hence Case 1 is an impossibility.

{\it Case 2: ($a_2={p_1}^{\alpha_1}{p_2}^{\beta_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$)} Since $lcm(a_1,a_2)=lcm(a_3,a_4)=n$, we have ${p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}\mid a_1$ and ${p_1}^{\alpha_1}\mid a_4$. But this implies $lcm(a_1,a_4)=n$, a contradiction and hence Case 2 is an impossibility.

{\it Case 3: ($a_2={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\beta_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$)}   Since $lcm(a_1,a_2)=n$, we have ${p_3}^{\alpha_3}{p_4}^{\alpha_4} \mid a_1$. Also, since $lcm(a_t,a_1)=n$, either ${p_1}^{\alpha_1}\mid a_1$ or ${p_1}^{\alpha_1}\mid a_t$. If ${p_1}^{\alpha_1}\mid a_1$, then we have ${p_1}^{\alpha_1}{p_3}^{\alpha_3}{p_4}^{\alpha_4} \mid a_1$ which implies $lcm(a_1,a_3)=n$, i.e., $a_1\sim a_3$ in $\overline{G(\mathbb{Z}_n)}$, a contradiction. On the other hand, if ${p_1}^{\alpha_1}\mid a_t$, we have $lcm(a_t,a_3)=n$, i.e., $a_t \sim a_3$ in $\overline{G(\mathbb{Z}_n)}$, a contradiction. Thus combining both the possibilities, Case 3 is an impossibility.

{\it Case 4: ($a_2={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\beta_4};a_3={p_1}^{\beta_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$)}   Since $lcm(a_1,a_2)=n$, we have ${p_4}^{\alpha_4} \mid a_1$. Also, since $lcm(a_t,a_1)=n$, either ${p_1}^{\alpha_1}\mid a_1$ or ${p_1}^{\alpha_1}\mid a_t$. If ${p_1}^{\alpha_1}\mid a_1$, then we have ${p_1}^{\alpha_1}{p_4}^{\alpha_4} \mid a_1$ which implies $lcm(a_1,a_3)=n$, i.e., $a_1\sim a_3$ in $\overline{G(\mathbb{Z}_n)}$, a contradiction. On the other hand, if ${p_1}^{\alpha_1}\mid a_t$, we have $lcm(a_t,a_3)=n$, i.e., $a_t \sim a_3$ in $\overline{G(\mathbb{Z}_n)}$, a contradiction. Thus combining both the possibilities, Case 4 is an impossibility.

Thus, combining all the cases we conclude that $\overline{G(\mathbb{Z}_n)}$ does not contain any induced cycle of length greater than $4$.\hfill \rule{2mm}{2mm}

Finally, with Theorems \ref{spgt}, \ref{not-perfect}, \ref{no-odd-hole} and \ref{no-odd-antihole} in hand, we are now in a position to prove the main result of this paper.
\begin{theorem*}
The intersection graph of ideals of $\mathbb{Z}_n$ is perfect if and only if  $n={p_1}^{\alpha_1}{p_2}^{\alpha_2}{p_3}^{\alpha_3}{p_4}^{\alpha_4}$ where $p_i$'s are distinct primes and $\alpha_i \in \mathbb{N}\cup\{0\}$, i.e., the number of distinct prime factors of $n$ is less than or equal to $4$.
\end{theorem*}
\noindent {\bf Proof: } Clearly, Theorem \ref{not-perfect} shows that the condition is necessary. For the sufficiency part, first with the help of Theorems \ref{no-odd-hole} and \ref{no-odd-antihole}, along with Theorem \ref{spgt}, we conclude that the intersection graph of ideals of $\mathbb{Z}_n$ is perfect if $n$ has exactly four distinct prime factors. The proofs for the cases when $n$ has exactly three, two or one distinct prime factors follows similarly by suitably taking some of the $\alpha_i$'s to be zero. \hfill \rule{2mm}{2mm}

\section*{Acknowledgement}
The author is thankful to Sabyasachi Dutta and Jyotirmoy Pramanik for some fruitful discussions on the paper. The research is partially funded by NBHM Research Project Grant, (Sanction No. 2/48(10)/2013/ NBHM(R.P.)/R\&D II/695), Govt. of India.

redpajama_set_name: RedPajamaArXiv


The French three-strikes anti-piracy law "Hadopi" is heralded by copyright holders as an effective way to curb piracy. However, in France the legislation has often been criticized and in a surprise move against the will of the Government, the National Assembly has now voted to dismantle it in a few years.
In recent years many anti-piracy initiatives have emerged and in several countries so-called "graduated response" schemes have been implemented.
France is seen as the pioneer of so-called "three strikes" anti-piracy legislation, in which repeated file-sharing offenders face fines of up to 1,500 euros.
Since 2010 the French Hadopi agency has handed out millions of warning notices. A few thousand account holders received more than three notices, of which a few hundred of the worst cases were referred for prosecution.
Copyright holders around the world have cited Hadopi as one of the success stories, hoping to establish similar legislation elsewhere. However, in France the law hasn't been without controversy and in a total surprise the lower house of the French Parliament has now voted in favor of killing it.
Interestingly, the vote late last week went down under quite unusual circumstances.
In a nearly empty chamber, the French National Assembly voted to end the Hadopi institution and law in 2022, Next Inpact reports. What's noteworthy is that only 7 of the 577 Members of Parliament were present at the vote, and the amendment passed with four in favor and three against.
The decision goes against the will of the sitting Government, which failed to have enough members present at the vote. While it's being seen as quite an embarrassment, the amendment still has to pass the senate, which seems unlikely without Government support.
The 'coup,' orchestrated by the Green party has caused quite a media stir, not least because French President François Hollande called for the end of Hadopi before his election, a position he later retracted.
"Related Greens" MP Isabelle Attard says that it's time to end the "schizophrenic" behavior of the Government on the matter. "A choice has to be made at some point. We can't call out Hadopi as useless and, years later, still let it linger on," she says.
While it's doubtful that the amendment means the definite end of Hadopi, it certainly puts it back on the political agenda. Whether this will lead to actual change will become apparent in the future.


Many of us have been directly affected by breast cancer, either through family members or friends. October is Breast Cancer Awareness Month, so Waterworks Aquatics would like to invite you to participate in the American Cancer Society Making Strides Against Breast Cancer walk! This event will bring the community together to honor breast cancer survivors and raise awareness and funds to save lives from breast cancer!
It is our goal to raise at least $200 to help support one of the leading organizations in the fight to end breast cancer. We are honored to be participating in this great opportunity to help raise money for a cure, and we hope you can join us too!


Armstrong Series 4300 split coupled Vertical In line pumps- A trusted Series of Armstrong Armstrong's vertical in line pumps is a design innovation that continues to lead the industry. Armstrong has improved the efficiency and utility of commercial pumps on a large scale by designing the vertical in line pump in the early 1970s. Role in Inspiring Engineers Armstrong continues to play a leading role in educating engineers and inspiring them all around the world on the value of Vertical in line Installations and offers a wide range of sizes and options. Armstrong vertical in line pumps combine it'sexcellent operating efficiency with very low vibration. In installations in all around the world, substituting vertical in line pumps has led to significant cost savings in piping and system efficiencyon a large scale. Phenomenal design features •	It has a radially split volute with discharge flange sizes and equal suction. •	Its motor shaft run out is limited to 0.0001 TR. •	Its industry standard motor is designed for vertical in line operation. •	It consists of a dynamically balanced impeller which assures smooth and vibration free operation. •	The grand plate has a flush connection which ensures positive venting of seal chamber and liquid at seal faces. •	It has a heavy cylindrical bracket with 360 degree providing a rigid union of pump and motor. •	Its axially split, spacer type rigid coupling permits seal maintenance without disturbing the pump. •	It has a ribs cast integral with a valute. •	It's confined casing gasket is designed to meet the stringent industrial temperature and pressure. •	The coupling guard fully encloses access openings. Advantages of Armstrong Series 4300split coupled verticalin line pump 1.	It requires a smaller footprint than that of horizontal pumps. 2.	It is suitable for spaces where the ground area surface is restricted. 3.	With a vertical centrifugal pump, the NPSH Available can be increased. 4.	Such pumps are suitable for higher temperature and high pressure fluids. 5.	Armstrong vertical in line pumps combine excellent operating efficiency with very low vibration and noise. 6.	It offers a very valuable design/ program whereby an internal designer will work with contractors and engineers to 'lay out' and mechanical room using vertical technology. Reasons to buy Armstrong Series 4300 split coupled vertical in line pumps •	Low installation costs 1.	Vertical in line design eliminates the need for inertia, housekeeping pads, flex connectors and field alignmentor grouting requirements. 2.	Vertical in line design minimizes space and pipe requirements. 3.	Isolated mechanical seal design with optional sediment filters or separators available to extend seal life in contaminated systems. •	Low energy consumption and operating cost 1.	Armstrong Series 4300 Split coupled Vertical In line pumps are available with high efficiency motors for significant reduction in energy consumption and cost. 2.	All motors are suitable for inverter fed service 3.	Vertical in line design minimizes pipe requirements and related friction losses resulting in higher system efficiency 4.	Split coupling allows replacement of mechanical seal without disturbing the pump. 5.	No lubrication or seal adjustments required. •	Low operating risk 1.	Universal replacement seals, bearing and motors available locally worldwide. 2.	Optional isolated mechanical seal with sediment filters or separators for extended seal life in contaminated systems. 3.	Vertical in line configuration allows for less disrupting pump service. Vertical in line pumps being integral components of the pipework, eliminate need of inertia bases, inertia based springs, flexible pipe connectors, filed grouting and alignment.


text:
Amazing Kid! of the Month – Mariah Bailey – September 2014
Posted in AKOM Archives 2014, Amazing Kids! of the Month, September 2014 Issue
Mariah Bailey, 12-year-old Inspiring Trendsetter
By Kasey Dallman, AKOM Writer
"Sometimes people are starving for more than food. Sometimes they a starving for love."
-Mariah Bailey, 12-year-old Inspiring Trendsetter
Amazing Kid! of the Month
You don't have to be a comic book approved superhero to save the lives of others. Sometimes all it takes is a sprinkle of inspiration to set out on a course to make a difference. For 12-year-old Mariah Bailey, her spark was watching her father make a difference in the lives of the homeless. Now, Mariah has her own organization, Trending Change, which aims to care for the homeless by showing them that they are loved.
While Mariah is an inspiration for all the lives she has been touching, she is an even bigger inspiration for us! Amazing Kids is beyond thrilled to announce Mariah as September's Amazing Kid! of the Month. Keep reading to find out more about Mariah's story and how you too can make a difference!
Trending Change
Mariah Bailey's vision to make a difference started with her dad and his organization, Sidewalk Angels, Inc. Mariah's dad would go out on the streets anonymously to help the homeless in her community, and sometimes Mariah would go out with him. "After being with him on so many occasions, I started to get to know the people that he helped," Mariah said. "I started to learn about who they were as people and started building a friendship with them."
While Mariah's dad liked to do his good deeds anonymously, Mariah thought that she would like to start something similar that was better known to raise even more awareness. After years of making a difference, Trending Change was officially launched in October 2013 through her dad's organization.
What do you trend for? Mariah trends for change!
How does a 12-year-old start her organization? Well, for Mariah, she gave up everything she would have received for Christmas and used the money to start Trending Change. Now Mariah has helped countless numbers of people feel loved and comforted through her organization. Congrats Mariah! Keep up the excellent work!
Spreading Cheer and Love
Over the last year, Mariah has done a few campaigns, raising over $1,100. Her big goal for last December was to provide blessing bags for hundreds of homeless people in Myrtle Beach. Each blessing bag not only had staple necessities, but Mariah also added notes to each bag reminding each person that they were loved and she would be there for them until things got better.
Mariah putting together items for the homeless. We can't thank Mariah enough for her selfless work!
When Christmas day came, Mariah spent the morning with the Red Cross feeding the homeless and then spent the afternoon walking the streets to make sure that those who couldn't make it to the feeding event were also reached.
"My goal is to let the homeless know that they have not been forgotten and that they are loved," Mariah said. For people who are homeless, there are a few outlets for them to find food and a shelter. However, no one at these homeless shelters are obligated to show compassion. "Sometimes people are starving for more than food. Sometimes they are starving for love. "
Through Trending Change, Mariah has been able to show people that even though they may be struggling they still have a friend and are not invisible to the world. Thanks so much for your inspiring work, Mariah! Everybody needs to be reminded just how much we care for each other, and we love the work you are doing to accomplish just that!
More Than Just a Trend
When sharing her stories of helping those that are homeless, Mariah's favorite encounter has been with a disabled veteran named Fred. She first met Fred on Thanksgiving Day in Chapin Park. Fred was sitting on a bench eating food that a person had given him, when Mariah went and started talking with him.
"Fred always starts his stories off with, 'Back when I was someone.' I knew right then that he needed me," Mariah said. "He did so much for our country. He is and always will be someone. Every time I go to Chapin Park I look for Fred. He makes my day a little brighter, and I can only hope I make his days brighter too."
"The most rewarding part for me is seeing them smile," Mariah said.
Mariah hopes that Trending Change will be more than just a trend. She hopes that others start to see those who are homeless like she does and make it a part of everyday life to spread compassion to them.
You too can be a trendsetter in your own town! Be a friend to someone in need. Go out of your way to bring compassion into someone else's life. Spread the word about Trending Change so more and more people become inspired! We can't wait to see the difference you can make.


...in our neighbors' yard this weekend!
And then we found a chipmunk. And he's not actually in our neighbors' yard. He's in our driveway looking at their yard. He lives in our downspout and feeds at our bird feeder.
We wonder if you have found anything interesting in your neighbors' yards?
A few weeks ago, we asked people if they wanted to put their pets faces into a common scenario over at Face In Hole. Two of our bloggie friends sent us their photos. Adorable! Here's our little mini-gallery of puggies in the King Tut mask.
Pearl here! I don't know if anyone remembers but last week I reported that we had gone to the Montezuma National Wildlife Refuge to look at birds. Yeah, it's as exciting as it sounds.
Mommy took oodles of pictures but since this is a puggie blog, we figured we'd balance each bird photo with a puggie pic then we'll decide at the end who is cuter - us or the birds.
Whew! Well, that was pretty easy. I mean, birds don't even have curly tails!!! What's up with that?! So I'd say Daisy and I won this round.


Here you will find a selection of my maternity portfolio. Visit The Blog to view featured session galleries.
For full information on my maternity photography sessions please head to the Maternity Session Info page.


7/02/2012 · Wipes clean and never collects hair, obviously. It's great and has held up very well to our two crazy 50+ lb dogs and the bountiful amount of people that sit on it. It's great and has held up very well to our two crazy 50+ lb dogs and the bountiful amount of people that sit on it.... 27/12/2009 · Faux leather can be found on a variety of items these days. It's fantastic for families with children, as is far less prone to staining than real leather.
Staining on and along your nylon jacket's cuffs is a common issue simply because the area is constantly exposed to food, dirt, mud and anything else you touch or handle throughout the day. Before laundering your nylon jacket, pretreat the stained cuffs with the proper store-bought product.
27/12/2009 · Faux leather can be found on a variety of items these days. It's fantastic for families with children, as is far less prone to staining than real leather.
The method for cleaning an Alpinestars jacket remains the same as other jackets. The detailed method is best described in this article Riding Jacket Maintenance Guide Please go through the article and let me know if there are any further questions.
FAUX SUEDE BODY WITH Pleather (. ALFANI MOTO JACKET. Fully Lined including Sleeves (100% Polyester). Wipe Clean-Do NOT Machine Wash or Dry Clean.


Fyodorov began his acting career in provincial Russian theatres, moving eventually to Moscow, where he also developed an interest in photography. In the summer of 1896 he acquired a Vitagraphe, a combined camera, projector and printer constructed by the French firm Clément et Gilmer, and on 1 September 1896 his first films were announced, which all showed scenes of actors at Moscow's Fyodor Korsh theatre in the course of rehearsals, in their dressing rooms, during intervals, eating and leaving the theatre. Fyodorov also produced similar 'behind the scenes' films for the suburban theatre of Pushkino in the summer of 1897, and again at the Korsh theatre on 1 September 1897, at the opening of the theatre season. At the same time he also produced several films of general interest: Free Fire Brigade of Bogorodsk in Moscow, Horse-Drawn Railway in Moscow and the fiction pieces Playing the Ball and an imitation of Lumière's Arroseur Arrosé with Fyodorov playing the gardener. Although his films received much acclaim in Moscow, by 1898 his name disappears completely from the press reports of film activities. He continued his career as a theatre actor, but returned to films in the 1910s, acting in five films for Aleksandr Drankov and the Moscow branch of Pathé Frères. In 1915 he established the First Sign film company, designed for the film reproduction of famous theatre performances, but whose debut production of Gogol's The Inspector General, though boasting an outstanding cast, was filmed in a wholly archaic manner. Illness overtook Fyodorov after 1916, his film business failed and he left the stage. None of his films is known to survive.


Sibel Edmonds is the author of the Memoir Classified Woman: The Sibel Edmonds Story. She is the founder & director of National Security Whistleblowers Coalition (NSWBC), and the recipient of the 2006 PEN Newman's Own First Amendment Award Ms. Edmonds has a MA in Public Policy and International Commerce from George Mason University, a BA in Criminal Justice and Psychology from George Washington University, and AS degree in Biology from NVCC. She is a certified linguist for Turkish, Farsi, Dari and Azerbaijani languages.
Transcript: "Target Iran" Operations on Pause- Here's Why!


JEEP COMPASS keyless remotes with savings up to 75% off dealer price. Purchase your replacement Jeep Compass key fobs from us and save. We also carry many accessories for Jeep Compass Remote controls like covers, remote cases, and transmitters - all at low, discount prices. We are the Better Business Bureau member.


text:
package org.covito.kit.utility.bean;

import org.covito.kit.FrameworkException;

public class CastorNotFoundException extends FrameworkException {
	private static final long serialVersionUID = 1L;

	public CastorNotFoundException(String message) {
		super(message);
	}

	public CastorNotFoundException(Exception e) {
		super(e);
	}
}


text:
Q: Install a Program from a Directory with All of Its Dependencies Suppose I have a directory in which I have .deb packages for two programs with all there dependencies. I want to know if there is a apt or dpkg command, or another program to install one out of these two programs with all their dependencies. The dependencies should be installed automatically as I install the program.

A: Use sudo dpkg -i *.deb to install all .deb packages

A: Try using apt-offline for this. I am using Ubuntu 12.04 (precise) for this!

*

*Download apt-offline package on your Networked PC from packages.ubuntu.com. wget http://ubuntu.hitsol.net/ubuntu//pool/universe/a/apt-offline/apt-offline_1.1.1build1_all.deb

*Copy apt-offline_1.1.1build1_all.deb to your NON-Networked Notebook
using an USB-Stick

*Next install it there: sudo dpgk -i apt-offline_1.1.1build1_all.deb

*Now say you want to install a package called cuckoo on the NON-Networked Notebook execute this command on the NON-Networked Notebook:sudo apt-offline set cuckoo-offline.sig --install-packages cuckoo

*Copy the file cuckoo-offline.sig to the Networked PC and run this command:sudo apt-offline get cuckoo-offline.sig --no-checksum --bundle cuckoo-offline.zip This will fetch the package and all it's dependencies.

*Finally copy the zip file cuckoo-offline.zip using an USB-Stick to your NON-Networked Notebook and run this command there: sudo apt-offline install cuckoo-offline.zip
Of course you can do this for updates/upgrades and for other packages.
How to Update/Upgrade Your Ubuntu Without Internet Connection

redpajama_set_name: RedPajamaStackExchange


text:
ViewsBusiness
By CartoonArts International
ViewsBusiness by CartoonArts International for July 11, 2019
Radish Premium Member 4 days ago
To find a time when the planet's air was consistently above 400 ppm you have to look much farther back to the warm part of the Miocene, some 16 million years ago, or the Early Oligocene, about 25 million years ago, when Earth was a very different place and its climate totally dissimilar from what we might expect today.
There's a lot of debate about both temperatures and CO2 levels from millions of years ago. But the evidence is much firmer for the last 800,000 years, when ice cores show that CO2 concentrations stayed tight between 180 and 290 ppm, hovering at around 280 ppm for some 10,000 years before the industrial revolution hit.
How the World Passed a Carbon Threshold and Why It Matters
BY NICOLA JONES • JANUARY 26, 2017
Mauna Loa Observatory | Atmospheric CO2 Concentrations
411.58 ppm
NOAA-ESRL
© Copyright ViewsBusiness, © CartoonArts International/CWS


text:
Q: Why Axios response doesn't console log result? I'm working on an backend API but at some point I need to get user data from another API. I am trying to use Axios to make http request in order to do that. The request return the result in the browser as expected but the problem is that I can't display console log in the terminal. It doesn't show anything even though I asked the program to do so. Is there a problem probably with my code?
Here is my code :
const axios = require('axios');
const AxiosLogger = require('axios-logger');

const instance = axios.create();

module.exports = (router) => {
    router.get('/profile', function(req, res) {
       //random fake profile info
        axios.get('https://randomuser.me/api/')
            .then(response => {
                console.log(response.data);
                console.log(response.data);
                return response.data
            })
            .catch(error => {
                console.log(error);
            });

    });
};

A: I would suggest trying response.send to forward the axios response to your client like so:
module.exports = (router) => {
    router.get('/profile', function(req, res) {
        //random fake profile info
        axios.get('https://randomuser.me/api/')
            .then(response => {
                console.log(response.data);
                // Send the axios response to the client...
                res.send(response.data)
            })
            .catch(error => {
                console.log(error);
            });
    });
};


After the cryptocurrency market boom of 2017, many digital coin and tokens have been developed with each seeking to solve a specific problems. Some have thrived to become global brand hits while others have become misses. CyberMiles (CMT) is among coins that are industry specific and their use real life use cases have catapulted it to its current popularity.
The platform offers the e-commerce industry access to templates that enable them easily compatible with blockchain technologies. Cybermiles borrows a lot from Ethereum in the provision of smart contracts only that it does it better. Most large scale e-commerce companies prefer working with Cybermiles due to its low transaction fees.
Cybermiles is preparing to usher in their MainNet and are looking for 19 validators during the genesis phase that will help run the ecosystem to ensure its stability during the first year after its launch. Applications are open and upon application interested parties will go through an evaluation process before getting started on the blockchain.
To be considered as a validator, you need to provide your company name and logo for the community to identify you. You must have a website and the team that is knowledgeable about matters blockchain. You are also required o provide the maximum amount of CMT you are willing to stake.
Other requirements include technical information about your data center. This includes the server type, how it is configured, its location and security you are indenting to provide. You should also have a dedicated budget for community outreach.
Last but more importantly, you need to showcase plans for your social media exposure for better engagement with the community. You need to have at least 500+ followers to be considered as a validator.
CyberMiles (CMT) smart contracts come with a wide range of programming languages that is bug proof. This has been developed to improve on the Bitcoin and Ethereum ecosystems that are not 100% bug free. It is for this reason the platform has come up with cryptocurrency online market place, Blocktronic.io that only accepts digital coins as the only form of payment.
Blocktronic.io only accepts CMT coins as the only way to make payments. At the moment, the platform has over 70 products and more are being added to its portfolio. Some popular products on sale include electronic and their first ever mining machine for CMT, The Cybermiles platform is fully optimized for most commercial applications.
Cybermiles (CMT) has been steadily growing and is now ranked 71 with a market cap of $145.6millionm 692.5million CMT in circulation and the last 24 hours a total of $16,920,200 worth of CMT tokens have changed hands.
Cybermiles (CMT) is so focused that many crypto analysts think the platform is becoming a threat to e-commerce movers like Alibaba and Amazon. Will the coin is still in its beta stages and will soon start making inroads into the ever growing market.
The growth of the platform and its value traction is its route to market strategy and has already hit the ground running after getting listed on Binance, OkEx, Huobi, Zebpay and Cobinhood among the major exchanges. The pairing so far is a good mix and aimed at bringing the token to the masses towards its MainNet launch.


SHS-CAS graphic machines is seller and buyer of used printing machines with over 20 years experience in the market. Our stock of over 2500 sqm is loacted 15 minutes from the Hamburg Airport. We have our own technicians and own dry ice cleaning sytem.