text:
The 2016 Service Auctions Are Almost Here!
Chocolate Is the Answer
Silent: Sunday, November 6, Noon-2:00 p.m.
Charleston Day School
Live: Saturday, November 12, 6:30-9:00 p.m.
Gage Hall
Give It to Us – We Can Take It – And You Can Deduct It!
There is still time to be creative and resourceful in coming up with donations for each of the service auctions. Consider what's in your household, your expertise or passion that can be shared. Also think of your network of friends and associates, and the community of businesses you patronize.
With chocolate as the theme, we hope to appeal to all of our members—new and old, couples, young singles and families. Please consider donating a service or item that would be reasonably priced and of particular interest to our expanding and wonderfully diverse church family. Some examples include: "cater" a meal for 2 families at buyer's home; paint a room; host a pool party; host a child's birthday party; provide a gift certificate to a popular music venue; make arrangements for a surprise birthday party for someone's significant other. You get the idea – let your imagination run wild!
Click HERE for the online donation form.
*Donations due by Tuesday, November 1
Click HERE for a list of silent auction current offerings!
Click HERE for a list of live auction current offerings!
Mark Your Calendars to Attend!
Our auction events are fun! Both offer food, fellowship and a terrific assortment of auction items. The Live Auction is on a
Saturday night and offers food, wine and other beverages along with lively bidding. Childcare is available by request.


text:
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subset: commoncrawl


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Law Office of La'Donna Harlan, PLLC
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Nasty NymphosFavorites [+]
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When you purchase DRM-free downloadable content, that content is available for you to download on any computer and save to a disk whenever you like. You can also download to multiple computers. There is no timeframe associated with your purchase and the content will not expire.
Running Time: 140 minutes
Released: 03/2003
Studio Name: Anabolic Digital
Series: Nasty Nymphos
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text:
Jim Geraghty points out that Romney just needed 407,000 additional votes in four states to be president — out of 120 million cast nationally. Keep in mind, though, that with 121,000 additional votes in two states, Democrats would have kept Republicans out of the White House for 24 years (1993–2017).
Close Is a Relative Term
The Corner
The one and only.


text:
Staff Recognition Nominations Due by March 31
It's time for the Outstanding Educator award; we have the great opportunity to publicly recognize educators who make a difference for our special kids. The winner will receive an award and be recognized at an exclusive Washington State PTA dinner. Do you know an educator who's just smashing? Think of… Read more »


text:
i have been working on this not knowing there was a CAL going on here!!! some i have you guys to motivate me!! here is no progress so far...
i followed the pattern as it was written, but i did a round of double crochet around to tie them all together
do you guys think this would make a good baby blanket? the mom is a little non-conventional
the first pic shows the colors the truest


text:
Site Navigation
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Student Administration
Issue 35
25 November 2016
2016 timetables are available using the 2016 Online Timetable link.
2017 timetables are available using the 2017 Online Timetable link.
Important Dates
|2 December
|Tutorial Direct opens to staff for buffering and application of constraints
The Semester 1 2017 Draft Timetable has now been published for students and staff and the Change Request Form (CRF) is open to accept requests for changes to this timetable. As students have now viewed this timetable, it is imperative that requests for changes are kept to a minimum and comply with the conditions set out in the University's Timetabling Policy.
Due to the cut over and testing of Tutorial Direct in the single sign-on portal, the opening of the system for buffering and posting of messages has had to be delayed until Friday 2 December.
Please note:
Timetabling staff members hold a weekly team meeting every Wednesday afternoon between 2pm and 3pm.


text:
Hair Salon / Shopping Mall Cardboard POP Displays For Natural Hair Care Products
Natural Hair Care Products Cardboard POP Display For Hair Salon / Shopping Mall
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After-treatment: glossing, laminating, spot. UV, embossing, stamping, water-proof;
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4. Customized Size: Our normal size for free standing display is W 50cm X D 45cm X H 157cm. But we have designing department for structure designing and plane designing, professional graphics & structural design can be offered.
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|Natural Hair Care Products Cardboard POP Display For Hair Salon / Shopping Mall
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40% of Total Amount as deposit, 60% of Total Amount as Balance by TT
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Express, Air or Sea
Flat packaging saving shipping costs or as per Customers' request
Lead time
About 15-20days after the final sample is confirmed and deposit reaches
our account
Sample Fee
200usd for per sample with inkjet printing(shipping cost is not included),
the sample free will be refunded after the final order confirmed(above the MOQ)
Production Time
3-5 days after the artwork is confirmed and the sample fee reaches our account.
Inkjet printing
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text:
: Value of:
John Gibson
View Single Post
Advanced Stagnostic
Join Date: Aug 2008
Location: Bergisch Gladbach
Posts: 19,233
vCash: 500
Well, this took a bit longer than I expected.
Originally Posted by
Gibson was great. Before annointing him anything, I recall Jack Campbell being great in a WJHC. Where's he now? I could name several Canadian goalies I could say the same about over the years. He looks like a good prospect. But he hasn't proven a thing professionally.
He doesn't need to be "annointed" anything. Not having proven a thing professionally is inherently true for any prospect at that level - it doesn't change anything about the subjective value to a hockey club. And when looking at Gibson's play, I feel more confident in his style than in Campbell's, in terms of how good of a basis he has to move his game to the next level. There's no guarantees, obviously. But there's no question whatsoever that he's a premier goaltending prospect, and those hold very real value around the league.
Originally Posted by
Frederik Andersen (born 2 October 1989) is a Danish ice hockey goaltender, who currently plays for Norfolk Admirals.
That doesn't really contradict what he says, since Andersen can't really be considered a "top notch blue chip goaltending prospect". He's an intruiging project, but he certainly has significant holes in his game that he'll have to work on to make it to the NHL. His numbers in the AHL indicate that even with the struggles he did face, there's some upside, but there shouldn't be any doubts about him being a long-shot, and not coming close to the value of Gibson.
Ultimately, the Ducks most likely won't trade him. They have been looking to achieve some sort of depth in the goalie pipeline for quite a while, and Gibson is the safest bet (he's not a sure bet, obviously, he's just the safest in comparision) to take over from Hiller when he's ready. It's a very comfortable situation for the Ducks, and since we don't really lack prospect depth in any area far more than in net, it's very hard to imagine a deal the Ducks would be tempted with.
Organisations tend to not give away prospects that they have every reason to be excited with.
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text:
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text: Seems like a rather elementry question, but I see so many problems with the main shut off valves most people have used, thought I'd ask what other guy's use... Most systems here are on wells and do not have double checks or backflow preventers. There is usually a main shut off valve in a round box that's 2 to 4 feet down. Many have went to the glue-in plastic ball valves with the red handle, they seem to work well but if someone breaks off the ears on the handle you are boned... The other valves are usually brass threaded gate valves, they are pretty unrelieable, either they won't close tight, the stem leaks, or the handle rusts and comes off. Brass ball valves work the best, but the long handles on those have to be either bent or cut to fit into a 4" pipe. Many Iv'e seen the handle was galvinized steel and in a few years the handle has rusted and dosn't work right too. So what do the pro's use? BTW, this is a 1" or 1 1/4" valve, less than 20 gpm. Would only be turned off for winter then back on again for spring.


text:
What is the purpose of the dateline?
The Inverted Pyramid
What is the title of the news article called?
What is the name of the subheading of an article?
What is the lead of a news article and what does it do?
The Inverted Pyramid
Information in a News Article
What part of the newspaper identifies the author?
To identify the location of
the author is writing from
The lead is the first paragraph of a news article and it contains the most important information: the 5W's of the event.
What part of the paper identifies the title of the paper, its editors and its location?
The Masthead
The Lead - 5 Ws
Additional Info
Least Important Info
The Lead - The most critical and newsworthy
information: Who, what, where, why and when
- Information that must be conveyed in
order to make communication successful
Body - Helpful, though not critical information. Contains additional facts and details that are revealed in order of importance. Information is useful to an
understanding of the story but not necessary.
Contains supporting information,
interviews, overviews or references
Least important information, unnecessary to the understanding of the
story. Details for
those most
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text: Project Description The kitchen of this 1890's Victorian home, on the National Register of Historic Places, didn't fit the needs of the clients or match the historic flavor of the home. Remodeled in the 1970's, the kitchen was poorly designed, in disrepair, and lacked adequate storage. Generally, the owners wanted to keep with the current floor plan of the kitchen, including space for an eat-in table. The architect decided on a pre-WWII design style with modern updates. The entire kitchen was gutted, and the awkward furr-downs that hid the air conditioning ducts were reorganized. The outdated cabinets were replaced with new face framed cabinets topped with crown molding. New open storage surrounds the range and sink. The kitchen is illuminated by period sconces over the farmhouse sink and window niche, pendant lights throughout the room, and under cabinet lighting. Carrera marble counters and off-white subway tiles in the backsplash contribute to the austere, clean period look. In order to maximize storage opportunities, new shallow tower cabinets were built on either side of the original kitchen window. The cabinets next to the exterior door function as a kids' drop zone while those on the other side of the window provide much needed dish storage. A window seat, built of long leaf pine to match the table, provides extra seating as needed. An original brick vertical element from a former wood burning stove remains next to the range, and the Saltillo tile from the 1970's remodel goes well with the new look. As might be expected with a more than a century old home, there were plenty of repairs that had to be addressed before the work was able to progress. Additional blocking was added to the home's original balloon framing. Spaces were sealed where rodents and insects were entering. The original double hung window was rebuilt and the ropes and counter weights were repaired. The hardware of the transoms above the doors was replaced and repaired. Plumbing was reorganized and upgrades to the electrical system were made. New fiberglass insulation was added in the walls and floor. The family is enjoying the updated kitchen with its enhanced cabinetry, display shelves, and storage for kids' gear. Once mismatched and tired, the kitchen is now an inspired space with plenty of room to create meals, gather, and spend family time together.


text: Meetings & Meeting Reports The Practical Side of Mosquito Extermination + See all authors and affiliations Science 09 Mar 1906:Vol. 23, Issue 584, pp. 379-385DOI: 10.1126/science.23.584.379 HENRY CLAY WEEKSFind this author on Google Scholar Find this author on PubMed Search for this author on this site Article Info & Metrics eLetters PDF This is a PDF-only article. The first page of the PDF of this article appears below.


text:
Product code: 23374
Despatched in 5-10 days.
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eChemist.co.uk © 2017. All rights reserved.


text:
Beauty Secret From Within by French Secret®
Beauty Secret From Within is a unique blend combining powerful nutrients like Phosphatidylserine, Choline, Alpha-GPC, L-Carnosine and an ORAC blend, along with Resveratrol, Tremella, Ipriflavone and low molecular weight Hyaluronic Acid, Vitamins C & E, Green Tea, Grape Seed, PABA, Biotin, Silica, and Pantothenic acid.
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text:
Your answer is one click away!
I have a number of files 15GB+ in size, non of them can be compressed as the content is an encrypted container.
I have many of these files where only a slight different exists between them, so 90%+ of the data is common.
Using Winrar I can set a dictionary size of 1gb, but I believe this means only 1gb of the 15gb thats common to each file will be efficiently compressed. So two files equals 29gb best case.
Does any software exist to compress multiple large and similar files.
If the files are correctly encrypted there will be no similarity in the encrypted data. A properly encrypted file is indistinguishable from random data.
If there is any similarity, even small sections, between the files the encryption incorrect and compromised. If an encryption mode such as ECB or CTR mode with the same key and nonce.
Note: If there are repeats in the encrypted data that leaks information about the underlying data, that is in general a security problem.
Solid compression of WinRAR works different as you think. The dictionary size is just a memory space allocated for compressing similar small files used dynamically for each file.
For example I have a folder with 366 files. 30 files are text files with less than 12 KB. The others are binary files with a file size between 40 KB and 450 KB. Total size of all files is 48 MB. Solid compression with a dictionary size of just 4 MB with correct configured
RarFiles.lst for those files results in a RAR archive with just 205 KB using RAR4 format. Most files are stored with less than 500 bytes in the archive including header for the file as it can be seen on opening the RAR archive in WinRAR. So although the total number of bytes is more than 10 times of dictionary size, the solid compression is nevertheless impressive. The RAR archive file can be made even smaller by using RAR5 format and use a dictionary size of 64 MB resulting in a file size of 163 KB for the solid RAR archive file.
But solid compression of WinRAR is not designed for compressing similar very large files.
The technique to best compress such files is first putting all those files into a single archive file using store for compression method, i.e. producing one huge file with uncompressed data. Then this huge archive file is compressed using normal, good or even best compression without creating a solid archive.
Note: By default WinRAR just stores files containing usually already compressed data in an archive file. So after selecting the huge RAR archive with data of all files just stored in the archive and clicking on button Add, it is necessary to remove on tab Files the file name pattern *.rar from list of files to store without compression to get the select


text:
Property ID : 1505Cadfel
Rented / Occupied More coming soon - Condominium, Single Family
This is a prime, perfect location in the Wilmington area. This location is in the heart of Wilmington, being minutes away from Wilmington's historic downtown, Greenfield Lake/ Amphitheatre, and New Hanover Regional Medical Center. Downtown has around 230 blocks of beautiful Victorian style architecture, local trendy shops, local restaurants, and fun for all ages. The beautiful Greenfield Lake offers an endless list of enjoyable activities, along with the newly built Amphitheatre. Since this location is central, you are short 10-minute ride from Wrightsville Beach as well as Carolina Beach. This popular area is quiet and lined with cherished neighborhoods. Enjoy biking, jogging, and walking paths that conveniently lead you safely around the area. Take advantage of the prestige, award winning, local restaurants that are minutes away! Entertain all your guest without going within miles from your home. The home itself includes…
Amazing community with unmatched amenities including… Huge swimming pool with equipment, stunning clubhouse, state of the art fitness center, 24 seat theater, grilling and playground areas, tennis courts and a basketball court!
Beautiful stainless steel appliances
Undermount microwave
Gorgeous tile floor
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Ceiling fans throughout
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Large closets
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Sqft is approx and taken from the tax records, not measured. Rate and security deposit are usually, but not always based on 2 year occupancy, highly qualified tenant, and 30 day move in or less. Subject to change according to circum. We try to match competitor pricing so feel free to send them and we'll consider each case. Equal opportunity housing provider.
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text:
About half of the populations of many countries live in rural areas and in these countries about half of the rural hospitals do not have X-ray services. Among the reasons for such a deficiency are the high cost and sophistication of radiological equipment, requiring fully trained technicians, and the high standard required for a radiologist's training, none of which a rural hospital can afford. Simplified equipment makes it possible for locally recruited auxiliary personnel to obtain radiographs of consistently acceptable quality after a short training period. The training of physicians in radiographic interpretation could also be abbreviated if it were directed at a limited range of basic X-ray examinations that provide essential diagnostic information in most clinical conditions. The activities of the World Health Organization in these areas is discussed.
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text:
Ostrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias
- Primrose Gilbert
- 2 years ago
- Views:
1 Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology, PO Box 448, Melbourne City, MC 800, Victori, Austrli E-mil ddress, SS Drgomir: URL: TM Rssis) Deprtment of Mthemtics, Ntionl Technicl University of Athens, Zogrfou Cmpus, 5780 Athens, Greece
2 99 Mthemtics Subject Clssifiction Primry 6D5, 6D0; Secondry 4A55, 4A99 Abstrct Ostrowski type inequlities for univrite nd multivrite rel functions nd their nturl pplictions for numericl qudrtures re presented
3 Contents Prefce v Chpter Generlistions of Ostrowski Inequlity nd Applictions Introduction Generlistions for Functions of Bounded Vrition 3 3 Generlistions for Functions whose Derivtives re in L 5 4 Generlistion for Functions whose Derivtives re in L p 8 5 Generlistions in Terms of L norm 4 Bibliogrphy 53 Chpter Integrl Inequlities for n Times Differentible Mppings 55 Introduction 55 Integrl Identities 56 3 Integrl Inequlities 64 4 The Convergence of Generl Qudrture Formul 7 5 Grüss Type Inequlities 75 6 Some Prticulr Integrl Inequlities 80 7 Applictions for Numericl Integrtion 04 8 Concluding Remrks 8 Bibliogrphy 9 Chpter 3 Three Point Qudrture Rules 3 Introduction 3 Bounds Involving t most First Derivtive 3 33 Bounds for n Time Differentible Functions 86 Bibliogrphy Chpter 4 Product Brnches of Peno Kernels nd Numericl Integrtion 5 4 Introduction 5 4 Fundmentl Results 7 43 Simpson Type Formule 4 44 Perturbed Results 6 45 More Perturbed Results Using Seminorms Concluding Remrks 4 Bibliogrphy 43 Chpter 5 Ostrowski Type Inequlities for Multiple Integrls 45 5 Introduction 45 iii
4 CONTENTS iv 5 An Ostrowski Type Inequlity for Double Integrls Other Ostrowski Type Inequlities Ostrowski s Inequlity for Hölder Type Functions 73 Bibliogrphy 8 Chpter 6 Some Results for Double Integrls Bsed on n Ostrowski Type Inequlity 83 6 Introduction 83 6 The One Dimensionl Ostrowski Inequlity Mpping Whose First Derivtives Belong to L, b) Numericl Results Appliction For Cubture Formule Mpping Whose First Derivtives Belong to L p, b) Appliction For Cubture Formule Mppings Whose First Derivtives Belong to L, b) Integrl Identities Some Integrl Inequlities Applictions to Numericl Integrtion 3 Bibliogrphy 35 Chpter 7 Product Inequlities nd Weighted Qudrture 37 7 Introduction 37 7 Weight Functions Weighted Interior Point Integrl Inequlities Weighted Boundry Point Lobtto) Integrl Inequlities Weighted Three Point Integrl Inequlities 339 Bibliogrphy 35 Chpter 8 Some Inequlities for Riemnn-Stieltjes Integrl Introduction Some Trpezoid Like Inequlities for Riemnn-Stieltjes Integrl Inequlities of Ostrowski Type for the Riemnn-Stieltjes Integrl Some Inequlities of Grüss Type for Riemnn-Stieltjes Integrl 39 Bibliogrphy 40
5 v Prefce It ws noted in the prefce of the book Inequlities Involving Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, 99, by DS Mitrinović, JE Pečrić nd AM Fink; since the writing of the clssicl book by Hrdy, Littlewood nd Poly 934), the subject of differentil nd integrl inequlities hs grown by bout 800% Ten yers on, we cn confidently ssert tht this growth will increse even more significntly Twenty pges of Chpter XV in the bove mentioned book re devoted to integrl inequlities involving functions with bounded derivtives, or, Ostrowski type inequlities This is now itself specil domin of the Theory of Inequlities with mny powerful results nd lrge number of pplictions in Numericl Integrtion, Probbility Theory nd Sttistics, Informtion Theory nd Integrl Opertor Theory The min im of this present book, jointly written by the members of the Victori University node of RGMIA Reserch Group in Mthemticl Inequlities nd Applictions, is to present selected number of results on Ostrowski type inequlities Results for univrite nd multivrite rel functions nd their nturl pplictions in the error nlysis of numericl qudrture for both simple nd multiple integrls s well s for the Riemnn-Stieltjes integrl re given In Chpter, uthored by SS Drgomir nd TM Rssis, generlistions of the Ostrowski integrl inequlity for mppings of bounded vrition nd for bsolutely continuous functions vi kernels with n brnches including pplictions for generl qudrture formule, re given Chpter, uthored by A Sofo, builds on the work in Chpter He investigtes generlistions of integrl inequlities for n-times differentible mppings With the id of the modern theory of inequlities nd by use of generl Peno kernel, explicit bounds for interior point rules re obtined Firstly, he develops integrl equlities which re then used to obtin inequlities for n-times differentible mppings on the Lebesgue spces L, b, L p, b, < p < nd L, b Secondly, some prticulr inequlities re obtined which include explicit bounds for perturbed trpezoid, midpoint, Simpson s, Newton-Cotes, left nd right rectngle rules Finlly, inequlities re lso pplied to vrious composite qudrture rules nd the nlysis llows the determintion of the prtition required for the ccurcy of the result to be within prescribed error tolernce In Chpter 3, uthored by P Cerone nd SS Drgomir, unified tretment of three point qudrture rules is presented in which the clssicl rules of mid-point, trpezoidl nd Simpson type re recptured s prticulr cses Riemnn integrls re pproximted for the derivtive of the integrnd belonging to vriety of norms The Grüss inequlity nd number of vrints re lso presented which provide vriety of inequlities tht re suitble for numericl implementtion Mppings tht re of bounded totl vrition, Lipschitzin nd monotonic re lso investigted with reltion to Riemnn-Stieltjes integrls Explicit priori bounds re provided llowing the determintion of the prtition required to chieve prescribed error tolernce
6 CONTENTS vi It is demonstrted tht with the bove clsses of functions, the verge of midpoint nd trpezoidl type rule produces the best bounds In Chpter 4, uthored by P Cerone, product brnches of Peno kernels re used to obtin results suitble for numericl integrtion In prticulr, identities nd inequlities re obtined involving evlutions t n interior nd t the end points It is shown how previous work nd rules in numericl integrtion re recptured s prticulr instnces of the current development Explicit priori bounds re provided llowing the determintion of the prtition required for chieving prescribed error tolernce In the min, Ostrowski-Grüss type inequlities re used to obtin bounds on the rules in terms of vriety of norms In Chpter 5, uthored by NS Brnett, P Cerone nd SS Drgomir, new results for Ostrowski type inequlities for double nd multiple integrls nd their pplictions for cubture formule re presented This work is then continued in Chpter 6, uthored by G Hnn, where n Ostrowski type inequlity in two dimensions for double integrls on rectngle region is developed The resulting integrl inequlities re evluted for the clss of functions with bounded first derivtive They re employed to pproximte the double integrl by one dimensionl integrls nd function evlutions using different types of norms If the one-dimensionl integrls re not known, they themselves cn be pproximted by using suitble rule, to produce cubture rule consisting only of smpling points In ddition, some generlistions of n Ostrowski type inequlity in two dimensions for n - time differentible mppings re given The result is n integrl inequlity with bounded n - time derivtives This is employed to pproximte double integrls using one dimensionl integrls nd function evlutions t the boundry nd interior points In Chpter 7, uthored by John Roumeliotis, weighted qudrture rules re investigted The results re vlid for generl weight functions The robustness of the bounds is explored for specific weight functions nd for vriety of integrnds A comprison of the current development is mde with trditionl qudrture rules nd it is demonstrted tht the current development hs some dvntges In prticulr, this method llows the nodes nd weights of n n point rule to be esily obtined, which my be preferentil if the region of integrtion vries Other explicit error bounds my be obtined in dvnce, thus mking it possible to determine the weight dependent prtition required to chieve certin error tolernce In the lst chpter, SS Drgomir presents recent results in pproximting the Riemnn-Stieltjes integrl by the use of Trpezoid type, Ostrowski type nd Grüss type inequlities Applictions for certin clsses of weighted integrls re lso given This book is intended for use in the fields of integrl inequlities, pproximtion theory, pplied mthemtics, probbility theory nd sttistics nd numericl nlysis
7 vii The Editors, Melbourne nd Athens, December 000
9 CHAPTER Generlistions of Ostrowski Inequlity nd Applictions by SS DRAGOMIR nd TM RASSIAS Abstrct Generliztions of Ostrowski integrl inequlity for mppings of bounded vrition nd for bsolutely continuous functions vi kernels with n brnches plus pplictions for generl qudrture formule re given Introduction The following result is known in the literture s Ostrowski s inequlity see for exmple, p 468) Theorem Let f :, b R be differentible mpping on, b) with the property tht f t) M for ll t, b) Then ) f x) b ) f t) dt b x +b 4 + b ) b ) M for ll x, b The constnt 4 is the best possible in the sense tht it cnnot be replced by smller constnt A simple proof of this fct cn be done by using the identity: ) f x) b where f t) dt + b t if t x p x, t) : t b if x < t b p x, t) f t) dt, x, b, which lso holds for bsolutely continuous functions f :, b R The following Ostrowski type result for bsolutely continuous functions holds see 7, 0 nd 8)
10 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS Theorem Let f :, b R be bsolutely continuous on, b Then, for ll x, b, we hve: f x) 3) f t) dt b 4 + x +b ) b b ) f if f L, b ; where r nd p+) p x b ) p+ + + x +b b f ; The constnts 4, Theorem b x b ) p+ p b ) p f q if f L q, b, r, ) re the usul Lebesgue norms on L r, b, ie, g : ess sup g t) t,b g r : g t) r dt p+) p nd ) r, r, ) p + q, p > ; respectively re shrp in the sense presented in The bove inequlities cn lso be obtined from the Fink result in on choosing n nd performing some pproprite computtions If one drops the condition of bsolute continuity nd ssumes tht f is Hölder continuous, then one my stte the result see 5) Theorem 3 Let f :, b R be of r H Hölder type, ie, 4) f x) f y) H x y r, for ll x, y, b, where r 0, nd H > 0 re fixed Then for ll x, b we hve the inequlity: f x) 5) f t) dt b b ) r+ ) r+ H x x + b ) r r + b b The constnt r+ is lso shrp in the bove sense Note tht if r, ie, f is Lipschitz continuous, then we get the following version of Ostrowski s inequlity for Lipschitzin functions with L insted of H) due to SS Drgomir 3, see lso ) 6) f x) b Here the constnt 4 is lso best f t) dt x +b 4 + b ) b ) L
11 3 SS Drgomir nd TM Rssis Moreover, if one drops the condition of the continuity of the function, nd ssumes tht it is of bounded vrition, then the following result due to Drgomir my be stted see lso 4 or ) Theorem 4 Assume tht f :, b R is of bounded vrition nd denote by b its totl vrition Then 7) f x) b for ll x, b The constnt is the best possible f t) dt + x +b b b f) If we ssume more bout f, ie, f is monotoniclly incresing, then the inequlity 7) my be improved in the following mnner see lso ) Theorem 5 Let f :, b R be monotonic nondecresing Then for ll x, b, we hve the inequlity: 8) f x) f t) dt b { x + b) f x) + b sgn t x) f t) dt {x ) f x) f ) + b x) f b) f x)} b + x +b f b) f ) b } All the inequlities in 8) re shrp nd the constnt is the best possible For recent generlistion of this result see 6 where further extensions were given The min im of the present chpter is to provide number of generlistions for kernels with N brnches of the bove Ostrowski type inequlity Nturl pplictions for qudrture formule re lso given Generlistions for Functions of Bounded Vrition Some Inequlities We strt with the following theorem 4 Theorem 6 Let I k : x 0 < x < < x k < x k b be division of the intervl, b nd α i i 0,, k + ) be k+ points so tht α 0, α i x i, x i i,, k) nd α k+ b If f :, b R is of bounded vrition on, b, then
12 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 4 we hve the inequlity: 9) f x) dx ν h) + mx ν h) b f), k α i+ α i ) f x i ) { α i+ x } i + x i+ b, i 0,, k f) where ν h) : mx {h i i 0,, k }, h i : x i+ x i i 0,, k ) nd b f) is the totl vrition of f on the intervl, b Proof Define the kernel K :, b R given by see lso 4) t α, t, x ) t α, t x, x ) K t) : t α k, t x k, x k ) t α k, t x k, b Integrting by prts in Riemnn-Stieltjes integrl, we hve successively K t) df t) k xi+ k k K t) df t) x i t α i+ ) f t) xi+ x i xi+ x i xi+ x i t α i+ ) df t) f t) dt k α i+ x i ) f x i ) + x i+ α i+ ) f x i+ ) k α ) f ) + α i+ x i ) f x i ) k + i x i+ α i+ ) f x i+ ) + b α n ) f b) k k α ) f ) + α i+ x i ) f x i ) + x i α i ) f x i ) + b α n ) f b) i f t) dt k α ) f ) + α i+ α i ) f x i ) + b α n ) f b) i k α i+ α i ) f x i+ ) f t) dt, i f t) dt f t) dt f t) dt
13 5 SS Drgomir nd TM Rssis nd then we hve the integrl equlity which is of interest in itself too: 0) f t) dt k α i+ α i ) f x i ) Using the modulus properties, we hve k K t) df t) k However, xi+ t α i+ ) f t) dt sup x i t x i,x i+ xi+ x i xi+ x i K t) df t) K t) df t) K t) df t) : T x i+ t α i+ f) x i+ mx {α i+ x i, x i+ α i+ } f) x i Then T x i+ x i ) + α i+ x i + x i+ x i x i+ f)dt k h i + α i+ x x i + x i+ i+ f) x i mx,,k h i + α i+ x i + x i+ k x i+ f) x i { ν h) + mx α i+ x } i + x i+ b, i 0,, k f) : V x i Now, s then mx nd, consequently, α i+ x i + x i+ h i, { α i+ x i + x i+ The theorem is completely proved V ν h) }, i 0,, k ν h) b f) Now, if we ssume tht the points of the division I k re given, then the best inequlity we cn obtin from Theorem 6 is embodied in the following corollry:
14 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 6 Corollry 7 Let f nd I k be s bove Then we hve the inequlity: f x) dx ) x ) f ) k + x i+ x i ) f x i ) + b x k ) f b) i b ν h) f) Proof We choose in Theorem 6, In this cse we get α 0, α + x, α x + x,, α k x k + x k, α k x k + x k k α i+ α i ) f x i ) α α 0 ) f ) + α α ) f x ) + nd α k+ b + α k α k f x k ) + b α k ) f b) ) + x x + x f ) + + x ) f x ) xk + b + + x ) k + x k f x k ) + b x ) k + b f b) k x ) f ) + x i+ x i ) f x i ) + b x k ) f b) i Now, pplying the inequlity 9), we get ) The following corollry for equidistnt prtitioning lso holds Corollry 8 Let I k : x i : + b ) i i 0,, k) k be n equidistnt prtitioning of, b If f is s bove, then we hve the inequlity: b k f ) + f b) b ) k i) + ib ) f x) dx b ) + f k k k b k b ) f) i
15 7 SS Drgomir nd TM Rssis A Generl Qudrture Formul Let n : x n) 0 < x n) < < x n) n < xn) n b be sequence of division of, b nd consider the sequence of numericl integrtion formule I n f, n, w n ) : n j0 w n) j f x n) j where w n) j j 0,, n) re the qudrture weights nd n j0 wn) j b The following theorem provides sufficient condition for the weights w n) j I n f, n, w n ) pproximtes the integrl f x) dx see lso 4) ) so tht Theorem 9 Let f :, b R be function of bounded vrition on, b If the qudrture weights w n) j stisfy the condition i 3) x n) i w n) j x n) i+ for ll i 0,, n, j0 then we hve the estimte I 4) n f, n, w n ) f x) dx h ν n)) i + mx + ν h n)) b f), j0 w n) j xn) i + x n) b i+, i 0,, n f) where ν h n)) { } : mx h n) i i 0,, n nd h n) i : x n) i+ xn) i In prticulr, 5) lim I n f, n, w n ) νh n) ) 0 uniformly by rpport of the w n Proof Define the sequence of rel numbers Note tht α n) α n) nd observe lso tht α n) i+ i+ : + i n+ + n x n) i j0 j0, x n) i+ f x) dx w n) j, i 0,, n w n) j + b b, Define α n) 0 : nd compute α n) α n) 0,
16 Then GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 8 α n) i+ αn) i + n i j0 α n) n+ αn) n ) α n) i+ αn) i f w n) j i j0 w n) j n b + x n) i ) n j0 w n) i i,, n ), w n) j w n) i f x n) i Applying the inequlity 9), we get the estimte 4) w n) n ) I n f, n, w n ) The uniform convergence by rpport of qudrture weights w n) j lst inequlity is obvious by the Now, consider the equidistnt prtitioning of, b given by E n : x n) i : + i b ) i 0,, n) n nd define the sequence of numericl qudrture formule n I n f, w n ) : w n) i f + in b ) The following corollry which cn be more useful in prctice holds: Corollry 0 Let f be s bove If the qudrture weight w n) j condition: 6) i n b i j0 then we hve: I 7) n f, w n ) f x) dx b n + mx i + b ) n b f) j0 In prticulr, we hve the limit uniformly by rpport of w n w n) j i +, i 0,, n ; n w n) j i + lim I n f, w n ) n b ) n f x) dx, stisfy the b, i 0,, n f)
17 9 SS Drgomir nd TM Rssis 3 Prticulr Inequlities The following proposition holds 4 see lso 4) Proposition Let f :, b R be function of bounded vrition on, b Then we hve the inequlity: 8) f x) dx α ) f ) + b α) f b) b ) + α + b b f) for ll α, b The proof follows by Theorem 6 choosing x 0, x b, α 0, α α, b nd α b Remrk ) If in 8) we put α b, then we get the left rectngle inequlity 9) b f x) dx b ) f ) b ) f); b) If α, then by 8) we get the right rectngle inequlity b 0) f x) dx b ) f b) b ) f); c) It is esy to see tht the best inequlity we cn get from 8) is for α +b obtining the trpezoid inequlity see lso 4) ) f x) dx f ) + f b) b ) b b ) f) Another proposition with mny interesting prticulr cses is the following one 4 see lso 4): Proposition Let f be s bove nd x b, α x α b Then we hve ) f x) dx α ) f ) + α α ) f x ) + b α ) f b) b ) + x + b + α + x + α x + b + α + x α x + b b f) b ) + x + b b b f) b ) f)
18 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 0 Proof Consider the division x 0 x x b nd the numbers α 0, α, x, α x, b nd α 3 b Now, pplying Theorem 6, we get f x) dx α ) f ) + α α ) f x ) + b α ) f b) { mx {x, b x } + mx α + x, α x } + b b f) 4 b ) + x + b + α + x + α x + b + α + x α x + b b f) nd the first inequlity in ) is proved Now, let observe tht α + x x, α x + b b x Consequently, mx { α + x nd the second inequlity in ) is proved The lst inequlity is obvious, α x } + b mx {x, b x } Remrk ) If we choose bove α, α b, then we get the following Ostrowski type inequlity obtined by Drgomir in the recent pper : 3) f x) dx b ) f x ) b ) + x + b b f) for ll x, b We note tht the best inequlity we cn get in 3) is for x +b obtining the midpoint inequlity see lso ) ) + b 4) f x) dx f b ) b b ) f) b) If we choose in ) α 5+b 6, α +5b 6 nd x 5+b 6, +5b 6, then we get f x) dx b f ) + f b) + f x ) 5) 3 b ) + x + b { + mx x + b } 3, + b b x f) 3
19 SS Drgomir nd TM Rssis In prticulr, if we choose in 5), x +b, then we get the following Simpson s inequlity 9 f x) dx b ) f ) + f b) + b 6) + f 3 b 3 b ) f) 4 Prticulr Qudrture Formule Let us consider the prtitioning of the intervl, b given by n : x 0 < x < < x n < x n b nd put h i : x i+ x i i 0,, n ) nd ν h) : mx {h i i 0,, n } The following theorem holds 4: Theorem 3 Let f :, b R be bsolutely continuous on, b nd k Then we hve the composite qudrture formul 7) where 8) A k n, f) : T n, f) + k nd f x) dx A k n, f) + R k n, f), n k k j) xi + jx i+ f k j 9) T n, f) : n f x i ) + f x i+ ) h i is the trpezoid qudrture formul The reminder R k n, f) stisfies the estimte 30) R k n, f) b k ν h) f) h i get Proof Applying Corollry 8 on the intervls x i, x i+ i 0,, n ) we xi+ x i x k h i+ i f) f x) dx k x i f x i ) + f x i+ ) h i + h i k k k j) xi + jx i+ f k j
20 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS Now, using the generlized tringle inequlity, we get: R k n, f) n xi+ f x) dx x i k f x i) + f x i+ ) h i + h k i k j) xi + jx i+ f k k n k f) ν h) n k h i x i+ x i nd the theorem is proved x i+ x i f) ν h) k b f) j The following corollries hold: Corollry 4 Let f be s bove Then we hve the formul: 3) f x) dx T n n, f) + M n n, f) + R n, f) where M n n, f) is the midpoint qudrture formul, n ) xi + x i+ M n n, f) : f h i nd the reminder R n, f) stisfies the inequlity: 3) R n, f) b 4 ν h) f) Corollry 5 Under the bove ssumptions we hve b f x) dx n ) xi + x i+ 33) T n n, f) + f h i 3 3 n ) xi + x i+ + f h i + R 3 n, f) 3 The reminder R 3 n, f) stisfies the bound: 34) R 3 n, f) b 6 ν h) f) The following theorem holds 4 see lso 4): Theorem 6 Let f nd n be s bove nd ξ i x i, x i+ i 0,, n ) Then we hve the qudrture formul: 35) n f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) + R ξ, n, f)
21 3 SS Drgomir nd TM Rssis The reminder R ξ, n, f) stisfies the estimtion: 36) R ξ, n, f) { ν h) + mx ξ i x } i + x i+ b, i 0,, n f) ν h) for ll ξ i s bove b f), get Proof Apply Proposition on the intervl x i, x i+ i 0,, n ) to xi+ f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) x i { h i + mx ξ i x } x i + x i+ i+ f) x i Summing over i from 0 to n, using the generlized tringle inequlity nd the properties of the mximum mpping, we get 36) Corollry 7 Let f nd n be s bove Then we hve ) the left rectngle rule 37) n f x) dx f x i ) h i + R l n, f) ; ) the right rectngle rule 38) n f x) dx f x i+ ) h i + R r n, f) ; 3) the trpezoid rule 39) where nd f x) dx T n, f) + R T n, f) b R l n, f) R r n, f) ν h) f) R T n, f) b ν h) f) The following theorem lso holds 4
22 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 4 Theorem 8 Let f nd n be s bove nd ξ i x i, x i+, x i α ) i x i+, then we hve the qudrture formul: α ) i 40) n f x) dx n + α ) i x i ) f x i ) + x i+ α ) i n ) f x i+ ) + R ) α ) i α ) i f ξ i ) ) ξ, α ), α ), n, f ξ i The reminder R ξ, α ), α ), n, f ) stisfies the estimtion R ξ, α ), α ) 4), n, f) { ν h) + mx,,n ξ i x i + x i+ { + mx mx,,n α) i x i + ξ i, mx,,n α) i ξ }} i + x i+ b f) ν h) + mx,n ξ i x i + x i+ b f ν h) f) Proof Apply Proposition on the intervl x i, x i+ to obtin xi+ f x) dx x i ) ) ) α ) i x i f x i ) + α ) i α ) i f ξ i ) + x i+ α ) i f x i+ ) h i + ξ i x i + x i+ { + mx α ) i x i + ξ i, α ) i ξ } x i + x i+ i+ f) x i Summing over i from 0 to n nd using the properties of modulus nd mximum, we get the desired inequlity We shll omit the detils The following corollry is the result of Drgomir from the recent pper Corollry 9 Under the bove ssumptions, we hve the Riemnn s qudrture formul: 4) n f x) dx f ξ i ) h i + R R ξ, n, f)
23 5 SS Drgomir nd TM Rssis The reminder R R ξ, n, f) stisfies the bound 43) R R ξ, n, f) { ν h) + mx ξ i x } i + x i+ b, i 0,, n f) ν h) b f) for ll ξ i x i, x i+ i 0,, n) Finlly, the following corollry which generlizes Simpson s qudrture formul holds Corollry 0 Under the bove ssumptions nd if ξ i xi++5x i 6, xi+5xi+ 6 i 0,, n ), then we hve the formul: 44) f x) dx n f x i ) + f x i+ ) h i + n f ξ 6 3 i ) h i + S f, n, ξ) The reminder S f, n, ξ) stisfies the estimte: 45) S f, n, ξ) { { ν h) + mx ξ i x } i + x i+,,n { + mx mx,,n ξ i x i + x i+ 3, mx,,n x i + x i+ 3 The proof follows by the inequlity 5) nd we omit the detils ξ i }} b Remrk 3 Now, if we choose in 44), ξ i xi+xi+, then we get Simpson s qudrture formul 9 46) f x) dx n f x i ) + f x i+ ) h i 6 + n ) xi + x i+ f h i + S f, n ) 3 where the reminder term S f, n ) stisfies the bound: 47) S f, n ) b 3 ν h) f) f) 3 Generlistions for Functions whose Derivtives re in L 3 Some Inequlities We strt with the following result 5 Theorem Let I k : x 0 < x < < x k < x k b be division of the intervl, b, α i i 0,, k + ) be k + points so tht α 0, α i
24 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 6 x i, x i i,, k) nd α k+ b If f :, b R is bsolutely continuous on, b, then we hve the inequlity: 48) k f x) dx α i+ α i ) f x i ) k k h i + α i+ x ) i + x i+ f 4 k f h i b ) f ν h), where h i : x i+ x i i 0,, k ) nd ν h) : mx {h i i 0,, k } The constnt 4 in the first inequlity nd the constnt in the second nd third inequlity re the best possible Proof Define the mpping K :, b R given by see the proof of Theorem 6) K t) : t α, t, x ) ; t α, t x, x ) ; t α k, t x k, x k ) ; t α k, t x k, b Integrting by prts, we hve successively: K t) f t) dt k xi+ k k K t) f t) dt x i t α i+ ) f t) xi+ x i xi+ x i xi+ x i t α i+ ) f t) dt f t) dt k α i+ x i ) f x i ) + x i+ α i+ ) f x i+ ) f t) dt k k α ) f ) + α i+ x i ) f x i ) + x i+ α i+ ) f x i+ ) + b α n ) f b) i f t) dt
25 7 SS Drgomir nd TM Rssis k k α ) f ) + α i+ x i ) f x i ) + x i α i ) f x i ) + b α n ) f b) i f t) dt k α ) f ) + α i+ α i ) f x i ) + b α n ) f b) i k α i+ α i ) f x i ) f t) dt nd then we hve the integrl equlity: k 49) f t) dt α i+ α i ) f x i ) Using the properties of modulus, we hve K t) f 50) t) dt k xi+ k K t) f t) dt x i k xi+ x i A simple clcultion shows tht 5) xi+ x i t α i+ dt for ll i 0,, k xi+ x i t α i+ f t) dt f αi+ i K t) f t) dt K t) f t) dt k xi+ x i xi+ f t) dt t α i+ dt α i+ t) dt + t α i+ ) dt x i α i+ x i+ α i+ ) + α i+ x i ) 4 h i + α i+ x ) i + x i+ Now, by 49) 5), we get the first inequlity in 48) Assume tht the first inequlity in 48) holds for constnt c > 0, ie, k 5) f x) dx α i+ α i ) f x i ) k k c h i + α i+ x ) i + x i+ f If we choose f :, b R, f x) x, α 0, α b, x 0, x b in 5), we obtin b ) c b ) b ) +, 4
26 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 8 from where we get c 4, nd the shrpness of the constnt 4 is proved The lst two inequlities s well s the shrpness of the constnt we omit the detils re obvious nd Now, if we ssume tht the points of the division I k re given, then the best inequlity we cn get from Theorem is embodied in the following corollry: Corollry Let f, I k be s bove Then we hve the inequlity 53) f x) dx k x ) f ) + x i+ x i ) f x i ) + b x k ) f b) k 4 f h i i The constnt 4 is the best possible one Proof Similr to the proof of Corollry 7 The cse of equidistnt prtitioning is importnt in prctice Corollry 3 Let I k : x i +i b k i 0,, k) be n equidistnt prtitioning of, b If f is s bove, then we hve the inequlity b k f ) + f b) b ) k i) + ib 54) f x) dx b ) + f k k k 4k b ) f The constnt 4 is the best possible one Remrk 4 If k, then we hve the inequlity see for exmple 4) f ) + f b) 55) f x) dx b ) 4 b ) f Choose f :, b R, f x) x +b, which is L-Lipschitzin with L nd if t ), +b f x) if t +b, b Then f nd f x) dx f ) + f b) b ) i b ), 4 nd the equlity is obtined in 55), showing tht the constnt 4 is shrp
27 9 SS Drgomir nd TM Rssis 3 A Generl Qudrture Formul Let n : x n) 0 < x n) < < x n) n < xn) n b be sequence of divisions of, b nd consider the sequence of numericl integrtion formule where w n) j b I n f, n, w n ) : n j0 w n) j f x n) j j 0,, n) re the qudrture weights nd ssume tht n j0 wn) j The following theorem contins sufficient condition for the weights w n) j so tht I n f, n, w n ) pproximtes the integrl f x) dx with n error expressed in terms of f see lso 5) Theorem 4 Let f :, b R be n bsolutely continuous mpping on, b If the qudrture weights w n) j j 0,, n) stisfy the condition i 56) x n) i w n) j x n) i+ for ll i 0,, n ; j0 then we hve the estimtion I 57) n f, n, w n ) f x) dx n n i h n) i f n h n) i j0 j0 w n) j xn) i ), f b ) ν h n)), where ν h n)) : mx{h n) i : i 0,, n } nd h n) i In prticulr, if f <, then lim I n f, n, w n ) νh n) ) 0 uniformly by the influence of the weights w n + x n) i+ f : x n) i+ xn) i f x) dx Proof Similr to the proof of Theorem 9 nd we omit the detils The cse when the prtitioning is equidistnt is importnt in prctice Consider, then, the prtitioning E n : x n) i : + i b i 0,, n), n nd define the sequence of numericl qudrture formule n I n f, w n ) : w n) i f + i b ) n, w n) j b n j0
28 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 0 The following result holds: Corollry 5 Let f :, b R be bsolutely continuous on, b If the qudrture weights w n) i stisfy the condition: i n i w n) j i + i 0,, n ); b n j0 then we hve the estimte I n f, w n ) f x) dx f n 4n b i ) + n f b ) In prticulr, if f <, then uniformly by the influence of w n lim I n f, w n ) n j0 w n) j i + f x) dx b n 33 Prticulr Inequlities In this sub-section we point out some prticulr inequlities which generlize some clssicl results such s: rectngle inequlity, trpezoid inequlity, Ostrowski s inequlity, midpoint inequlity, Simpson s inequlity nd others in terms of the sup-norm of the derivtive 5 see lso 4) Proposition 6 Let f :, b R be bsolutely continuous on, b nd α, b Then we hve the inequlity: 58) f x) dx α )f ) + b α)f b) 4 b ) + α + b ) f b ) f The constnt 4 is the best possible one Proof Follows from Theorem by choosing x 0, x b, α 0, α α, b nd α b Remrk 5 ) If in 58) we put α b, then we get the left rectngle inequlity : 59) f x) dx b ) f ) b ) f
29 SS Drgomir nd TM Rssis b) If α, then by 58) we obtin the right rectngle inequlity 60) f x) dx b ) f ) b ) f 6) c) It is cler tht the best estimtion we cn hve in 58) is for α +b getting the trpezoid inequlity see lso ) f x) dx f ) + f b) b ) 4 b ) f Another prticulr integrl inequlity with mny pplictions is the following one 5: Proposition 7 Let f :, b R be n rbitrry bsolutely continuous mpping on, b nd α x α b Then we hve the inequlity: 6) f x) dx α )f ) + α α )fx ) + b α ) f b) 8 b ) + x + b ) + α + x ) + α x ) + b f Proof Follows by Theorem nd we omit the detils Corollry 8 Let f be s bove nd x, b Then we hve Ostrowski s inequlity: 63) f x) dx b ) fx ) 4 b ) + x + b ) f Remrk 6 If we choose x +b in 63), we obtin the midpoint inequlity 64) ) + b f x) dx b ) f 4 b ) f The following corollry generlizing Simpson s inequlity holds: Corollry 9 Let f be s bove nd x 5+b 6, +5b 6 Then we hve the inequlity f x) dx b f ) + f b) + fx ) 65) b ) + x + b ) f Proof Follows by Proposition 7 using simple computtion nd we omit the detils
30 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS Remrk 7 Let us observe tht the best estimtion we cn obtin from 65) is tht one for which x +b, obtining the Simpson s inequlity 0 66) f x) dx b 3 f ) + f b) + b + f ) 5 36 b ) f The following corollry lso holds Corollry 30 Let f be s bove nd α +b α b Then we hve the inequlity 67) 8 b ) + f x) dx α )f ) + α α )f α 3 + b ) + α + 3b b ) ) + b α )f b) f The proof is obvious by Proposition 7 by choosing x +b Remrk 8 The best estimtion we cn obtin from 67) is tht one for which α 3+b 4 nd α +3b 4, obtining the inequlity 3 68) f x) dx b f ) + f b) + b + f ) 8 b ) f The following proposition generlizes the three-eights rule of Newton-Cotes: Proposition 3 Let f be s bove nd x x b nd α, x, α x, x, α 3 x, b Then we hve the inequlity 69) f x) dx α )f ) + α α )fx ) +α 3 α )fx ) + b α 3 )f b) 4 x ) + x x ) + b x ) + α + x ) + α x ) + x + α 3 x ) + b f The proof is obvious by Theorem The next corollry contins generliztion of the three-eights rule of Newton- Cotes in the following wy:
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Advertisement [CLICK THE BUTTON ON THE RIGHT TO CONTINUE]
Author | Nitin Pandey | Friday, Aug 27,2010 8:23 AM
The Madhya Pradesh market has thrown up mixed results as per the IRS Q2 2010 numbers. The state has once again witnessed a huge difference in AIR (Average Issue Readership) between the top two players. The decline story of magazines has continued in Madhya Pradesh as well, with four of the top five magazines seeing negative growth in AIR. However, a very close fight is seen among the top three magazines.
Dainik Bhaskar leads in the state among dailies with an AIR of 3,201,000, witnessing a marginal dip of 0.77 per cent. NaiDunia, the No. 2 daily, lags far behind the leader with an AIR of 982,000 in Q2, as against 880,000 in Q1, a double digit growth of 11.59 per cent. Patrika, to, has seen double digit growth of 24.36 per cent, recording an AIR of 725,000 in Q2.
Dainik Agniban, Hari Bhoomi and Nav Dunia are the other dailies among the Top 10 to have seen positive growth in AIR. With the massive growth of 160 per cent, Nav Dunia has emerged as the fastest growing daily in Madhya Pradesh. Its AIR in Q2 stands at 143,000.
Among the top five magazines in Madhya Pradesh, Meri Saheli is the only one to have recorded a positive growth of 1.25 per cent in AIR. Pratiyogita Darpan leads the field with an AIR of 111,000, followed closely by India Today (Hindi) and Grehlakshmi with AIRs of 105,000 and 101,000, respectively.
Other Headlines on IRS
IRS research methodology still a concern for the industry
IRS Q2 2010: Growth story for newspaper and magazines in Uttar Pradesh; Dainik Jagran leads
IRS Q2 2010: Some cheer for Thanthi and Dinakaran in Chennai
IRS Q2 2010: Magazines perform better than dailies in Bangalore market
IRS Q2 2010: Dailies do exceptionally well in Kolkata; magazines fail to keep pace
IRS Q2 2010: Magazines display strong growth potential in Andhra Pradesh
IRS Q2 2010: Most dailies, magazines witness slump in Kerala
IRS Q2 2010: In a case of reversals, dailies lose out in Himachal Pradesh; magazines do well
IRS Q2 2010: Poor results continue to haunt dailies in Jammu & Kashmir
The great IRS debate: Industry divided on single currency, but upbeat about proposed new agency
Guest Article: IRS and the elusive credibility: Manajit Ghoshal, Mid-Day Infomedia
IRS Q2 2010: Publications in Maharashtra see some light
IRS Q2 2010: Positive fallout of Jharkhand print war – four out of top five dailies see growth
IRS Q2 2010: Hindi publications dominate Bihar market; magazines witness decline trend
IRS Q2 2010: Many gainers in Karnataka market; magazines reverse the decline trend
IRS Q2 2010: Publications in West Bengal do well with positive readership trends
IRS Q2 2010: Assam market looks grim for dailies and magazines; all see dip in AIR
IRS Q2 2010: TOI's might continues in Mumbai; positive growth boosts other players too
IRS Q2 2010: Dismal shows for English magazines; numbers head south
IRS Q2 2010: Four of the top five Hindi magazines witness decline; Saras Salil maintains lead
IRS Q2 2010: Mixed reactions from South publishers; call for different pattern of estimation for magazines
IRS Q2 2010: English dailies show positive growth; TOI remains undisputed leader
IRS Q2 2010: Positive growth for Top 3 Hindi dailies in Delhi NCR; NBT leads
IRS Q2 2010: Hindi dailies witness several ups and down in Mumbai
IRS Q2 2010: Pecking order of Tamil publications remains intact despite decline in AIR
IRS Q2 2010: Top order remains unchanged for Marathi publications; dailies soar, but magazines despair
IRS Q2 2010: Kannada publications chalk up a growth story; double digit growth for 3 Kannada dailies
IRS Q2 2010: Four of Top 10 dailies see decline in AIR; no change in order seen among top 5 dailies; Dainik Jagran, Dainik Bhaskar lead again
IRS Q2 2010: Decline trend continues to plague magazines; top order remains unchanged
IRS Q2 2010: HT continues lead in Delhi & Delhi NCR; negative growth for top 2 players
IRS Q2 2010: Business dailies put up a decent show; magazines still fighting it out
IRS Q2 2010: Top 10 Hindi dailies witness ups and downs; Dainik Jagran remains the leader
IRS Q2 2010: Daily Thanthi shows growth amid declining AIRs among Tamil publications


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