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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 567401, 15 pages
Local Fractional Fourier Series with Application to Wave Equation in Fractal Vibrating String
1Institute of Software Science, Zhengzhou Normal University, Zhengzhou 450044, China
2Department of Mathematics, Texas A and M University, Kingsville, TX 78363-8202, USA
3Department of Mathematics and Mechanics, China University of Mining and Technology, Jiangsu, Xuzhou 221008, China
Received 14 August 2012; Revised 25 October 2012; Accepted 8 November 2012
Academic Editor: Lan Xu
Copyright © 2012 Ming-Sheng Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag-Leffler function.
1. Introduction
Fractional calculus arises in many problems of physics, continuum mechanics, visco-elasticity, and quantum mechanics, and other branches of applied mathematics and nonlinear dynamics have been studied [1–7]. In general, the fractional analogues are obtained by changing the classical time derivative by a fractional one, which can be Riemann-Liouville, Caputo, or another one. Many classical partial differential equations possess a fractional analogue, like the fractional diffusion-wave equation [8–12], the fractional diffusion equation [13–16], the fractional wave equation [17, 18], the fractional Schrödinger equation [19, 20], the fractional heat equation [21], the fractional KdV equation [22], the fractional Fokker-Planck equations [23], the fractional Fick's law [24], the fractional evolution equation [25], the Fractional Heisenberg equation [26], the fractional Ginzburg-Landau equation [27], Fractional hydrodynamic equation [28], the fractional seepage flow equation [29], and the fractional KdV-Burgers equation [30].
There also are other methods for solving fractional differential equations, for example, the fractional variational iteration method [31, 32] and the fractional complex transform [33–37]. In all of the methods mentioned above, the solutions of the fractional differential equations should be analytical if the fractional derivative is in the Caputo or Riemann-Liouville sense. However, some solutions to ordinary and partial differential equations are fractal curves. As a result, we cannot employ the classical Fourier series, which requires that the defined functions should be differentiable, to describe some solutions to ordinary and partial differential equations in fractal space. However, based on the modified Riemann-Liouville derivative, Jumarie structured a Jumurie's calculus of fractional order [38] (which is one of useful tools to deal with everywhere continuous but nowhere differentiable functions) and its applications were taken into account in Probability calculus of fractional order [39], Laplace transform of fractional order via the Mittag-Leffler function (in convenient Hilbert space) [40], and adomian decomposition method for nonsmooth initial value problems [41]. Local fractional calculus is revealed as one of useful tools to deal with everywhere continuous but nowhere differentiable functions in areas ranging from fundamental science to engineering [42–57]. For these merits, local fractional calculus was successfully applied in the local fractional Laplace problems [53, 54], local fractional Fourier analysis [53, 54], local fractional short time transform [53, 54], local fractional wavelet transform [53–55], fractal signal [55, 56], and local fractional variational calculus [57].
In this paper we introduce a local fractional wave equation in fractal vibrating string which is described as with fractal boundary conditions where ,,, and are local fractional partial differential operator, and where is local fractional continuous (for more details, see [53, 54]). We study the technique of the local fractional Fourier series for treating the local fractional wave equation in fractal vibrating string. This paper is organized as follows. In Section 2, we specify and investigate the concepts of local fractional calculus and local fractional Fourier series. In Section 3, we present the solving process for local fractional wave equation with local fractional derivative. In Section 4, we study the expression solution with Mittag-Leffler functions in fractal space. Finally, Section 5 is conclusions.
2. Preliminaries
In this section we start with local fractional continuity of functions, and we introduce the notions of local fractional calculus and local fractional Fourier series.
2.1. Local Fractional Continuity of Functions
In order to discuss the local fractional continuity of nondifferential functions on fractal sets, we first consider the following results.
Lemma 2.1 (see [57]). Let be a subset of the real line and be a fractal. If is a bi-Lipschitz mapping, then there are for constants , and , such that for all ,
As a direct result of Lemma 2.1, we have, [57], such that where is fractal dimension of . The result that is directly deduced from fractal geometry is related to fractal coarse-grained mass function , which reads, [57], with where is dimensional Hausdorff measure.
Notice that we consider the dimensions of any fractal spaces (e.g., Cantor spaces or like-Cantor spaces) as a positive number. It looks like Euclidean space because its dimension is also a positive number. The detailed results had been considered in [53, 54, 57].
Definition 2.2. If there exists, [53, 57], with , for and , then is called local fractional continuous at , denoted by . is called local fractional continuous on the interval , denoted by if (2.7) is valid for .
Definition 2.3. If a function is called a nondifferentiable function of exponent , , which satisfies Hölder function of exponent , then for such that, [54, 57],
Definition 2.4. A function is called to be continuous of order , , or shortly continuous, when we have that, [54, 57],
Remark 2.5. Compared with (2.10), (2.7) is standard definition of local fractional continuity. Here (2.9) is unified local fractional continuity [57].
2.2. Local Fractional Derivatives and Integrals
Definition 2.6 (let ). Local fractional derivative of of order at is given, [53–57], where .
For any , there exists, [53–57], denoted by Local fractional derivative of high order is derived as, [57], and local fractional partial derivative of high order, [57],
Definition 2.7 (let ). Local fractional integral of of order in the interval is given by, [53–57], where ,, and , , , , is a partition of the interval .
For convenience, we assume that For any , we get, [53, 54, 57], denoted by
Remark 2.8. If , or , we have that, [46, 47, 50],
2.3. Special Functions in Fractal Space
Definition 2.9. The Mittag-Leffler function in fractal space is defined by, [53, 57],
Definition 2.10. The sine function in fractal space is given by the expression, [54, 57],
Definition 2.11. The cosine function in fractal space is given, [54, 57], The following rules hold [54, 57]:
Remark 2.12. is fractal imaginary unit, for more details, see [53–57].
2.4. Local Fractional Fourier Series
Definition 2.13. Suppose that and be -periodic. For , local fractional Fourier series of is defined as, [53–55], where are the local fractional Fourier coefficients.
For local fractional Fourier series (2.25), the weights of the fractional trigonometric functions are calculated as
Definition 2.14. Suppose that and be -periodic. For , complex generalized Mittag-Leffler form of local fractional Fourier series of is defined as, [53, 54], where the local fractional Fourier coefficients is The above generalized forms of local fractional series are valid and are also derived from the generalized Hilbert space [53, 54].
For local fractional Fourier series (2.28), the weights of the Mittag-Leffler functions are written in the form Above is generalized to calculate local fractional Fourier series.
3. Solutions to Wave Equation with Fractal Vibrating String
Now we look for particular solutions of the form and arrive at the equations with the boundary conditions
Equation has the solution According to (3.4), for and we derive as
Assuming that , since otherwise is identically zero, we find that where is an integer; we write For equation (3.3) leads to and therefore
To solve our problem, we form the local fractional Fourier series and require that A calculation of local fractional Fourier coefficients of and with respect to the system is given by But and therefore
Thus, the solution of our problem is given by formula (3.11), where local fractional Fourier coefficients are determined. From (3.14) and (3.16), we get the harmonic vibrations where
4. Expression Solutions with Mittag-Leffler Functions in Fractal Space
Taking into account the relations, [57], we obtain the harmonic vibration with the Mittag-Leffler functions in fractal space where its coefficients are
Hereby, we always find that with , for and .
Hence the boundary conditions are fractal and solution with Mittag-Leffler functions in fractal space is given by where its coefficients are derived as
5. Conclusions
We applied the technique of the local fractional Fourier series to treat with the local fractional wave equation in fractal vibrating string. When contrasted with other analytical methods, such as the heat-balance integral method, the homotopy perturbation method [11], the variational iteration method [29], the exp-function method [58], the fractional variational iteration method [31, 32], the fractional complex method [33–37], and others [59–61], the present method combines the following two advantages. The boundary conditions to the governing equations are local fractional continuous (the functions are nondifferential functions in fractal space) because we employ the local fractional Fourier series, derived from local fractional calculus, to deal with them. The governing equations with fractal behaviors in media are structured based on the local fractional calculus. The way plays a crucial role in local fractional calculus. This technique is efficient for the applied scientists to process these differential equations with the local fractional differential operators in fractal space. This paper that is an outstanding example of application of local fractional Fourier series to the local fractional differential operators is given to elucidate the solution processes and reliable results.
This paper is sponsored by the National Natural Science Foundation of China (NSFC, Grant U1204703), the Key Scientific and Technological Project of Henan Province (122102310004), the Fundamental Research Funds for the Central Universities (HUST: 2012QN087, 2012QN088), and the Innovation Scientists and Technicians Troop Construction Projects of Zhengzhou City (10LJRC190, 121PRKXF658-4).
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The Infrastructure Approach
Millions of LifeStraws or ceramic filters, however, will not necessarily solve the drinking water crisis. "I don't see that there's going to be one silver bullet solution for the world's water," Murcott says. And ultimately, "the solution to unsafe drinking water is bigger than a technology solution," she asserts.
The aspiration is to establish water-treatment plants and utilities like we have in the developed world throughout remote regions of Africa, Latin America, Southeast Asia, and elsewhere, maintains John Briscoe, of the Harvard School of Public Health, who has been working in the field of environmental health and engineering for four decades. But "that's incredibly unsexy," he confesses. It requires economic growth and relative stability—tall orders for many regions, but ones that will bring with them myriad other improvements, such as jobs, transportation, and medical care, in addition to clean water. Piecemeal approaches such as point-of-use filters, he notes, are addressing the symptoms rather than the causes of water insecurity. And that, he argues, can even distract from and slow down development of crucial larger water-infrastructure projects. It is a hard position for many to swallow, he says, but many government or aid- organization attempts to improve water access—with filters or one-off cleaning powders—fail to result in more lasting changes in the affected areas of the globe.
Saltwater and Wastewater Solutions
Where climate, geography, and other factors make fresh water scarce, many places are already running off the freshwater grid. For example, Saudi Arabia and Singapore are among the more than 120 countries using desalination to create some of their drinking water. The most basic method of desalination has been in practice for centuries and requires simply boiling salt water and gathering the purified water vapors to condense back down into potable water. A less energy-intensive process that is becoming more popular simply filters the salt out of the water using a super-fine mesh membrane, which has the added benefit of removing other contaminants at the same time. Utilities are already using this technique in Israel, Hong Kong, and El Paso, Texas. Research groups and forward-looking companies are now also enlisting artificially made proteins to extract the salt from water. These proteins work like those in our own bodies that move water out of cells; they have a positive charge on one end that repels salt molecules and prevents them from passing through the membrane. Earlier this year, a Danish company named Aquaporin acquired a patent in the United States for a version of this protein-based desalination technology.
For the landlocked, successful water reuse has finally arrived, which means that—if the public can stomach it—engineers can turn wastewater back into drinking water (by filtering out solids and then pushing the water through super-fine filters to remove bacteria and other contaminants before it is disinfected further via UV light, hydrogen peroxide, and other treatments). Some localities, such as Orange County, California, and El Paso, are pumping treated wastewater underground to replenish freshwater aquifers. Others, including cities such as San Diego and Windhoek, Namibia, are pumping this recycled water directly back into the main public supply.
Of course, climate change, with its contaminating floods and devastating droughts, will further stress the planet's resources. And the proliferation of new contaminants, such as pharmaceutical compounds from consumer and agricultural wastewater that have been found in the United States, Europe, and Southeast Asia, will continue to push us to develop new and smarter filtering approaches.
Here in the U.S., for example, agricultural runoff, industrial pollutants, and other manmade hazards—such as risks to well water near hydraulic fracturing sites—also play a part. Concerns over these threats and others have pushed at least 4 out of 10 American households to use some sort of home filtering, according to a 2005 study by the EPA.
Yet, says Patricia Toccalino, a water-quality expert with the U.S. Geological Survey, "the U.S. has one of the safest drinking water supplies in the world." And for the rest of the world, Briscoe notes, more than a quarter million people each day are already gaining access to better drinking and cooking water. Something to raise a glass to, indeed.
Katherine Harmon is a writer living in Brooklyn, New York. She is currently an associate editor at Scientific American. Her first book, about octopuses, will be published in 2013 by Current, a division of Penguin.


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September 7, 2009 9:12 PM
In Conversation With Steve Wozniak tagged "Video Game"
Episode #170 of Six Pixels of Separation - The Twist Image Podcast is now live and ready for you to listen to. First off, let me say that the audio quality of the phone call is pitiful in this Podcast.... Read more
July 12, 2009 9:22 AM
Augmented Reality - A Marketer's Primer tagged "Video Game"
If you're looking to stay slightly ahead of the Digital Marketing curve and wondering if virtual worlds like Second Life are going to make a comeback, Augmented Reality could well be "what's next" and a huge skip past virtual worlds.... Read more
November 15, 2008 6:50 PM
Will All Media Go Digital By 2014? tagged "Video Game"
"I want to make a bet with you today. By January 2014 I will wager that in the US almost all forms of tangible media will either be in sharp decline or completely extinct. I am not just talking about... Read more
October 28, 2008 6:57 PM
Blogs Have Influence Beyond Search Engine Optimization tagged "Video Game"
The default reason to listen and follow Blogs when all other convincing fails is that Blogs rank high in natural search engine results. If someone says something bad, it could well be that this highly ranked search engine result will... Read more


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- 268Promote
Art Print
Collect your choice of gallery quality Giclée, or fine art prints custom trimmed by hand in a variety of sizes with a white border for framing.
Letting go doesn't mean giving up... it means moving on. - Anonymous
Excellent! The textures and markings that you use fit together so well and makes your work unique every time. Keep it going!
wow Thank you so much guys!!! :) I drew it for my old characters! I am moving on to next step I hope!! Thank you so much for your supports!! I really appreciate it!:D you guys are the best!!!!!
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Dysfunction of basal forebrain cholinergic neurons (BFCNs) is an early pathological hallmark of Alzheimer's disease (AD). Numerous studies have indicated that nerve growth factor (NGF) supports survival and phenotypic differentiation of BFCNs. Consistent with a potential link to AD pathogenesis, TrkA, a NGF receptor, is expressed in cholinergic forebrain neuronal populations including those in BF and striatum, and is markedly reduced in individuals with mild cognitive impairment (MCI) without dementia and early-stage AD. To investigate the role of TrkA in the development, connectivity, and function of the BF cholinergic system and its contribution to AD pathology, we have generated a forebrain-specific conditional TrkA knock-out mouse line. Our findings show a key role for TrkA signaling in establishing the BF cholinergic circuitry through the ERK pathway, and demonstrate that the normal developmental increase of choline acetyltransferase expression becomes critically dependent on TrkA signaling before neuronal connections are established. Moreover, the anatomical and physiological deficits caused by lack of TrkA signaling in BFCNs have selective impact on cognitive activity. These data demonstrate that TrkA loss results in cholinergic BF dysfunction and cognitive decline that is reminiscent of MCI and early AD.
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Orc, Durotar (4e Race)
From D&D Wiki
Durotar Orcs[edit]
A proud race that once was controlled by the Burning Legion, but now looks to the future and the past for hope...
Racial Traits |
Average Height: 6'6" to 8'6" |
Average Weight: 198 to 257 lbs. |
Ability Scores: +2 to Strength, +2 to Wisdom |
Size: Medium |
Speed: 6 squares |
Vision: Normal |
Languages: Giant |
Skill Bonuses: +2 to Endurance and Athletics |
Axe Specialization: You gain Proficiency with Axes and Throwing Axes |
Hardiness: You gain +5 to saving throws against being Dazed or Weakened |
Blood Fury: You gain use of the Blood Fury encounter power. |
The savage, green-skinned Orcs are one of the most prolific races of Azeroth. They are commonly believed to be brutal and mindless, possessing no humanity or empathy for other races. Born on the hellish world of Draenor, the Orcs were brought into the kingdom of Stormwind through the dimensional gateway known as the Dark Portal and forced to wage war on the Humans. Although few are aware of their history, the Orcs once cultivated a noble, Shamanistic society on the world of Draenor. Tragically, the proud Orc clans were corrupted by the Burning Legion and used as pawns in the Legion's invasion of Azeroth. The Orcs managed to rebel, however, and they were ultimately able to help turn the tide against their demon masters. Led by the young Warchief, Thrall, the Orcs have reclaimed their strength and honor. Now, the Orcs stand ready to fight not for the sake of conquest, but for their right to survive in their adopted world.
Play a Durotar Orc if you want...
- To be a race of proud warriors.
- To tap into a shamanistic heritage with a marred past.
- To defy the common stereotypes of most fantasy orcs.
- To be a member of a race that favors the Fighter, Warlord and Barbarian classes.
Physical Qualities[edit]
Orcs are a hardy and strong race. They have a skin range from olive to emerald green, and on rare occasions red. Their hair color ranges from black to red, with older Orcs having white hair. They grow rapidly from birth to teenage years, maturing at around 14. They can live up to the late seventies, with some old orcs being in their late eighties. Orcs are active into their senior years, many times right until death.
Playing a Durotar Orc[edit]
Orcs are always ready to fight, as it is literally in thier blood. They have no trouble getting into melee and dishing up pain for their foes. Orcs are dependable and value honor and family above all. Orcs are self-reliant, but trully shine while in groups, where they are able to focus their rage onto directed enemies. Orcs in adventuring parties see their allies as brothers and sisters, and will protect them at all costs.
Durotar Orc Characteristics: Honorable, Dependable, Ferocious, Brutal, Feral, Willful, Austere.
Male Names: Rekgar, Thurgrim, Krag, Durgar, Loke'thegar, Brokk, Balnakk
Female Names: Thirga, Lira, Nalaa, Rega, Nalu, Giro, Thila
Durotar Orc Adventurers[edit]
Three sample Durotar Orc adventurers are described below.
Rekgar is a Warlord who fought in the third war. He treats all around him as his brothers-in-arms, and will do anything for them. He is always in the middle of the frey, fighting enemies with his adventuring group. He hopes to help Thrall to carve a new orcish civilization out of the harsh wilderness of Kalimdor.
Krag is a Fighter who has just recently joined the ranks of the Horde. He is agressive and quick to strike, and will do whatever it takes to win. He sees the Horde's new homeland of Durotar as a stepping stone to greater glory, and do anything to achieve that...
Thirga is a Warlock who wishes to use the powers of the Burning Legion against itself. Although she knows that being a Warlock is now frowned upon by her brothers and sisters in the Horde, she also knows that to wield such powers will giver her benefits that other professions wouldn't. She wishes to repay the Burning Legion for what it has done to her, and the Draenei even moreso, for it is they that drew the Legion to her ancestor's homeland of Draenor...
Back to Main Page → 4e Homebrew → Races


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We're getting there. The ABV is coming down while the flavour is getting ramped up. More people are getting turned on to sub-5% beers and, ahead of CAMRA Vancouver's Spring Sessional cask fest next month, here is maybe the finest … Continue reading
Stout reviews RSS
Desolation Imperial Oyster Stout (Lighthouse Brewing Company, 9.3%)
Here's my tenuous Valentine's Day tie-in.
Oysters have long known to be the food of love and booze the social lubricant par excellence, and long have the twain paired well. But yea, verily, here the twain shall also meet in … Continue reading
Ugly Sweater Milk Stout (Parallel 49 Brewing Company, 5%)
Personally speaking, I'm still unsure what all the fuss is about when it comes to St. Patrick's Day.
Considering 99% of the people who "celebrate" March 17 by swilling too much beer and recycling it down the gutter aren't Irish, … Continue reading
Black Mammoth Winter Ale (Fernie Brewing Co., 8%)
Fernie has been releasing solid brews for a decade now, and to celebrate it developed this pitch-black winter warmer. (Yes, it's still winter. To everyone out there wearing shorts already: Stop it.)
It's been called a winter ale, but to … Continue reading
Oatmeal Stout (Steamworks, 5%)
Say what you like about Steamworks (and a lot has been said recently), but the fact is they make consistently good beer.
You could call it the legacy of Conrad Gmoser, the long-time head brewer on Water Street who … Continue reading
French Oaked Stiff Stout (Dead Frog Brewery, 6.5%)
The next release in Dead Frog's Beermaster Series must have been cooked up around the time CBC's The Big Decision was filming at the Aldergrove brewery, but owner Derrick Smith says the two productions are unrelated, and that this … Continue reading
Pothole Filler Imperial Stout (Howe Sound Brewing, 9%)
Despite being one of those select beers that somehow seems to define B.C.'s craft brew scene, Howe Sound's imperial stout is pretty hard to find in my experience.
Certainly, stock of this limited-release beer is dwindling this year, so … Continue reading
Keepers Stout (Lighthouse Brewing Company, 5.1%)
I had planned a whole week of stout reviews leading up to St. Patrick's Day, but life got in the way so you'll have to make do with just the one.
I had planned on railing against the hegemony of … Continue reading
Natureland Organic Black Stout (Pacific Western Brewing, 6.2%)
The fifth organic beer by Pacific Western moves away from the Natureland series' German trend (hefeweizen, festbier) with a stout that's (shocker!) black.
Actually, that's not the whole truth because this brew, when poured into a glass, … Continue reading


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Davidoff Exquisitos (10) Reviews [view details]
Give us YOUR review!Showing 1 through 3 of 3 reviews
Really Smooth with just a little bite, they burn and smell great. Don't be fooled by the size!!
Grady in Tuscaloosa, AL June 30, 2013
"So cool! :)"
These are really good! There are like small cigars! Everything is really good and satisfied - the burning, the taste, like a big cigar! I really can see the deference between the Mini Silver cigarillos and a hand rolled ones! Thumbs up! ;)
Vasil in California June 19, 2013
"Not the best..."
OK, but certainly not to be mistaken for a small version of Davidoff's cigars. As one would expect from such a small cigar, they burn too fast and lack the complexity and flavor profile of their larger cousins. Definitely worth a shot for cigarillo fans, but as a cigar smoker who loves Davidoff, I just didn't find these very satisfying.
Mike in Seattle July 12, 2007


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Amon Carter Museum
- Souvenir of Fort Worth, Texas : photo-gravures
- A souvenir booklet with photos of Fort Worth. texashistory.unt.edu/ark:/67531/metapth41338/
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- A book promoting Fort Worth for business and development. texashistory.unt.edu/ark:/67531/metapth41337/
- Fort Worth, "The convention city" : and a city of beautiful homes
- A book promoting Fort Worth for business and conventions. texashistory.unt.edu/ark:/67531/metapth41336/
- Texans and their state; a newspaper reference work
- Biographies of prominent Texans. texashistory.unt.edu/ark:/67531/metapth41332/
- Makers of Fort Worth
- Each leaf contains a biographical note with photographic portrait and a caricature by Heppner Blackman, first cartoonist of the Fort Worth Star-Telegram, or W.K. Patrick, cartoonist of the Fort Worth Record. texashistory.unt.edu/ark:/67531/metapth41334/
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So my friend asked my what possible reason I could have for owning an AK-47. He says first of all, nothing is going to happen to me and Ill probably never have to defend myself and second of all, why would I need a "powerful assault rifle" for protecting myself, wouldn't a pistol be enough?
Haha humor me with some good responses


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Surely one of the most celebrated of all jazz pianists is 2010 NEA Jazz Master Kenny Barron. He has won every imaginable distinction among DownBeat and JazzTimes Readers and Critics polls. He is a six-time recipient of the Best Pianist award by the Jazz Journalists Association. And his recordings on the Verve label have earned a stunning nine Grammy nominations. Kenny has collaborated with many of the jazz giants from Stan Getz to Dizzy Gillespie, Freddie Hubbard, Charlie Haden, Stanley Turrentine and Eddie Harris, as well as a younger generation including Stefon Harris and Lionel Loueke. Veterans George Mraz and Lewis Nash round out the trio. Also featured in the group is phenomenal saxophonist Dayna Stephens, widely known as a mentor to young musicians at the Stanford Jazz Workshop over the past ten years. He has collaborated with jazz elite such as John Scofield, Roy Hargrove, Terrence Blanchard, Dr. Lonnie Smith and Eric Harland to become both a fiery bandleader and an in-demand sideman.
$15 / 40 / 60 / 90 / 120 premium
or call 650.725.2787
Kenny Barron, piano
Dayna Stephens, tenor saxophone
Lewis Nash, drums
George Mraz, bass
Note: this concert will feature acoustic instruments only.


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The question of whether cell phone use can lead to brain tumors (benign or malignant) has no definitive answer; studies so far have failed to find an association. Cell phones emit electromagnetic radiation close to the head that theoretically can damage DNA and lead to brain tumors. The leading researcher in the field, Henry Lai, Ph.D., of the University of Washington in Seattle, notes that 70 to 80 percent of the energy emitted by a cell phone antenna is absorbed by the head. However, because brain tumors can take 30 to 40 years to develop, we just don't know yet whether, over time, we'll see an increase in brain tumors attributable to cell phones. We do know that the number of cell phones in use in the United States grew from 55 million in 1997 to more than 180 million in 2004.
Dr. Lai has warned against giving cell phones to young children simply because the long-term health effects are unknown. He has voiced concern that the growing brain in children may be more susceptible to radiation than the brains of adults. For the same reasons, the chairman of Britain's National Radiological Protection Board has advised parents not to give cell phones to children under the age of eight. Most of the studies done so far have focused on adults, not children, and most have looked at short term use (cell phones haven't been around long enough or used long enough by a large number of people to assess long-range health effects).
Until we know for sure whether cell phones present any danger (to adults as well as children), it makes sense to take the following precautions:
Cell phones have become an indispensable convenience for people around the world, but until we know more about their risks, you may want to follow the advice to use them prudently and not give them to your kids.
Andrew Weil, M.D.
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Fabric and paper blooms are always in season. The best part? You can make these faux flowers, suitable for formal and casual weddings, in advance. They're easy; all you need are a few supplies and your own loving hands.
Think of the joy this scene would evoke: flower girls skipping down the aisle, bearing beribboned wands that mimic the fluttering and falling of cherry blossoms. We wrapped a dowel in ribbon, thumbtacked on ribbons of various lengths, and finished off by gluing on confetti flowers. (To make the flowers, gather a bunch of the confetti petals together and start gluing.) The "Blossom" flower girl dresses are from Ash & Robbins, matthewrobbinsdesign.com.
Browse Through More Floral Ideas -
These handmade silk flowers make great boutonnieres; just start with the smallest templates. Make them following the directions below, and pin in place with a corsage pin or a daisy-head quilting pin.
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Silk flowers add elegance to a ring pillow. A crisp fabric holds its shape well; we used silk dupioni, shantung, and taffeta.
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A natty nosegay of fabric blossoms that manages to look festive yet restrained can be pinned to his (or the groomsman's) lapel; make a larger, brighter clutch of these faux flowers for the bridesmaids' bouquets.
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Joyous bursts of color dance above a table, imparting a cheerful radiance to a rehearsal dinner or casual reception. The dahlialike pom-poms appear to float in the air; in reality they are hung from the ceiling with monofilament.
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Do guests a favor and present them with tokens of affection that you've planted inside these petite flower-topped boxes. Wrap each small box with a belt of layered ribbons or a strip of woven fabric, and cap it off with a single fabric blossom.
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Guests will cheer you on when you zip away after the ceremony in a car covered with pom-pom garlands. These decorations are easy to make with tissue paper and twine, and they stay on the car with suction cups, which won't damage the auto's exterior.
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A crepe-paper lily makes a beautiful boutonniere or could be worked into a larger lily bouquet.
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To decorate the tiers of your cake with the crepe-paper bands, cut the crepe paper into strips with the ribbing running top to bottom, and gently stretch the top edge to create a ruffle. Wrap the band snugly around the tier, and secure to itself in the back with double-sided tape.
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The luscious blooms in this bouquet are always in season. We chose shades of pink, but a bride could fashion them in any color she wishes. What's more, you can make these paper flowers, which are suitable for formal and casual weddings, in advance.
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Flowers crafted from colorful tissue or crepe paper are graceful, inexpensive to make, and always in season. Arrange them on tables, use as boutonnieres, or create a whole bouquet.
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Guests will be enchanted with a pale-blue wedding cake draped in garlands of tiny leaves and delicate paper bows.
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With their whispery petals and luminous silver centers, the tissue-paper flowers that bloom on this wintry wreath look as if they were plucked from a fairy tale. Perhaps more magical is the fact that this inexpensive pew decoration won't wilt or fade, so you can make as many as you need well ahead of time.
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Swizzle sticks topped with crepe-paper flowers -- a spider chrysanthemum and a sleek calla lily -- add zest to lime spritzers.
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Paper dahlia napkin rings in citrus shades adorn each place setting. Square glass vessels in various sizes line tables. Covered in sunset-hued tissue (cut to size and secured with double-sided tape), they cast a soft glow.
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These favors are made using just four different flower punches. Attach them to favor boxes, scatter them on a table, use them on the cover of your programs, or come up with your own creative way to use these pretty paper flowers.
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Wrap favors in brightly colored tissue paper and top them off with a paper flower -- a low-cost alternative to the real thing.
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In winter, when fresh blossoms may be harder to come by, these lush tissue-paper flowers are in full bloom. Their silver centers are actually Christmas balls, a nod to the season. Arranged at different heights in silver trumpet vases and mint julep cups, they bring whimsy to a formal reception table set in all white.
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Transform inexpensive silk flowers into delicate garlands by dismantling the blooms and separating the layers of petals, removing any greenery.
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To create blooms in a variety of textures, sizes, and shapes, use ribbons and rickrack in different widths and weaves. The folds of each flower are clustered around artificial stamens; stems are individually wrapped in floral tape before all are bundled together and bound in wide white grosgrain.
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Fabric flowers make a pretty, wilt-proof way for guests to find their seats -- and then live a double-life as a corsage or boutonniere. Simply use floral tape to attach a brooch pin to the stem of a fabric flower (these are from Dulken & Derrick). Have a calligrapher pen names onto strips of card stock, punch small holes on one end, and slip pins through.
Silk flowers spell out a couple's initials on a silk faille banner.
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Surprise your bridesmaids by making these pretty silk flower hair bands.
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A runner decorated with handmade ribbon flowers is an inviting decoration for a guest-book table. The blooms are easy to make using thick woven-cotton ribbon with a floral design and buttons. A mix of large and small flowers is affixed with safety pins to the raw-silk runner, which has a grosgrain-ribbon border. A piece of woven ribbon is the bookmark.
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Here is a playful way to distinguish the flower girl dresses from the bridesmaids' gowns. This adornment looks best on simple dresses with little other embellishment.
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Give your bridesmaids a fashionable floral accessory that will look fresh long after the wedding day: wrist corsages made with silk ribbon flowers. They can even be tied around a ponytail or worn as brooches. These corsages are easy to make: Tie elastic ribbon around your wrist in a big bow, then pin the bloom to the knot.
How to Make This Corsage advertisement
Who said you have to stay within the lines? If you're crazy for color, plant a dramatic dahlia in your hair -- it will guarantee a double take.
The Materials
Creating wearable art has never been simpler -- you might even have everything you need at home. Want proof? Here's your shopping list: fabric, needle, thread. All fabric from Mood Fabrics. "Mia" gown, The Steven Birnbaum Collection. Earrings, Mauboussin.Learn How to Make This -
A bodice bubbling over with blossoms is a colorful way to supplement a simple dress. The key: Use just one color in your bouquet. "Ashley" dress, Coren Moore. Studs, Mauboussin.
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Gift them pre-wedding so they'll adorn dresses, wrists, or hair. "Crystal Clear" Photo Boxes, ClearBags, clearbags.com.
Learn How to Make Shoe Clips, Boutonnieres, and Brooches
Looking for a fuss-free frill for your flower girl? Sew a row of fabric dahlias (in different sizes and colors) to a soft, stretchy headband. Dress (#105), Us Angels.
Learn How to Make This Headband advertisement
Make a deep impact by borrowing from the design of your dress to fashion your bloom. "Jillian" gown, Monique Lhuillier. "Poise" white gold and diamond earrings, Hearts on Fire.
Learn How to Make Fabric Flowers -
Your flower can be classic, like this one that makes an oversize statement, or modern -- try metallic fabric. "Shiloh" gown, Thread. Leverback earrings, Hearts on Fire.
Learn How to Make Fabric Dahlia advertisement
Shrinking violets they're not. If you want to make your grand entrance even grander, opt for this punch-drunk bouquet, made from bell-shaped paper ornaments that have been trimmed. Five- and 9-inch Westminster bells, partycheap.com.
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