Representer theorem - Revision history

84.227.252.224: /* Generalizations: Variations on a theme by Kimeldorf and Wahba */

2014-10-16T17:40:45Z

Generalizations: Variations on a theme by Kimeldorf and Wahba

← Older revision		Revision as of 19:40, 16 October 2014
Line 1:		Line 1:
	~~{{Orphan\|date=February 2013}}~~

	The solution to the [[Schrödinger equation]], the [[wavefunction]], describes the quantum mechanical properties of a particle on microscopic scales. Measurable quantities such as position, momentum and energy are all derived from the wavefunction.<ref name="Davies1">Davies, p. 1</ref>

	~~<math>\mathrm{i}\hbar\frac{\partial}{\partial t}\psi(x~~,~~t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(x~~,~~t) +V(x)\psi(x~~,~~t),<~~/~~math>~~		Teeth cleaning is one of the most common dental procedures and, if you've never been to your dentist from a while, its one of this first services you'll involve. Unlike other forms of healthcare, most dental cost are paid for of jean pocket. Even [https://www.youtube.com/watch?v=4To0cBm6IsM family dentistry] if you have dental insurance, you could finish up paying more up front than anticipate. So it pays to know up front what you should expect to cover the cost of this service.<br><br>Should you beloved this informative article in addition to you wish to obtain more info about [https://www.youtube.com/watch?v=4To0cBm6IsM family dentistry] i implore you to check out the web site. To avoid a tooth cavity, people must brush and floss day after day. Also visit a dentist regularly to clean plaque and acid layers on smiles. Avoid taking sweet foods and even the starchy foods, this may in reducing bacteria inside your mouth. Some chewing gum available is made to reduce acidity in teeth. It truly is going aid while we are avoiding tooth tooth cavity.Use fluoride toothpaste if necessary, and brush your teeth twice an event in a correct manner.<br><br>When problem metastasizes, talk to travels together with lymphatic systems. Infected cells that enter the lymphatic system are carried along by lymph, a clear, watery fluid.<br><br>Finally, patients will sometimes have temporary dental crowns placed within mouth in the dentist 's business office. These are made of stainless steel or acrylic and are widely-used to restore the tooth while anticipating the permanent crown to come from the laboratory. In case the deterioration or damage is severe or causing pain, your dentist will likely use a short lived crown.<br><br>One of this most memorable features of your face is the smile extensive you smile your teeth are on display for all to view. For this reason, several take superior care of the company's teeth to take care of a white smile. Shops have done great lengths to get whiter teeth, and a financially demanding visit towards the dentist is really a small price for some to look their most efficient.<br><br>If bleaching is not as effective anyone have minor chips within your teeth, you cosmetic dentist may suggest veneers. Veneers are made or porcelain or vinyl. Your dentist may use a veneer to cover over a rather difficult or stubborn stain or to cosmetically [http://www.adobe.com/cfusion/search/index.cfm?term=&boost+appearance&loc=en_us&siteSection=home boost appearance] of a broken for ladies chipped dentist. Crooked teeth or even gaps inside your teeth could be fixed with veneers.<br><br>Dentists advice that we should brush our teeth no less three times a afternoon. Apart from this, we may also do some flossing and gargling of mouth to. However, these are not enough as we often need specialist help in detaching the deep-seated dirt, bacteria and plaque in teeth. This is where the skills of dental hygienists and dentists will be of big benefit.<br><br>Halloween is actually an possible opportunity to create memories that lasts forever, whether you are the one handing the actual candy insect killer one collecting the snack. I encourage you to go the extra mile to make it memorable for one kid, in conjunction with return, they'll remember individuals.
	~~where <math>\hbar<~~/~~math> is the [[reduced Planck constant]~~], <~~math~~>m<~~/math~~> ~~is the~~ [~~[mass~~]~~] of~~ the ~~particle~~, ~~<math>\mathrm{i}</math> is the [[imaginary unit]]~~ and ~~<math>t</math> is time~~.

	~~One peculiar potential that can be solved exactly is when~~ the ~~electric quadrupole moment~~ is ~~the dominant term of an infinitely long cylinder of charge~~.

	It ~~can be shown that the [[Schrödinger equation]]~~ is ~~solvable for a cylindrically symmetric electric quadrupole~~, ~~thus indicating that the quadrupole term of~~ an ~~infinitely long cylinder can be quantized~~.
	~~In the [[physics]] of [[classical electrodynamics]]~~, ~~it can be easily shown~~ that the ~~scalar potential and associated mechanical potential energy of~~ a ~~cylindrically symmetric quadrupole is as follows:~~

	:<~~math~~>~~\mathbf{V}_\mathrm{quad} = \frac{\lambda d^2 Cos[2 \phi]}{4 \pi \epsilon_0 s^2}~~ <~~/math~~> ~~([[SI]] units)~~
	~~:<math>\mathbf{V}_\mathrm{quad} = \frac{Q \lambda d^2 Cos[2 \phi]}{4 \pi \epsilon_0~~ s~~^2} </math> ([[SI]] units)~~

	~~Cylindrical symmetry should be~~ used ~~when solving~~ the ~~equation~~. ~~The time independent [[Schrödinger equation]] becomes~~ the ~~following in cylindrical symmetry~~.

	<~~math~~>E \psi(x) = -\frac{\hbar^2}{2m s}\frac{\partial}{\partial s} (s \frac{\partial}{\partial s}) \psi(s,\phi)-\frac{\hbar^2}{2m s^2}\frac{\partial^2}{\partial \phi^2}\psi(s,\phi) +\frac{Q \lambda d^2 Cos[2 \phi]}{4 \pi \epsilon_0 s^2} \psi(s,\phi),<~~/math~~>

	~~Using~~ the ~~technique [[Separation~~ of ~~Variables]], the above equation can be written as two ordinary differential equations in both~~ the ~~radial and azimuthal directions. The radial equation is [[Bessel~~'s ~~equation]] as can be seen below~~. ~~If one changes variables~~ to ~~<math>x= k s</math>~~, ~~Bessel's equation~~ is ~~exactly obtained~~.

	<~~math~~>~~\frac{1}{x} \frac{\partial}{\partial x} (x \frac{\partial}{\partial x}) S(x)+(1-\frac{\nu^2}{x^2}) S(x)=0~~<~~/math~~>

	~~==Azimuthal equation==~~
	~~The azimuthal equation~~ is [~~[Mathieu equation]], is as follows~~:

	~~<math> \frac{\partial^2}{\partial \phi^2} \Phi(\phi)+(\nu^2-\frac{\lambda q m d^2}{2 \pi \epsilon_0 \hbar} Cos[2 \phi]) \Phi[\phi]=0<~~/~~math>~~

	~~Since the canonical form of Mathieu's equation can be written as follows, it can be shown that <math>\nu^2<~~/~~math> corresponds to a and <math>\frac{\lambda q m d^2}{2 \pi \epsilon_0 \hbar} <~~/~~math> corresponds to q~~.

	~~:<math> \frac{d^2y}{dx^2}~~+~~[a-2q\cos (2x) ]y~~=~~0. </math>~~

	~~The solution of the [[Mathieu equation]~~] ~~is expressed in terms~~ of ~~'''Mathieu cosine''' <math>C(~~a~~,q,x)~~<~~/math~~> ~~'''Mathieu sine'''~~ <~~math~~>S(a,q,~~x)</math> for a unique a~~ and q. This ~~indicates that~~ the ~~quadrupole moment can~~ be ~~quantized in order~~ of ~~the Mathieu characteristic values~~ <~~math~~>~~a_n~~<~~/math~~> ~~and <math>b_n</math>.~~

	~~In general~~, ~~Mathieu functions~~ are ~~not periodic~~. ~~The term q must be that of a characteristic value in order~~ for ~~Mathieu functions to be periodic. Immediately~~, ~~it can be shown that the solution of the radial equation highly depends on what characteristic values are seen~~ in ~~this case.~~

	~~== References ==~~
	~~{{reflist}}~~

	~~==External links==~~
	* [[Cylindrical multipole moments\|- Cylindrical Multipole Moments]]
	* [http://www.~~jpier.org/PIERB/pierb26/09.10063008.pdf MULTIPOLE EXPANSION]~~
	* [http://www.sciencedirect.com/science/article/pii/0022285268900027 The nonvanishing coefficients of the dipole moment expansion in axially symmetric molecules]

	~~[[Category:Quantum mechanics]]~~

24.6.2.83: /* Generalizations: Variations on a theme by Kimeldorf and Wahba */ Fixed a minor typo

2013-09-12T04:04:28Z

Generalizations: Variations on a theme by Kimeldorf and Wahba: Fixed a minor typo

← Older revision		Revision as of 06:04, 12 September 2013
Line 1:		Line 1:
	~~When thinking about flexibility~~, ~~a lot~~ of ~~people who speak to~~ a ~~personal trainer from Huntington Beach will talk about performing splits or remarkable activities~~ and ~~won't aware of the fact that there~~ are ~~many adaptability elements that have to be known. It's a well-known fact that versatility could be improved but this is not always~~ the ~~situation with many elements~~. ~~By looking at anatomical elasticity factors we could understand what can be worked [http://tinyurl~~.~~com~~/~~pch83be uggs on sale] and what cannot through constant exercises.~~ <br><br>~~Anatomical Elasticity Elements~~<br><br>Bones are normally stated when debating about flexibility because a lack of versatility will normally harm bones. They are enclosed by articular cartilage and synovial membranes. Those will nourish and bolster your bones. When you acquire muscular flexibility in the ~~movement length of a joint~~, ~~elasticity is enhanced.~~ <br><br>~~Ligaments~~<br><br>Ligaments are composed of two tissues: yellow and white. You need to comprehend that white tissues cannot extend however they are really effective. Even if you are faced with a bone break, tissue will remain unharmed. White tissue gives you subjective action liberty. Yellow flexible tissue is ~~stretchable and could stretch greatly while at~~ the ~~same time being able~~ [~~http://tinyurl.com/pch83be uggs on sale~~] ~~to go back to original size.~~ <br><br>~~Areolar Cells<br><br>The personal trainer from Huntington Beach will seldom be asked about this factor but it's one~~ that ~~could~~ be ~~[http://Www.Alexa.com/search?q=crucial&r=topsites_index&p=bigtop crucial]. Areolar cells~~ is ~~always permeable and significantly distributed in your system. That~~ is ~~mainly a cell that could function as other tissue binder. <br><br>Muscles<br><br>When pertaining to elasticity, muscles are normally brought into~~ the ~~conversation. Contrary to popular theory, they cannot expand because they are not elastic. We could categorize muscles as connective tissues~~. It is ~~responsible~~ for ~~supporting~~, ~~holding and surrounding muscle mass~~. ~~Normal ligament involves non flexible~~ and ~~flexible tissues but tendons do not. <br><br>Muscle Fibers<br>Your tendon is composed~~ of ~~muscle cell, which~~ is ~~flexible.~~ <br><br>~~Stretch Receptors~~<br><br>~~Generally only~~ the ~~personal trainer from Huntington Beach~~ [~~http://tinyurl~~.~~com~~/~~pch83be ugg boots usa~~] ~~will describe stretch receptors simply because people today don't even know that they exist. A stretch receptor is made out of~~ two ~~elements: Golgi muscles and Spindle tissues. The Spindle cells are situated~~ in the ~~middle of the tendon~~ and ~~send messages so that your muscle could contract~~. ~~Golgi muscles are receptors are situated at the end part of the muscle tissue~~. ~~It will send messages~~ to ~~ensure the muscles could relax. Receptors would be practiced by using them consistently so that stretching will become less difficult.~~ <br><br>It is usually essential to take all those elements into consideration. Our recommendation is to talk with a really good personal trainer in Huntington Beach in order that you will properly understand how you should improve flexibility. <br>Simply by studying the phrases above you will notice that there are elements that you probably didn'~~t know about till today. Consider the fact that the details you get from the personal trainer~~ is ~~a lot vaster and there are so many other mistakes that you could find yourself doing without even realizing that they are errors~~. <br><br>~~Personal Trainer Huntington Beach Tips~~: ~~Comprehending The Benefits Of Strength Training~~<br><br>~~The personal trainer in Huntington Beach normally has to talk about~~ the ~~many advantages~~ of ~~resistance training because most people don't really understand what they get into. There~~'s ~~this popular opinion~~ that this exercise plan will make strong muscles but this is actually not correct. Resistance training is actually customized towards increasing force and also several other things are rarely taken into account. Then again, there are numerous advantages included and ~~you must know all of them prior to starting your exercise routine.~~ <br><br>~~Improving Physical Attractiveness~~<br><br>The ~~most consistently discussed benefits~~ of ~~using resistance training are~~ the ~~growth~~ in ~~physical force~~, ~~appearance~~, stamina and muscle mass. Addititionally there is the advantage of an enhanced bone density. Normally the personal trainer from Huntington Beach deals with individuals who start exercising to be able to enhance personal appeal. <br><br>~~The men will improve big muscles but the females cannot because they lack of testo-sterone. However~~, a ~~lady can quickly build-up~~ a ~~toned, strong physique~~ and ~~maximize strength~~ in ~~a similar part with what men develop. Individual genetics will dictate strength training stimuli reaction to some degree.~~ <br><br>~~Basal Metabolism~~<br><br>~~You~~ are ~~going~~ to ~~notice that your basal metabolism will maximize as muscle mass raises~~. ~~A personal trainer from Huntington Beach will highlight this factor as~~ it ~~immediately produces~~ the ~~promotion~~ of ~~fat burning~~ on ~~a long term basis~~. ~~You may actually stay away from yo~~-~~yo dieting due to this. The severe exercise will certainly improve~~ [http://~~tinyurl~~.~~com~~/~~pch83be uggs on sale~~] ~~your metabolic rate for many hours following the workout~~ [http://~~tinyurl~~.com/pch83be uggs on sale] is complete. This will increase weight loss potential. <br><br>Useful Advantages<br><br>Weight training allows you to get much stronger body and that will develop your attitude, offer better joint support as well as minimize [http://~~tinyurl.com~~/pch83be cheap ugg boots] accident risk while performing regular, daily tasks. Much older individuals are suggested to speak to a personal trainer in Huntington Beach so as to hinder muscle tissue loss that's usual during aging. Even functional intensity could be renewed. You could actually prevent several physical disabilities which sometimes show up while being able to vigorously prevent osteoporosis. <br><br>Fast Recovery<br><br>When you're in rehabilitation or you have to deal with a disability including orthopedic operation and stroke, weight training is definitely an important aspect in optimizing the recovery of weaker muscles. If you are afflicted by this sort of illness, a proper, certified personal trainer from Huntington Beach that is specialized in recuperation has to be consulted. Physiotherapists will be preferred. <br><br>Improved Sports Effectiveness<br><br>The ~~majority~~ of ~~competitors utilizes some sort of strength training plan that's specific to~~ the sport that is done. Stronger muscle tissues will certainly help you to improve your functionality. The only issue is that you need to make sure that your training has muscle contraction that occurs with the precise velocity that is utilized in that particular sport. <br><br>As a result, when you begin resistance training under the help of a qualified personal trainer from Huntington Beach, you could expect all the benefits mentioned previously. You will surely value the fact that you will always feel great following the workout and you will feel vitalized after it. In general, people that exercise in the fitness center will live a much longer and happier life.		{{Orphan\|date=February 2013}}

			The solution to the [[Schrödinger equation]], the [[wavefunction]], describes the quantum mechanical properties of a particle on microscopic scales. Measurable quantities such as position, momentum and energy are all derived from the wavefunction.<ref name="Davies1">Davies, p. 1</ref>

			<math>\mathrm{i}\hbar\frac{\partial}{\partial t}\psi(x,t) = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi(x,t) +V(x)\psi(x,t),</math>
			where <math>\hbar</math> is the [[reduced Planck constant]], <math>m</math> is the [[mass]] of the particle, <math>\mathrm{i}</math> is the [[imaginary unit]] and <math>t</math> is time.

			One peculiar potential that can be solved exactly is when the electric quadrupole moment is the dominant term of an infinitely long cylinder of charge.

			It can be shown that the [[Schrödinger equation]] is solvable for a cylindrically symmetric electric quadrupole, thus indicating that the quadrupole term of an infinitely long cylinder can be quantized.
			In the [[physics]] of [[classical electrodynamics]], it can be easily shown that the scalar potential and associated mechanical potential energy of a cylindrically symmetric quadrupole is as follows:

			:<math>\mathbf{V}_\mathrm{quad} = \frac{\lambda d^2 Cos[2 \phi]}{4 \pi \epsilon_0 s^2} </math> ([[SI]] units)
			:<math>\mathbf{V}_\mathrm{quad} = \frac{Q \lambda d^2 Cos[2 \phi]}{4 \pi \epsilon_0 s^2} </math> ([[SI]] units)

			Cylindrical symmetry should be used when solving the equation. The time independent [[Schrödinger equation]] becomes the following in cylindrical symmetry.

			<math>E \psi(x) = -\frac{\hbar^2}{2m s}\frac{\partial}{\partial s} (s \frac{\partial}{\partial s}) \psi(s,\phi)-\frac{\hbar^2}{2m s^2}\frac{\partial^2}{\partial \phi^2}\psi(s,\phi) +\frac{Q \lambda d^2 Cos[2 \phi]}{4 \pi \epsilon_0 s^2} \psi(s,\phi),</math>

			Using the technique [[Separation of Variables]], the above equation can be written as two ordinary differential equations in both the radial and azimuthal directions. The radial equation is [[Bessel's equation]] as can be seen below. If one changes variables to <math>x= k s</math>, Bessel's equation is exactly obtained.

			<math>\frac{1}{x} \frac{\partial}{\partial x} (x \frac{\partial}{\partial x}) S(x)+(1-\frac{\nu^2}{x^2}) S(x)=0</math>

			==Azimuthal equation==
			The azimuthal equation is [[Mathieu equation]], is as follows:

			<math> \frac{\partial^2}{\partial \phi^2} \Phi(\phi)+(\nu^2-\frac{\lambda q m d^2}{2 \pi \epsilon_0 \hbar} Cos[2 \phi]) \Phi[\phi]=0</math>

			Since the canonical form of Mathieu's equation can be written as follows, it can be shown that <math>\nu^2</math> corresponds to a and <math>\frac{\lambda q m d^2}{2 \pi \epsilon_0 \hbar} </math> corresponds to q.

			:<math> \frac{d^2y}{dx^2}+[a-2q\cos (2x) ]y=0. </math>

			The solution of the [[Mathieu equation]] is expressed in terms of '''Mathieu cosine''' <math>C(a,q,x)</math> '''Mathieu sine''' <math>S(a,q,x)</math> for a unique a and q. This indicates that the quadrupole moment can be quantized in order of the Mathieu characteristic values <math>a_n</math> and <math>b_n</math>.

			In general, Mathieu functions are not periodic. The term q must be that of a characteristic value in order for Mathieu functions to be periodic. Immediately, it can be shown that the solution of the radial equation highly depends on what characteristic values are seen in this case.

			== References ==
			{{reflist}}

			==External links==
			* [[Cylindrical multipole moments\|- Cylindrical Multipole Moments]]
			* [http://www.jpier.org/PIERB/pierb26/09.10063008.pdf MULTIPOLE EXPANSION]
			* [http://www.sciencedirect.com/science/article/pii/0022285268900027 The nonvanishing coefficients of the dipole moment expansion in axially symmetric molecules]

			[[Category:Quantum mechanics]]

en>Magioladitis: Fix category spacing + general fixes using AWB (8232)

2012-08-07T16:55:22Z

Fix category spacing + general fixes using AWB (8232)

New page

When thinking about flexibility, a lot of people who speak to a personal trainer from Huntington Beach will talk about performing splits or remarkable activities and won't aware of the fact that there are many adaptability elements that have to be known. It's a well-known fact that versatility could be improved but this is not always the situation with many elements. By looking at anatomical elasticity factors we could understand what can be worked [http://tinyurl.com/pch83be uggs on sale] and what cannot through constant exercises. Anatomical Elasticity Elements Bones are normally stated when debating about flexibility because a lack of versatility will normally harm bones. They are enclosed by articular cartilage and synovial membranes. Those will nourish and bolster your bones. When you acquire muscular flexibility in the movement length of a joint, elasticity is enhanced. Ligaments Ligaments are composed of two tissues: yellow and white. You need to comprehend that white tissues cannot extend however they are really effective. Even if you are faced with a bone break, tissue will remain unharmed. White tissue gives you subjective action liberty. Yellow flexible tissue is stretchable and could stretch greatly while at the same time being able [http://tinyurl.com/pch83be uggs on sale] to go back to original size. Areolar Cells The personal trainer from Huntington Beach will seldom be asked about this factor but it's one that could be [http://Www.Alexa.com/search?q=crucial&r=topsites_index&p=bigtop crucial]. Areolar cells is always permeable and significantly distributed in your system. That is mainly a cell that could function as other tissue binder. Muscles When pertaining to elasticity, muscles are normally brought into the conversation. Contrary to popular theory, they cannot expand because they are not elastic. We could categorize muscles as connective tissues. It is responsible for supporting, holding and surrounding muscle mass. Normal ligament involves non flexible and flexible tissues but tendons do not. Muscle Fibers Your tendon is composed of muscle cell, which is flexible. Stretch Receptors Generally only the personal trainer from Huntington Beach [http://tinyurl.com/pch83be ugg boots usa] will describe stretch receptors simply because people today don't even know that they exist. A stretch receptor is made out of two elements: Golgi muscles and Spindle tissues. The Spindle cells are situated in the middle of the tendon and send messages so that your muscle could contract. Golgi muscles are receptors are situated at the end part of the muscle tissue. It will send messages to ensure the muscles could relax. Receptors would be practiced by using them consistently so that stretching will become less difficult. It is usually essential to take all those elements into consideration. Our recommendation is to talk with a really good personal trainer in Huntington Beach in order that you will properly understand how you should improve flexibility. Simply by studying the phrases above you will notice that there are elements that you probably didn't know about till today. Consider the fact that the details you get from the personal trainer is a lot vaster and there are so many other mistakes that you could find yourself doing without even realizing that they are errors. Personal Trainer Huntington Beach Tips: Comprehending The Benefits Of Strength Training The personal trainer in Huntington Beach normally has to talk about the many advantages of resistance training because most people don't really understand what they get into. There's this popular opinion that this exercise plan will make strong muscles but this is actually not correct. Resistance training is actually customized towards increasing force and also several other things are rarely taken into account. Then again, there are numerous advantages included and you must know all of them prior to starting your exercise routine. Improving Physical Attractiveness The most consistently discussed benefits of using resistance training are the growth in physical force, appearance, stamina and muscle mass. Addititionally there is the advantage of an enhanced bone density. Normally the personal trainer from Huntington Beach deals with individuals who start exercising to be able to enhance personal appeal. The men will improve big muscles but the females cannot because they lack of testo-sterone. However, a lady can quickly build-up a toned, strong physique and maximize strength in a similar part with what men develop. Individual genetics will dictate strength training stimuli reaction to some degree. Basal Metabolism You are going to notice that your basal metabolism will maximize as muscle mass raises. A personal trainer from Huntington Beach will highlight this factor as it immediately produces the promotion of fat burning on a long term basis. You may actually stay away from yo-yo dieting due to this. The severe exercise will certainly improve [http://tinyurl.com/pch83be uggs on sale] your metabolic rate for many hours following the workout [http://tinyurl.com/pch83be uggs on sale] is complete. This will increase weight loss potential. Useful Advantages Weight training allows you to get much stronger body and that will develop your attitude, offer better joint support as well as minimize [http://tinyurl.com/pch83be cheap ugg boots] accident risk while performing regular, daily tasks. Much older individuals are suggested to speak to a personal trainer in Huntington Beach so as to hinder muscle tissue loss that's usual during aging. Even functional intensity could be renewed. You could actually prevent several physical disabilities which sometimes show up while being able to vigorously prevent osteoporosis. Fast Recovery When you're in rehabilitation or you have to deal with a disability including orthopedic operation and stroke, weight training is definitely an important aspect in optimizing the recovery of weaker muscles. If you are afflicted by this sort of illness, a proper, certified personal trainer from Huntington Beach that is specialized in recuperation has to be consulted. Physiotherapists will be preferred. Improved Sports Effectiveness The majority of competitors utilizes some sort of strength training plan that's specific to the sport that is done. Stronger muscle tissues will certainly help you to improve your functionality. The only issue is that you need to make sure that your training has muscle contraction that occurs with the precise velocity that is utilized in that particular sport. As a result, when you begin resistance training under the help of a qualified personal trainer from Huntington Beach, you could expect all the benefits mentioned previously. You will surely value the fact that you will always feel great following the workout and you will feel vitalized after it. In general, people that exercise in the fitness center will live a much longer and happier life.