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In [[mathematics]] and its applications, a classical '''Sturm–Liouville equation''', named after [[Jacques Charles François Sturm]] (1803–1855) and [[Joseph Liouville]] (1809–1882), is a real second-order linear [[differential equation]] of the form
{{AccountNotLive}}
The '''Internet''' is a term with many meanings, depending on the context of its use <ref name=Internet>{{cite book | last = Comer | first = Douglas E.  | authorlink = | coauthors =  | title = Computer Networks and Internets | publisher = Pearson Prentice-Hall | date = 2009 | location = Upper Saddle River, NJ | pages = | url = | doi = | id = | isbn = 978-0-13-606127-3 }}</ref>. To the general public, the term is often used synonymously with the [[World Wide Web]], its best-known application <ref name=WWW>{{cite book | last = Okin | first = J. R.  | authorlink = | coauthors =  | title = The Information Revolution: The Not-for-dummies Guide to the History, Technology, And Use of the World Wide Web | publisher = Ironbound Press | date = 2005 | location = Winter Harbor, ME | pages = | url = | doi = | id = | isbn = 0-9763857-4-0 }}</ref>. But the internet supports many other applications, such as [[electronic mail]], [[streaming media]], such as internet radio and video, a large percentage of [[telephone traffic]], [[system monitoring]] and [[real-time control]] applications, to name a few. In one respect the Internet is similar to an iceberg. The vast majority of it is out of sight. While these [[distributed applications]] allow users to utilize [[internet services]], they require a large suite of technologies visible only to the enterprises that provide them. To [[Internet Service Providers]], the '''Internet''' identifies these underlying services. There are internet services that are accessible to the general public, while the same technologies providing similar services are available in restricted environments, such as those in an enterprise [[intranet]], in military and government [[private internet]]s and in local [[home networks]]. Further complicating the notion of an Internet is is the frequent interconnection of public and private networks in ways that allow limited interaction. This article and the subgroup it describes uses the term Internet in the broadest sense. That is, it identifies the applications that provide an interface between users and [[communications services]], those services themselves, public and private instances of application and communications services and the aggregation of private and public networks into a global communications and application resource.


<div style="text-align: right;">
__TOC__
<div style="float: left;"><math> -\frac{d}{dx}\left[p(x)\frac{dy}{ dx}\right]+q(x)y=\lambda w(x)y.</math></div>
<span id="(1)">(1)</span>
</div>


==The architecture of the Internet==


where ''y'' is a function of the free variable ''x''. Here the functions ''p''(''x'')&nbsp;>&nbsp;0 has a ''continuous derivative'', ''q''(''x''), and ''w''(''x'')&nbsp;>&nbsp;0 are specified at the outset, and in the simplest of cases are continuous on the finite closed interval [''a'',''b'']. In addition, the function ''y'' is typically required to satisfy some [[boundary condition]]s at ''a'' and ''b''. The function ''w''(''x''), which is sometimes called ''r''(''x''), is called the "weight" or "density" function.  
The [[history of the internet]] shows it as the culmination of significant developments in both the commercial world as well as within government sponsored programs. While the main development occurred in the United States, there were major contributions from researchers and engineers in both the U.K., France and other parts of Europe. This work led to the existing architectural model.


The value of &lambda; is not specified in the equation; finding the values of &lambda;  for which there exists a non-trivial solution of [[#(1) | (1)]] satisfying the boundary conditions is part of the problem called the '''Sturm–Liouville problem''' (S&nbsp;L).  
In order to engineer the internet, internet designers and engineers place its services into one of several layers, which in total comprise the [[internet protocol architecture]]<ref name=Arch>{{cite web |title=RFC 1958: Architectural Principles of the Internet |url=http://www.ietf.org/rfc/rfc1958.txt |date=June 1996 |work= |publisher=Internet Engineering Task Force |accessdate=}}</ref>. While there have been several different protocol architecture designs, the one with the strongest support consists of 5 layers: 1) the application layer, 2) the transport layer, 3) the network layer, 4) the link-layer, and 5) the physical layer<ref name=ProtLayers>{{cite web |title=The TCP/IP network architecture |url=http://www.iusmentis.com/technology/tcpip/architecture/ |date=Dec. 1, 2006 |work= |publisher=Ius Mentis |accessdate=Sept. 16, 2009}}</ref>. Each protocol layer utilizes the services of the next lower layer (except the lowest, the physical layer) to provide a value-added service to the layer above it (except for the application layer, which provides services to users). Utilizing this protocol architecture, it is possible to describe how the Internet works.


Such values of &lambda; when they exist are called the [[eigenvalues]] of the boundary value problem defined by [[#(1) | (1)]] and the prescribed set of boundary conditions. The corresponding solutions (for such a &lambda;) are the [[eigenfunction]]s of this problem. Under normal assumptions on the coefficient functions ''p''(''x''), ''q''(''x''), and ''w''(''x'') above, they induce a [[hermitian operator|Hermitian]] [[differential operator]] in some [[function space]] defined by [[boundary value problem|boundary conditions]]. The resulting theory of the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their [[Complete metric space|completeness]] in a suitable [[function space]] became known as '''Sturm–Liouville theory'''. This theory is important in applied mathematics, where S–L problems  occur very commonly, particularly when dealing with linear [[partial differential equation]]s that are [[separation of variables|separable]].
[[Web browser]]s are the most common user interface in the Internet. Such browsers translate human requests to the [[Hypertext Transfer Protocol (HTTP]]), which actually moves data between the browser and a [[Web server]]. Consequently, measured solely in terms of percentage of use, the World Wide Web is the most frequently used Internet application (However, this is expected to change. Forecasts of Internet bandwidth utilization suggest that video traffic will make up over 90% of Internet traffic by 2013<ref name=TrafficGrowth>{{cite web |title=Cisco Visual Networking Index:Forecast and Methodology, 2008–2013 |url=http://www.cisco.com/en/US/solutions/collateral/ns341/ns525/ns537/ns705/ns827/white_paper_c11-481360.pdf |date=June 9, 2009 |work= |publisher=Cisco Systems, Inc. |accessdate=Sept. 16, 2009}}</ref>. ). The communications services provided by the Internet have no direct human interfaces; every user-visible function must go through a program resident on a client or server computer. There are literally hundreds of different [[protocol (computer)|protocols]], applications and services that run over the Internet.  [[Virtual private network]]s interconnecting the parts of individual enterprises, or sets of cooperating enterprises, overlay the Internet. As mentioned previously a wide range of interconnected networks using the same protocols as the public Internet, but isolated from it, provide services ranging from passing orders to launch nuclear weapons, authorizing credit card purchases, collecting intelligence information, controlling the electric power grid (see [[System Control And Data Acquisition]]), [[telemedicine]] such as transferring medical images and even allowing remote surgery, etc. Many of these applications utilize custom [[application interface]]s that do not involve a web browser. Consequently, internet distributed applications comprise a much larger set than those visible to the general public.


== Sturm–Liouville theory ==
In addition to applications that are directly experienced by Internet customers, there are a wide-range of internet applications that exist to provide [[infrastructure services]] to the internet. Examples of infrastructure services are the [[Doman Name System (DNS]]), which associates computers connected to the Internet with human friendly names. The movement of data through the internet requires that it visit intermediate systems called [[router]]s. The activity of directing the data through the internet, called [[routing]], utilizes an infrastructure application that distributes routing data to routers. The [[secure identification]] of users to applications requires the use of [[authentication servers]], such as [[RADIUS]] and [[Kerberos]], each of which is a distributed application in and of itself. These are just a few of the internet infrastructure applications that support the provision of internet service.


Under the assumptions that the S–L problem is regular, that is, ''p''(''x'')<sup>&minus;1</sup>&nbsp;>&nbsp;0, ''q''(''x''), and ''w''(''x'')&nbsp;>&nbsp;0 are real-valued [[Lebesgue integrable|integrable]] functions over the finite interval [''a'',&nbsp;''b''],
Internet applications are distributed<ref name=Dist>{{cite web |title=Distributed Computing: An Introduction |url=http://www.extremetech.com/article2/0%2C1697%2C11769%2C00.asp |work= |publisher=ExtremeTech |accessdate=16 Sept., 2009}}</ref>. That is, they normally are comprised of pieces that reside at different locations. That means they must exchange data through communications equipment that is subject to various failure modes. Furthermore, one element may have the capability to send data faster than the receiver can process. The next layer in the protocol architecture, the transport layer, provides services that address these issues. [[Transport layer]] protocols, like the [[Transmission Control Protocol (TCP)]] provide [[end-to-end error management]] and [[flow-control]] services that ensure application elements can exchange data in an [[error-tolerant]] and synchronized manner. Instead of relying on the error and flow-control services provided by TCP, some applications handle these services themselves. Those that do utilize a [[datagram]] service also provided by the transport layer. For example the [[Unreliable Datagram Protocol (UDP]]) moves packets between application parts without the provision of either error-control or flow-control services.
with ''separated boundary conditions'' of the form


<div style="text-align: right;">
The next layer of internet service, the [[network layer]] moves data between [[end-systems]] (normally customer computers, but in some cases infrastructure systems) through an interconnected set of systems, routers, which are mentioned above. Routers come in all shapes and sizes. Some, normally located at the periphery of the internet such as those in a home or small business, are known as [[edge routers]]. Others are service provider equipment with varying capabilities, from modest performance [[border routers]] to high performance [[core routers]]. These routers are interconnected, moving data across the Internet in a way that increases the probability of successful transit. There are two types of routing schemes. [[Virtual circuit routing]] reserves resources over a fixed path between two end-systems. [[Packet routing]] operates in a way whereby individual [[packet]]s of data may take different paths through the systems that interconnect end-systems. The network layer also supports specialized data services, such as [[multicast]], [[broadcast]], and [[anycast]] routing.
<div style="float: left;"><math> y(a)\cos \alpha - p(a)y^\prime (a)\sin \alpha = 0,</math></div>
<span id="(2)">(2)</span>
</div>


<div style="text-align: right;">
Routers and end systems directly connect to each other through [[physical channels]] (addressed below) that introduce [[communications errors]] and that are themselves not flow-controlled. Each of these systems is called an [[intermediate system]]. It is the function of the [[link-layer]] to provide services that correct most of the errors that occur on physical channels and to provide the two directly communicating intermediate systems with flow-controlled data exchange. The characteristics of the physical channel may vary widely from the fairly reliable [[ethernet]], less reliable [[wireless]] channels, to the very unreliable deep space radio channels. Each type of physical channel may require a different link-layer protocol to accommodate its characteristics.
<div style="float: left;"><math> y(b)\cos \beta - p(b)y^\prime (b)\sin \beta = 0,</math></div>
<span id="(3)">(3)</span>
</div>


where <math>\alpha, \beta \in [0, \pi),</math>
Physical channels, which populate the [[physical layer]], encode data utilizing various techniques, thereby providing the basic data transmission service between directly connected equipment. There are a wide variety of physical channels, each utilizing its own data encoding scheme. Examples of physical channels used in the Internet are wire-based channels, such as those used by [[low-bandwidth ethernet]]; [[wireless broadcast channels]], such as those used in [[Wi-Fi]], also known as [[802.11]], as well as in [[cell phone service]]; [[optical channels]], such as those used by [[high-bandwidth ethernet]]; and wireless point-to-point [[radio channels]], such as those used by [[microwave link]]s and [[satellite communications]].
the main tenet of '''Sturm–Liouville theory''' states that:


* The eigenvalues &lambda;<sub>1</sub>, &lambda;<sub>2</sub>, &lambda;<sub>3</sub>, ... of the regular Sturm–Liouville problem [[#(1) | (1)]] -[[#(2) | (2)]] -[[#(3) | (3)]] are real and can be ordered such that
The Internet utilizes not only technology acting within a single layer of its protocol architecture, but also mechanisms that are spread over several protocol layers. As mentioned previously, routing is one such technology using application services to move routing data to routers in order to provide the network-layer routing service. Another example is the provision of [[network security]] within the Internet. For example, providing a [[secure transport service]] requires encrypting of packets at end-systems This requires [[encryption keys]] that are distributed by a logically separate application. [[Internet management]] may utilize an application layer protocol, such as the [[Simple Network Management Protocol (SNMP)]] in concert with a network-layer protocol, such as the [[Internet Message Control Protocol (ICMP)]].


:: <math>\lambda_1 < \lambda_2 < \lambda_3 < \cdots < \lambda_n < \cdots \to \infty; \, </math>
==Professional societies and organizations==


* Corresponding to each eigenvalue &lambda;<sub>''n''</sub> is a unique (up to a normalization constant) eigenfunction ''y''<sub>''n''</sub>(''x'') which has exactly ''n''&nbsp;&minus;&nbsp;1 zeros in (''a'',&nbsp;''b''). The eigenfunction ''y''<sub>''n''</sub>(''x'') is called the ''n''-th ''fundamental solution'' satisfying the regular Sturm–Liouville problem [[#(1) | (1)]] -[[#(2) | (2)]] -[[#(3) | (3)]].
:(See External Links subpage for website homepages)


* The normalized eigenfunctions form an [[orthonormal basis]]
*International: Internet Society (ISOC), IEEE Communications Society (IEEE ComSoc), World Wide Web Consortium (W3C), Internet Technical Committee (ITC), Association for Computer Machinery Special Interest Group on Data Communications ( ACM SIGCOMM), International Telecommunications Union (ITU), International Electrotechnical Commission (IEC).
*[[North America]]: North American Network Operators Group (NANOG)
*[[Europe]]: European Telecommunications Standards Institute (ETSI)
*[[Asia]]: South Asian Network Operators Group (SANOG)
*[[Middle East]]: Middle East Network Operators Group (MENOG)
*[[Africa]]:  African Network Operators Group (AfrNOG)
*[[Pacific]]: The Pacific Network Operators Group (PacNOG)
*[[Latin America]]: Latin America and Caribbean Region Network Operators Group (LACNOG)
*[[France]]: FRench Network Operators Group (FRnOG)
*[[United States of America|United States]]: Telecommunications Industry Association (TIA)


::<math> \int_a^b y_n(x)y_m(x)w(x)\,dx = \delta_{mn},</math> 
==References==
 
:in the [[Hilbert space]] [[Lebesgue space|''L''<sup>2</sup>([''a'',&nbsp;''b''],''w''(''x'')&nbsp;''dx'')]]. Here &delta;<sub>''mn''</sub> is a [[Kronecker delta]].


Note that, unless  ''p''(''x'') is continuously differentiable and ''q''(''x''), ''w''(''x'') are continuous, the equation has to be understood in a [[weak solution|weak sense]].
{{reflist}}
 
== Sturm–Liouville form ==
The differential equation [[#(1) | (1)]] is said to be in '''Sturm–Liouville form''' or '''self-adjoint form'''. All second-order linear [[ordinary differential equation]]s can be recast in the form on the left-hand side of [[#(1) | (1)]] by multiplying both sides of the equation by an appropriate [[integrating factor]] (although the same is not true of second-order [[partial differential equation]]s, or if ''y'' is a vector.)
 
=== Examples ===
The [[Bessel equation]]:
 
: <math>x^2y''+xy'+(\lambda^2x^2-\nu^2)y=0\,</math>
 
can be written in Sturm–Liouville form as
 
: <math>(xy')'+(\lambda^2 x-\nu^2/x)y=0.\,</math>
 
The [[Legendre polynomials|Legendre equation]],
 
: <math>(1-x^2)y''-2xy'+\nu(\nu+1)y=0\;\!</math>
 
can easily be put into Sturm–Liouville form, since ''D''(1&nbsp;&minus;&nbsp;''x''<sup>2</sup>) = &minus;2''x'', so, the Legendre equation is equivalent to
 
: <math>[(1-x^2)y']'+\nu(\nu+1)y=0\;\!</math>
 
Less simple is such a differential equation as
 
: <math>x^3y''-xy'+2y=0.\,</math>
 
Divide throughout by ''x''<sup>3</sup>:
 
: <math>y''-{x\over x^3}y'+{2\over x^3}y=0</math>
 
Multiplying throughout by an [[integrating factor]] of
 
: <math>e^{\int -{x / x^3}\,dx}=e^{\int -{1 / x^2}\, dx}=e^{1 / x},</math>
 
gives
 
: <math>e^{1 / x}y''-{e^{1 / x} \over x^2} y'+ {2 e^{1 / x} \over x^3} y = 0</math>
 
which can be easily put into Sturm–Liouville form since
 
: <math>D e^{1 / x} = -{e^{1 / x} \over x^2} </math>
 
so the differential equation is equivalent to
 
: <math>(e^{1 / x}y')'+{2 e^{1 / x} \over x^3} y =0.</math>
 
In general, given a differential equation
 
: <math>P(x)y''+Q(x)y'+R(x)y=0\,</math>
 
dividing by ''P''(''x''), multiplying through by the integrating factor 
 
: <math>e^{\int {Q(x) / P(x)}\,dx},</math>
 
and then collecting gives the Sturm–Liouville form.
 
== Sturm–Liouville equations as self-adjoint differential operators ==
 
The map
 
: <math>L  u  = {1 \over w(x)} \left(-{d\over dx}\left[p(x){du\over dx}\right]+q(x)u \right)</math>
 
can be viewed as a [[linear operator]] mapping a function ''u'' to another function ''Lu''. We may study this linear operator in the context of [[functional analysis]]. In fact, equation [[#(1) | (1)]] can be written as
 
: <math>L  u  = \lambda u \,.</math>
 
This is precisely the [[eigenvalue]] problem; that is, we are trying to find the eigenvalues &lambda;<sub>1</sub>, &lambda;<sub>2</sub>, &lambda;<sub>3</sub>, ... and the corresponding eigenvectors ''u''<sub>1</sub>, ''u''<sub>2</sub>, ''u''<sub>3</sub>, ... of the ''L'' operator. The proper setting for this problem is the [[Hilbert space]] [[Lp space#Weighted Lp spaces|''L''<sup>2</sup>([''a'',&nbsp;''b''],''w''(''x'')&nbsp;''dx'')]] with
scalar product
 
: <math> \langle f, g\rangle = \int_{a}^{b} \overline{f(x)} g(x)w(x)\,dx.</math> 
 
In this space ''L'' is defined on sufficiently smooth functions which satisfy the above [[boundary value problem|boundary condition]]s. Moreover, ''L'' gives rise to a [[self-adjoint]] operator.
This can be seen formally by using [[integration by parts]] twice, where the boundary terms vanish by virtue of the boundary conditions. It then follows that the eigenvalues of a Sturm–Liouville operator are real and that eigenfunctions of ''L'' corresponding to different eigenvalues are orthogonal. However, this operator is unbounded and hence existence of an orthonormal basis of eigenfunctions is not evident. To overcome this problem one looks at the [[resolvent]]
 
::<math> (L - z)^{-1}, \qquad z \in\mathbb{C},</math>
 
where ''z'' is chosen to be some real number which is not an eigenvalue. Then, computing the resolvent amounts to solving the inhomogeneous equation, which can be done using the [[variation of parameters]] formula. This shows that the resolvent is an [[integral operator]] with a continuous symmetric kernel (the [[Green's function]] of the problem). As a consequence of the [[Arzelà–Ascoli theorem]] this integral operator is compact and existence of a sequence of eigenvalues &alpha;<sub>''n''</sub> which converge to 0 and eigenfunctions which form an orthonormal basis follows from the [[compact operator on Hilbert space|spectral theorem for compact operators]]. Finally, note that <math>(L-z)^{-1} u = \alpha u</math> is equivalent to <math>L u = (z+\alpha^{-1}) u</math>.
 
If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls ''L'' singular. In this case the spectrum does no longer consist of eigenvalues alone and can contain a continuous component. There is still an associated eigenfunction expansion (similar to Fourier series versus Fourier transform). This is important in [[quantum mechanics]], since the one-dimensional [[Schrödinger equation]] is a special case of a S–L equation.
 
== Example ==
 
We wish to find a function ''u''(''x'') which solves the following Sturm–Liouville problem:
 
<div style="text-align: right;">
<div style="float: left;"><math> L  u  = \frac{d^2u}{dx^2} = \lambda u</math></div>
<span id="(1)">(1)</span>
</div>
 
 
where the unknowns are ''&lambda;'' and ''u''(''x''). As above, we must add boundary conditions, we take for example
 
:<math> u(0) = u(\pi) = 0 \, </math>
 
Observe that if ''k'' is any integer, then the function
 
:<math> u(x) = \sin kx \, </math>
 
is a solution with eigenvalue &lambda; = &minus;''k''<sup>2</sup>. We know that the solutions of a S–L problem form an orthogonal basis, and we know from Fourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases are always maximal (by definition) we conclude that the S–L problem in this case has no other eigenvectors.
 
Given the preceding, let us now solve the inhomogeneous problem
 
:<math>L  u  =x, \qquad x\in(0,\pi)</math>
 
with the same boundary conditions. In this case, we must write ''f''(''x'') = ''x'' in a Fourier series. The reader may check, either by integrating &int;exp(''ikx'')''x''&nbsp;d''x'' or by consulting a table of Fourier transforms, that we thus obtain
 
:<math>L  u  =\sum_{k=1}^{\infty}-2\frac{(-1)^k}{k}\sin kx.</math>
 
This particular Fourier series is troublesome because of its poor convergence properties. It is not clear ''a priori'' whether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are "square-summable", the Fourier series converges in ''L''<sup>2</sup> which is all we need for this particular theory to function. We mention for the interested reader that in this case we may rely on a result which says that Fourier's series converges at every point of differentiability, and at jump points (the function ''x'', considered as a periodic function, has a jump at &pi;) converges to the average of the left and right limits (see [[convergence of Fourier series]]).
 
Therefore, by using formula '''('''{{EquationNote|4}}''')''', we obtain that the solution is
 
:<math>u=\sum_{k=1}^{\infty}2\frac{(-1)^k}{k^3}\sin kx.</math>
 
In this case, we could have found the answer using antidifferentiation. This technique yields ''u'' =&nbsp;(''x''<sup>3</sup>&nbsp;&minus;&nbsp;''&pi;''<sup>2</sup>''x'')/6, whose Fourier series agrees with the solution we found. The antidifferentiation technique is no longer useful in most cases when the differential equation is in many variables.
 
== Application to normal modes ==
 
Suppose we are interested in the modes of vibration of a thin membrane, held in a rectangular frame, 0 < ''x'' < ''L''<sub>1</sub>, 0 < ''y'' < ''L''<sub>2</sub>. We know the equation of motion for the vertical membrane's displacement, ''W''(''x'', ''y'', ''t'') is given by the [[wave equation]]:
 
:<math>\frac{\partial^2W}{\partial x^2}+\frac{\partial^2W}{\partial y^2} = \frac{1}{c^2}\frac{\partial^2W}{\partial t^2}.</math>
 
The equation is [[separation of variables|separable]] (substituting ''W'' = ''X''(''x'') &times; ''Y''(''y'') &times; ''T''(''t'')), and the normal mode solutions that have [[harmonic]] time dependence and satisfy the boundary conditions ''W'' = 0 at ''x'' = 0, ''L''<sub>1</sub> and ''y'' = 0, ''L''<sub>2</sub> are given by
 
:<math>W_{mn}(x,y,t) = A_{mn}\sin\left(\frac{m\pi x}{L_1}\right)\sin\left(\frac{n\pi y}{L_2}\right)\cos\left(\omega_{mn}t\right)</math>
 
where ''m'' and ''n'' are non-zero [[integer]]s, ''A<sub>mn</sub>'' is an arbitrary constant and
 
: <math>\omega^2_{mn} = c^2 \left(\frac{m^2\pi^2}{L_1^2}+\frac{n^2\pi^2}{L_2^2}\right).</math>
 
Since the eigenfunctions ''W<sub>mn</sub>'' form a basis, an arbitrary initial displacement can be decomposed into a sum of these modes, which each vibrate at their individual frequencies <math>\omega_{mn}</math>. Infinite sums are also valid, as long as they [[convergence|converge]].
 
==See also==
 
* [[Normal mode]]
* [[Self-adjoint]]
 
== References ==
* P. Hartman, ''Ordinary Differential Equations'', SIAM, Philadelphia, 2002 (2nd edition). ISBN 978-0-898715-10-1
* A. D. Polyanin and V. F. Zaitsev, ''Handbook of Exact Solutions for Ordinary Differential Equations'', Chapman & Hall/CRC Press, Boca Raton, 2003 (2nd edition). ISBN 1-58488-297-2
* G. Teschl, ''Ordinary Differential Equations and Dynamical Systems'', http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ (Chapter 5)
* G. Teschl, ''Mathematical Methods in Quantum Mechanics and Applications to Schrödinger Operators'', http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ (see Chapter 9 for singular S-L operators and connections with quantum mechanics)
* A. Zettl, ''Sturm–Liouville Theory'', American Mathematical Society, 2005. ISBN 0-8218-3905-5.

Latest revision as of 16:02, 22 March 2024


The account of this former contributor was not re-activated after the server upgrade of March 2022.


The Internet is a term with many meanings, depending on the context of its use [1]. To the general public, the term is often used synonymously with the World Wide Web, its best-known application [2]. But the internet supports many other applications, such as electronic mail, streaming media, such as internet radio and video, a large percentage of telephone traffic, system monitoring and real-time control applications, to name a few. In one respect the Internet is similar to an iceberg. The vast majority of it is out of sight. While these distributed applications allow users to utilize internet services, they require a large suite of technologies visible only to the enterprises that provide them. To Internet Service Providers, the Internet identifies these underlying services. There are internet services that are accessible to the general public, while the same technologies providing similar services are available in restricted environments, such as those in an enterprise intranet, in military and government private internets and in local home networks. Further complicating the notion of an Internet is is the frequent interconnection of public and private networks in ways that allow limited interaction. This article and the subgroup it describes uses the term Internet in the broadest sense. That is, it identifies the applications that provide an interface between users and communications services, those services themselves, public and private instances of application and communications services and the aggregation of private and public networks into a global communications and application resource.

The architecture of the Internet

The history of the internet shows it as the culmination of significant developments in both the commercial world as well as within government sponsored programs. While the main development occurred in the United States, there were major contributions from researchers and engineers in both the U.K., France and other parts of Europe. This work led to the existing architectural model.

In order to engineer the internet, internet designers and engineers place its services into one of several layers, which in total comprise the internet protocol architecture[3]. While there have been several different protocol architecture designs, the one with the strongest support consists of 5 layers: 1) the application layer, 2) the transport layer, 3) the network layer, 4) the link-layer, and 5) the physical layer[4]. Each protocol layer utilizes the services of the next lower layer (except the lowest, the physical layer) to provide a value-added service to the layer above it (except for the application layer, which provides services to users). Utilizing this protocol architecture, it is possible to describe how the Internet works.

Web browsers are the most common user interface in the Internet. Such browsers translate human requests to the Hypertext Transfer Protocol (HTTP), which actually moves data between the browser and a Web server. Consequently, measured solely in terms of percentage of use, the World Wide Web is the most frequently used Internet application (However, this is expected to change. Forecasts of Internet bandwidth utilization suggest that video traffic will make up over 90% of Internet traffic by 2013[5]. ). The communications services provided by the Internet have no direct human interfaces; every user-visible function must go through a program resident on a client or server computer. There are literally hundreds of different protocols, applications and services that run over the Internet. Virtual private networks interconnecting the parts of individual enterprises, or sets of cooperating enterprises, overlay the Internet. As mentioned previously a wide range of interconnected networks using the same protocols as the public Internet, but isolated from it, provide services ranging from passing orders to launch nuclear weapons, authorizing credit card purchases, collecting intelligence information, controlling the electric power grid (see System Control And Data Acquisition), telemedicine such as transferring medical images and even allowing remote surgery, etc. Many of these applications utilize custom application interfaces that do not involve a web browser. Consequently, internet distributed applications comprise a much larger set than those visible to the general public.

In addition to applications that are directly experienced by Internet customers, there are a wide-range of internet applications that exist to provide infrastructure services to the internet. Examples of infrastructure services are the Doman Name System (DNS), which associates computers connected to the Internet with human friendly names. The movement of data through the internet requires that it visit intermediate systems called routers. The activity of directing the data through the internet, called routing, utilizes an infrastructure application that distributes routing data to routers. The secure identification of users to applications requires the use of authentication servers, such as RADIUS and Kerberos, each of which is a distributed application in and of itself. These are just a few of the internet infrastructure applications that support the provision of internet service.

Internet applications are distributed[6]. That is, they normally are comprised of pieces that reside at different locations. That means they must exchange data through communications equipment that is subject to various failure modes. Furthermore, one element may have the capability to send data faster than the receiver can process. The next layer in the protocol architecture, the transport layer, provides services that address these issues. Transport layer protocols, like the Transmission Control Protocol (TCP) provide end-to-end error management and flow-control services that ensure application elements can exchange data in an error-tolerant and synchronized manner. Instead of relying on the error and flow-control services provided by TCP, some applications handle these services themselves. Those that do utilize a datagram service also provided by the transport layer. For example the Unreliable Datagram Protocol (UDP) moves packets between application parts without the provision of either error-control or flow-control services.

The next layer of internet service, the network layer moves data between end-systems (normally customer computers, but in some cases infrastructure systems) through an interconnected set of systems, routers, which are mentioned above. Routers come in all shapes and sizes. Some, normally located at the periphery of the internet such as those in a home or small business, are known as edge routers. Others are service provider equipment with varying capabilities, from modest performance border routers to high performance core routers. These routers are interconnected, moving data across the Internet in a way that increases the probability of successful transit. There are two types of routing schemes. Virtual circuit routing reserves resources over a fixed path between two end-systems. Packet routing operates in a way whereby individual packets of data may take different paths through the systems that interconnect end-systems. The network layer also supports specialized data services, such as multicast, broadcast, and anycast routing.

Routers and end systems directly connect to each other through physical channels (addressed below) that introduce communications errors and that are themselves not flow-controlled. Each of these systems is called an intermediate system. It is the function of the link-layer to provide services that correct most of the errors that occur on physical channels and to provide the two directly communicating intermediate systems with flow-controlled data exchange. The characteristics of the physical channel may vary widely from the fairly reliable ethernet, less reliable wireless channels, to the very unreliable deep space radio channels. Each type of physical channel may require a different link-layer protocol to accommodate its characteristics.

Physical channels, which populate the physical layer, encode data utilizing various techniques, thereby providing the basic data transmission service between directly connected equipment. There are a wide variety of physical channels, each utilizing its own data encoding scheme. Examples of physical channels used in the Internet are wire-based channels, such as those used by low-bandwidth ethernet; wireless broadcast channels, such as those used in Wi-Fi, also known as 802.11, as well as in cell phone service; optical channels, such as those used by high-bandwidth ethernet; and wireless point-to-point radio channels, such as those used by microwave links and satellite communications.

The Internet utilizes not only technology acting within a single layer of its protocol architecture, but also mechanisms that are spread over several protocol layers. As mentioned previously, routing is one such technology using application services to move routing data to routers in order to provide the network-layer routing service. Another example is the provision of network security within the Internet. For example, providing a secure transport service requires encrypting of packets at end-systems This requires encryption keys that are distributed by a logically separate application. Internet management may utilize an application layer protocol, such as the Simple Network Management Protocol (SNMP) in concert with a network-layer protocol, such as the Internet Message Control Protocol (ICMP).

Professional societies and organizations

(See External Links subpage for website homepages)
  • International: Internet Society (ISOC), IEEE Communications Society (IEEE ComSoc), World Wide Web Consortium (W3C), Internet Technical Committee (ITC), Association for Computer Machinery Special Interest Group on Data Communications ( ACM SIGCOMM), International Telecommunications Union (ITU), International Electrotechnical Commission (IEC).
  • North America: North American Network Operators Group (NANOG)
  • Europe: European Telecommunications Standards Institute (ETSI)
  • Asia: South Asian Network Operators Group (SANOG)
  • Middle East: Middle East Network Operators Group (MENOG)
  • Africa: African Network Operators Group (AfrNOG)
  • Pacific: The Pacific Network Operators Group (PacNOG)
  • Latin America: Latin America and Caribbean Region Network Operators Group (LACNOG)
  • France: FRench Network Operators Group (FRnOG)
  • United States: Telecommunications Industry Association (TIA)

References

  1. ↑ Comer, Douglas E. (2009). Computer Networks and Internets. Upper Saddle River, NJ: Pearson Prentice-Hall. ISBN 978-0-13-606127-3. 
  2. ↑ Okin, J. R. (2005). The Information Revolution: The Not-for-dummies Guide to the History, Technology, And Use of the World Wide Web. Winter Harbor, ME: Ironbound Press. ISBN 0-9763857-4-0. 
  3. ↑ RFC 1958: Architectural Principles of the Internet. Internet Engineering Task Force (June 1996).
  4. ↑ The TCP/IP network architecture. Ius Mentis (Dec. 1, 2006). Retrieved on Sept. 16, 2009.
  5. ↑ Cisco Visual Networking Index:Forecast and Methodology, 2008–2013. Cisco Systems, Inc. (June 9, 2009). Retrieved on Sept. 16, 2009.
  6. ↑ Distributed Computing: An Introduction. ExtremeTech. Retrieved on 16 Sept., 2009.
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