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2014-11-30T18:37:21Z

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2014-02-28T04:48:21Z

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2014-02-01T23:16:22Z

71.234.59.55:

{{About|cryptographic signatures|simple signatures in digital form|Electronic signature}}
[[Image:Digital Signature diagram.svg|thumb|right|280px|A diagram showing how a digital signature is applied and then verified.]]
A '''digital signature''' is a mathematical scheme for demonstrating the authenticity of a digital message or document. A valid digital signature gives a recipient reason to believe that the message was created by a known sender, such that the sender cannot deny having sent the message ([[authentication]] and [[non-repudiation]]) and that the message was not altered in transit ([[Data integrity|integrity]]). Digital signatures are commonly used for software distribution, financial transactions, and in other cases where it is important to detect forgery or tampering.

== Explanation ==
Digital signatures are often used to implement [[electronic signature]]s, a broader term that refers to any electronic data that carries the intent of a signature,<ref>[http://frwebgate.access.gpo.gov/cgi-bin/getdoc.cgi?dbname=106_cong_public_laws&docid=f:publ229.106.pdf US ESIGN Act of 2000]</ref> but not all electronic signatures use digital signatures.<ref>[http://enterprise.state.wi.us/home/strategic/esig.htm State of WI]</ref><ref>[http://www.naa.gov.au/recordkeeping/er/Security/6-glossary.html National Archives of Australia]</ref> In some countries, including the United States, India, Brazil,<ref>{{cite book|first=Government of India|title=The Information Technology Act, 2000|url=http://www.dot.gov.in/sites/default/files/itbill2000_0.pdf}}</ref> and members of the [[European Union]], electronic signatures have legal significance.

Digital signatures employ a type of [[asymmetric key algorithm|asymmetric cryptography]]. For messages sent through a nonsecure channel, a properly implemented digital signature gives the receiver reason to believe the message was sent by the claimed sender. Digital signatures are equivalent to traditional handwritten signatures in many respects, but properly implemented digital signatures are more difficult to forge than the handwritten type. Digital signature schemes, in the sense used here, are cryptographically based, and must be implemented properly to be effective. Digital signatures can also provide [[non-repudiation]], meaning that the signer cannot successfully claim they did not sign a message, while also claiming their [[private key]] remains secret; further, some non-repudiation schemes offer a time stamp for the digital signature, so that even if the private key is exposed, the signature is valid. Digitally signed messages may be anything representable as a [[bitstring]]: examples include [[electronic mail]], [[contract]]s, or a message sent via some other [[cryptographic protocol]].

==Definition==
{{Main|Public-key cryptography}}

A digital signature scheme typically consists of three algorithms:
* A ''[[key generation]]'' algorithm that selects a ''private key'' [[Uniform distribution (discrete)|uniformly at random]] from a set of possible private keys. The algorithm outputs the private key and a corresponding ''public key''.
* A ''signing'' algorithm that, given a message and a private key, produces a signature.
* A ''signature verifying'' algorithm that, given a message, public key and a signature, either accepts or rejects the message's claim to authenticity.

Two main properties are required. First, a signature generated from a fixed message and fixed private key should verify the authenticity of that message by using the corresponding public key. Secondly, it should be computationally infeasible to generate a valid signature for a party without knowing that party's private key.

==History==
In 1976, [[Whitfield Diffie]] and [[Martin Hellman]] first described the notion of a digital signature scheme, although they only conjectured that such schemes existed.<ref>"New Directions in Cryptography", IEEE Transactions on Information Theory, IT-22(6):644–654, Nov. 1976.</ref><ref name=lysythesis>"[http://theory.lcs.mit.edu/~cis/theses/anna-phd.pdf Signature Schemes and Applications to Cryptographic Protocol Design]", Anna Lysyanskaya, PhD thesis, [[Massachusetts Institute of Technology|MIT]], 2002.</ref> Soon afterwards, [[Ronald Rivest]], [[Adi Shamir]], and [[Len Adleman]] invented the [[RSA (algorithm)|RSA]] algorithm, which could be used to produce primitive digital signatures<ref name="rsa">{{cite journal
| first = R. | last = Rivest
| coauthors = A. Shamir; L. Adleman
| url = http://people.csail.mit.edu/rivest/Rsapaper.pdf
| title = A Method for Obtaining Digital Signatures and Public-Key Cryptosystems
| journal = Communications of the ACM
| volume = 21 | issue = 2 | pages = 120–126 | year = 1978
| doi = 10.1145/359340.359342}}</ref> (although only as a proof-of-concept – "plain" RSA signatures are not secure<ref>For example any integer ''r'' "signs" ''m''=''r''''e'' and the product ''s''1''s''2 of any two valid signatures ''s''1, ''s''2 of ''m''1, ''m''2 is a valid signature of the product ''m''1''m''2.</ref>). The first widely marketed software package to offer digital signature was [[Lotus Notes]] 1.0, released in 1989, which used the RSA algorithm.{{Citation needed|date=March 2009}}

Other digital signature schemes were soon developed after RSA, the earliest being [[Lamport signature]]s,<ref>"Constructing digital signatures from a one-way function.", [[Leslie Lamport]], Technical Report CSL-98, SRI International, Oct. 1979.</ref> [[Merkle tree|Merkle signatures]] (also known as "Merkle trees" or simply "Hash trees"),<ref>"A certified digital signature", Ralph Merkle, In Gilles Brassard, ed., Advances in Cryptology – [[CRYPTO]] '89, vol. 435 of Lecture Notes in Computer Science, pp. 218–238, Spring Verlag, 1990.</ref> and [[Rabin signature]]s.<ref>"Digitalized signatures as intractable as factorization." [[Michael O. Rabin]], Technical Report MIT/LCS/TR-212, MIT Laboratory for Computer Science, Jan. 1979</ref>

In 1988, [[Shafi Goldwasser]], [[Silvio Micali]], and Ronald Rivest became the first to rigorously define the security requirements of digital signature schemes.<ref name="SJC 17(2)">"A digital signature scheme secure against adaptive chosen-message attacks.", Shafi Goldwasser, Silvio Micali, and Ronald Rivest. SIAM Journal on Computing, 17(2):281–308, Apr. 1988.</ref> They described a hierarchy of attack models for signature schemes, and also present the [[GMR (cryptography)|GMR signature scheme]], the first that can be proven to prevent even an existential forgery against a chosen message attack.<ref name="SJC 17(2)"/>

==How they work==
To create RSA signature keys, generate an RSA key pair containing a modulus ''N'' that is the product of two large primes, along with integers ''e'' and ''d'' such that ''e d'' [[Modular arithmetic|≡]] 1 (mod φ(''N'')), where φ is the [[Euler's totient function|Euler phi-function]]. The signer's public key consists of ''N'' and ''e'', and the signer's secret key contains ''d''.

To sign a message ''m'', the signer computes σ ≡ ''m''''d'' (mod ''N''). To verify, the receiver checks that σ''e'' ≡ ''m'' (mod ''N'').

As noted earlier, this basic scheme is not very secure. To prevent attacks, one can first apply a [[cryptographic hash function]] to the message ''m'' and then apply the RSA algorithm described above to the result. This approach can be proven secure in the so-called [[random oracle model]]{{Clarify|reason=Looks too general to me. Doesn't this only apply to hash functions with additional properties? (I.e. full domain hashes)|date=September 2010}}.
Most early signature schemes were of a similar type: they involve the use of a [[trapdoor permutation]], such as the RSA function, or in the case of the Rabin signature scheme, computing square modulo composite ''n.'' A trapdoor permutation family is a family of [[permutation]]s, specified by a parameter, that is easy to compute in the forward direction, but is difficult to compute in the reverse direction without already knowing the private key. However, for every parameter there is a "trapdoor" (private key) which when known, easily decrypts the message. Trapdoor permutations can be viewed as public-key encryption systems, where the parameter is the public key and the trapdoor is the secret key, and where encrypting corresponds to computing the forward direction of the permutation, while decrypting corresponds to the reverse direction. Trapdoor permutations can also be viewed as digital signature schemes, where computing the reverse direction with the secret key is thought of as signing, and computing the forward direction is done to verify signatures. Because of this correspondence, digital signatures are often described as based on public-key cryptosystems, where signing is equivalent to decryption and verification is equivalent to encryption, but this is not the only way digital signatures are computed.

Used directly, this type of signature scheme is vulnerable to a key-only existential forgery attack. To create a forgery, the attacker picks a random signature σ and uses the verification procedure to determine the message ''m'' corresponding to that signature.<ref>"Modern Cryptography: Theory & Practice", Wenbo Mao, Prentice Hall Professional Technical Reference, New Jersey, 2004, pg. 308. ISBN 0-13-066943-1</ref> In practice, however, this type of signature is not used directly, but rather, the message to be signed is first [[cryptographic hash function|hashed]] to produce a short digest that is then signed. This forgery attack, then, only produces the hash function output that corresponds to σ, but not a message that leads to that value, which does not lead to an attack. In the random oracle model, this [[Full domain hash|hash-then-sign]] form of signature is existentially unforgeable, even against a chosen-message attack.<ref name=lysythesis />{{Clarify|reason=Please give a page number or theorem number.|date=September 2010}}

There are several reasons to sign such a hash (or message digest) instead of the whole document.

*'''For efficiency:''' The signature will be much shorter and thus save time since hashing is generally much faster than signing in practice.
*'''For compatibility:''' Messages are typically bit strings, but some signature schemes operate on other domains (such as, in the case of RSA, numbers modulo a composite number ''N''). A hash function can be used to convert an arbitrary input into the proper format.
*'''For integrity:''' Without the hash function, the text "to be signed" may have to be split (separated) in blocks small enough for the signature scheme to act on them directly. However, the receiver of the signed blocks is not able to recognize if all the blocks are present and in the appropriate order.

==Notions of security==
In their foundational paper, Goldwasser, Micali, and Rivest lay out a hierarchy of attack models against digital signatures:<ref name="SJC 17(2)"/>

# In a ''key-only'' attack, the attacker is only given the public verification key.
# In a ''known message'' attack, the attacker is given valid signatures for a variety of messages known by the attacker but not chosen by the attacker.
# In an ''adaptive chosen message'' attack, the attacker first learns signatures on arbitrary messages of the attacker's choice.

They also describe a hierarchy of attack results:<ref name="SJC 17(2)"/>

# A ''total break'' results in the recovery of the signing key.
# A [[universal forgery]] attack results in the ability to forge signatures for any message.
# A [[selective forgery]] attack results in a signature on a message of the adversary's choice.
# An [[existential forgery]] merely results in some valid message/signature pair not already known to the adversary.

The strongest notion of security, therefore, is security against existential forgery under an adaptive chosen message attack.

==Applications of digital signatures==

As organizations move away from paper documents with ink signatures or authenticity stamps, digital signatures can provide added assurances of the evidence to provenance, identity, and status of an electronic document as well as acknowledging informed consent and approval by a signatory. The United States Government Printing Office (GPO) publishes electronic versions of the budget, public and private laws, and congressional bills with digital signatures. Universities including Penn State, [[University of Chicago]], and Stanford are publishing electronic student transcripts with digital signatures.

Below are some common reasons for applying a digital signature to communications:

===Authentication===
Although messages may often include information about the entity sending a message, that information may not be accurate. Digital signatures can be used to authenticate the source of messages. When ownership of a digital signature secret key is bound to a specific user, a valid signature shows that the message was sent by that user. The importance of high confidence in sender authenticity is especially obvious in a financial context. For example, suppose a bank's branch office sends instructions to the central office requesting a change in the balance of an account. If the central office is not convinced that such a message is truly sent from an authorized source, acting on such a request could be a grave mistake.

===Integrity===
In many scenarios, the sender and receiver of a message may have a need for confidence that the message has not been altered during transmission. Although encryption hides the contents of a message, it may be possible to ''change'' an encrypted message without understanding it. (Some encryption algorithms, known as [[Malleability (cryptography)|nonmalleable]] ones, prevent this, but others do not.) However, if a message is digitally signed, any change in the message after signature invalidates the signature. Furthermore, there is no efficient way to modify a message and its signature to produce a new message with a valid signature, because this is still considered to be computationally infeasible by most cryptographic hash functions (see [[collision resistance]]).

===Non-repudiation===
[[Non-repudiation]], or more specifically ''non-repudiation of origin'', is an important aspect of digital signatures. By this property, an entity that has signed some information cannot at a later time deny having signed it. Similarly, access to the public key only does not enable a fraudulent party to fake a valid signature.

Note that these authentication, non-repudiation etc. properties rely on the secret key ''not having been revoked ''prior to its usage. Public revocation of a key-pair is a required ability, else leaked secret keys would continue to implicate the claimed owner of the key-pair. Checking revocation status requires an "online" check, e.g. checking a "[[Certificate Revocation List]]" or via the "[[Online Certificate Status Protocol]]". Very roughly this is analogous to a vendor who receives credit-cards first checking online with the credit-card issuer to find if a given card has been reported lost or stolen. Of course, with stolen key pairs, the theft is often discovered only after the secret key's use, e.g., to sign a bogus certificate for espionage purposes.

==Additional security precautions==

===Putting the private key on a smart card===
All public key / private key cryptosystems depend entirely on keeping the private key secret. A private key can be stored on a user's computer, and protected by a local password, but this has two disadvantages:

* the user can only sign documents on that particular computer
* the security of the private key depends entirely on the [[computer insecurity|security]] of the computer

A more secure alternative is to store the private key on a [[smart card]]. Many smart cards are designed to be tamper-resistant (although some designs have been broken, notably by [[Ross J. Anderson (professor)|Ross Anderson]] and his students). In a typical digital signature implementation, the hash calculated from the document is sent to the smart card, whose CPU encrypts the hash using the stored private key of the user, and then returns the encrypted hash. Typically, a user must activate his smart card by entering a [[personal identification number]] or PIN code (thus providing [[two-factor authentication]]). It can be arranged that the private key never leaves the smart card, although this is not always implemented. If the smart card is stolen, the thief will still need the PIN code to generate a digital signature. This reduces the security of the scheme to that of the PIN system, although it still requires an attacker to possess the card. A mitigating factor is that private keys, if generated and stored on smart cards, are usually regarded as difficult to copy, and are assumed to exist in exactly one copy. Thus, the loss of the smart card may be detected by the owner and the corresponding certificate can be immediately revoked. Private keys that are protected by software only may be easier to copy, and such compromises are far more difficult to detect.

===Using smart card readers with a separate keyboard===
Entering a PIN code to activate the smart card commonly requires a [[numeric keypad]]. Some card readers have their own numeric keypad. This is safer than using a card reader integrated into a PC, and then entering the PIN using that computer's keyboard. Readers with a numeric keypad are meant to circumvent the eavesdropping threat where the computer might be running a [[keystroke logging|keystroke logger]], potentially compromising the PIN code. Specialized card readers are also less vulnerable to tampering with their software or hardware and are often [[Evaluation Assurance Level|EAL3]] certified.

===Other smart card designs===
Smart card design is an active field, and there are smart card schemes which are intended to avoid these particular problems, though so far with little security proofs.

===Using digital signatures only with trusted applications===
One of the main differences between a digital signature and a written signature is that the user does not "see" what he signs. The user application presents a hash code to be encrypted by the digital signing algorithm using the private key. An attacker who gains control of the user's PC can possibly replace the user application with a foreign substitute, in effect replacing the user's own communications with those of the attacker. This could allow a malicious application to trick a user into signing any document by displaying the user's original on-screen, but presenting the attacker's own documents to the signing application.

To protect against this scenario, an authentication system can be set up between the user's application (word processor, email client, etc.) and the signing application. The general idea is to provide some means for both the user application and signing application to verify each other's integrity. For example, the signing application may require all requests to come from digitally signed binaries.


===WYSIWYS===
{{Main|WYSIWYS}}
Technically speaking, a digital signature applies to a string of bits, whereas humans and applications "believe" that they sign the semantic interpretation of those bits. In order to be semantically interpreted, the bit string must be transformed into a form that is meaningful for humans and applications, and this is done through a combination of hardware and software based processes on a computer system. The problem is that the semantic interpretation of bits can change as a function of the processes used to transform the bits into semantic content. It is relatively easy to change the interpretation of a digital document by implementing changes on the computer system where the document is being processed. From a semantic perspective this creates uncertainty about what exactly has been signed. [[WYSIWYS]] (What You See Is What You Sign) <ref name="JPH2001">A. Jøsang, D. Povey and A. Ho. "What You See is Not Always What You Sign". Proceedings of the Australian Unix User Group Symposium (AUUG2002), Melbourne, September 2002. [http://www.unik.no/people/josang/papers/JPH2002-AUUG.pdf PDF]</ref> means that the semantic interpretation of a signed message cannot be changed. In particular this also means that a message cannot contain hidden information that the signer is unaware of, and that can be revealed after the signature has been applied. WYSIWYS is a necessary requirement for the validity of digital signatures, but this requirement is difficult to guarantee because of the increasing complexity of modern computer systems.

===Digital signatures vs. ink on paper signatures===

An ink signature could be replicated from one document to another by copying the image manually or digitally, but to have credible signature copies that can resist some scrutiny is a significant manual or technical skill, and to produce ink signature copies that resist professional scrutiny is very difficult.

Digital signatures cryptographically bind an electronic identity to an electronic document and the digital signature cannot be copied to another document. Paper contracts sometimes have the ink signature block on the last page, and the previous pages may be replaced after a signature is applied. Digital signatures can be applied to an entire document, such that the digital signature on the last page will indicate tampering if any data on any of the pages have been altered, but this can also be achieved by signing with ink and numbering all pages of the contract.

Additionally, most digital certificates provided by certificate authorities to end users to sign documents can be obtained by at most gaining access to a victim's email inbox.

==Some digital signature algorithms==
*[[RSA (algorithm)|RSA]]-based signature schemes, such as [[RSA-PSS]]
*[[Digital Signature Algorithm|DSA]] and its [[elliptic curve cryptography|elliptic curve]] variant [[Elliptic Curve DSA|ECDSA]]
*[[ElGamal signature scheme]] as the predecessor to DSA, and variants [[Schnorr signature]] and [[Pointcheval–Stern signature algorithm]]
*[[Rabin signature algorithm]]
*[[Pairing]]-based schemes such as [[BLS (cryptography)|BLS]]
*[[Undeniable signature]]s
*[[Aggregate signature]] - a signature scheme that supports aggregation: Given n signatures on n messages from n users, it is possible to aggregate all these signatures into a single signature whose size is constant in the number of users. This single signature will convince the verifier that the n users did indeed sign the n original messages.
*[[Signatures with efficient protocols]] - are signature schemes that facilitate efficient cryptographic protocols such as [[zero-knowledge proofs]] or [[secure computation]].

==The current state of use – legal and practical ==
{{Globalize|section|date=November 2009}}
All digital signature schemes share the following basic prerequisites regardless of cryptographic theory or legal provision:

#; Quality algorithms : Some public-key algorithms are known to be insecure, practical attacks against them having been discovered.
#
#; Quality implementations : An implementation of a good algorithm (or [[cryptographic protocol|protocol]]) with mistake(s) will not work.
#
#; The private key must remain private : If the private key becomes known to any other party, that party can produce ''perfect'' digital signatures of anything whatsoever.
#
#; The public key owner must be verifiable : A public key associated with Bob actually came from Bob. This is commonly done using a [[public key infrastructure]] (PKI) and the public key<math>\leftrightarrow</math>user association is attested by the operator of the PKI (called a [[certificate authority]]). For 'open' PKIs in which anyone can request such an attestation (universally embodied in a cryptographically protected [[identity certificate]]), the possibility of mistaken attestation is nontrivial. Commercial PKI operators have suffered several publicly known problems. Such mistakes could lead to falsely signed, and thus wrongly attributed, documents. 'Closed' PKI systems are more expensive, but less easily subverted in this way.
#
#; Users (and their software) must carry out the signature protocol properly.



Only if all of these conditions are met will a digital signature actually be any evidence of who sent the message, and therefore of their assent to its contents. Legal enactment cannot change this reality of the existing engineering possibilities, though some such have not reflected this actuality.

Legislatures, being importuned by businesses expecting to profit from operating a PKI, or by the technological avant-garde advocating new solutions to old problems, have enacted statutes and/or regulations in many jurisdictions authorizing, endorsing, encouraging, or permitting digital signatures and providing for (or limiting) their legal effect. The first appears to have been in [[Utah]] in the United States, followed closely by the states [[Massachusetts]] and [[California]]. Other countries have also passed statutes or issued regulations in this area as well and the UN has had an active model law project for some time. These enactments (or proposed enactments) vary from place to place, have typically embodied expectations at variance (optimistically or pessimistically) with the state of the underlying [[cryptographic engineering]], and have had the net effect of confusing potential users and specifiers, nearly all of whom are not cryptographically knowledgeable. Adoption of technical standards for digital signatures have lagged behind much of the legislation, delaying a more or less unified engineering position on [[interoperability]], [[algorithm]] choice, [[key length]]s, and so on what the engineering is attempting to provide.

:''See also'': [[ABA digital signature guidelines]]

==Industry standards==
{{cleanup|section|date=April 2011}}
Some industries have established common interoperability standards for the use of digital signatures between members of the industry and with regulators. These include the [[Automotive Network Exchange]] for the automobile industry and the [[SAFE-BioPharma Association]] for the healthcare industry.

===Using separate key pairs for signing and encryption===
In several countries, a digital signature has a status somewhat like that of a traditional pen and paper signature, like in the [http://europa.eu/legislation_summaries/information_society/l24118_en.htm EU digital signature legislation]. Generally, these provisions mean that anything digitally signed legally binds the signer of the document to the terms therein. For that reason, it is often thought best to use separate key pairs for encrypting and signing. Using the encryption key pair, a person can engage in an encrypted conversation (''e.g.'', regarding a real estate transaction), but the encryption does not legally sign every message he sends. Only when both parties come to an agreement do they sign a contract with their signing keys, and only then are they legally bound by the terms of a specific document. After signing, the document can be sent over the encrypted link. If a signing key is lost or compromised, it can be revoked to mitigate any future transactions. If an encryption key is lost, a backup or [[key escrow]] should be utilized to continue viewing encrypted content. Signing keys should never be backed up or escrowed.

==See also==
* [[Detached signature]]
* [[Digital certificate]]
* [[Public key infrastructure]]
* [[Digital signatures and law]]
* [[Electronic signature]]
* [[Global Trust Center]]
* [[GNU Privacy Guard]]
* [[21 CFR 11]]
* [[Electronic lab notebook]]
* [[Server-based signatures]]
* [[PAdES]]

==Notes==
<references/>

==Further reading==
* J. Katz and Y. Lindell, "Introduction to Modern Cryptography" (Chapman & Hall/CRC Press, 2007)
* Stephen Mason, Electronic Signatures in Law (3rd edition, Cambridge University Press, 2012)
* Lorna Brazell, Electronic Signatures and Identities Law and Regulation (2nd edn, London: Sweet & Maxwell, 2008);
* Dennis Campbell, editor, E-Commerce and the Law of Digital Signatures (Oceana Publications, 2005).
* M. H. M Schellenkens, Electronic Signatures Authentication Technology from a Legal Perspective, (TMC Asser Press, 2004).
* Jeremiah S. Buckley, John P. Kromer, Margo H. K. Tank, and R. David Whitaker, The Law of Electronic Signatures (3rd Edition, West Publishing, 2010).
* [http://journals.sas.ac.uk/deeslr/ ''Digital Evidence and Electronic Signature Law Review''] Free open source
* [https://trust1t.atlassian.net/wiki/display/OCSPublic/01.+Digital+signatures+in+PDF OpenCertificateSigning server] Open-source PDF Signing (EU, ETSI)

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Talk:Binomial coefficient

2014-01-04T10:37:10Z

71.234.213.31:

39 yrs old Senior Non-Commissioned Support Force Member Gibler from Victoria, has many hobbies which include country music, Silver Jewellery and bee keeping. Has signed up for a world contiki trip. Is incredibly excited specifically about going to Jewish Quarter and St Procopius' Basilica in Trebíc. my webpage; [http://www.pinterest.com/lizzielanejewel/hultquist-jewellery/ Hultquist]

Vector-valued function

2013-10-27T23:11:43Z

71.234.3.238: /* Partial derivative */

The '''Selberg zeta-function''' was introduced by {{harvs|txt|authorlink=Atle Selberg|first=Atle |last=Selberg|year=1956}}. It is analogous to the famous [[Riemann zeta function]]
:<math> \zeta(s) = \prod_{p\in\mathbb{P}} \frac{1}{1-p^{-s}} </math>
where <math> \mathbb{P} </math> is the set of prime numbers. The Selberg zeta-function uses the lengths of simple [[closed geodesic]]s instead of the primes numbers.　If <math>\Gamma</math> is a subgroup of SL(2,'''R''') Selberg zeta function is defined as follows,
:<math>\zeta_\Gamma(s)=\prod_p(1-N(p)^{-s})^{-1},</math>
or
:<math>Z_\Gamma(s)=\prod_p\prod^\infty_{n=0}(1-N(p)^{-s-n}),</math>
where p run all over the prime congruent class and N(p) is the norm of congruent class p, which is square of the bigger eigenvalue of p.

For any [[hyperbolic manifold | hyperbolic surface]] of finite area there is an associated '''Selberg zeta-function'''; this function is a [[meromorphic function]] defined in the [[complex plane]]. The zeta function is defined in terms of the closed [[geodesic]]s of the surface.

The zeros and poles of the Selberg zeta-function, ''Z''(''s''), can be described in terms of spectral data of the surface.

The zeros are at the following points:
# For every cusp form with eigenvalue <math>s_0(1-s_0)</math> there exists a zero at the point <math>s_0</math>. The order of the zero equals the dimension of the corresponding eigenspace. (A cusp form is an eigenfunction to the [[Laplace-Beltrami operator]] which has [[Fourier expansion]] with zero constant term.)
# The zeta-function also has a zero at every pole of the determinant of the scattering matrix, <math> \phi(s) </math>. The order of the zero equals the order of the corresponding pole of the scattering matrix.

The zeta-function also has poles at <math> 1/2 - \mathbb{N} </math>, and can have zeros or poles at the points <math> - \mathbb{N} </math>.

== Selberg zeta-function for the modular group ==
For the case where the surface is <math> \Gamma \backslash \mathbb{H}^2 </math>, where <math> \Gamma </math> is the [[modular group]], the Selberg zeta-function is of special interest. For this special case the Selberg zeta-function is intimately connected to the [[Riemann zeta function|Riemann zeta-function]].

In this case the determinant of the [[scattering matrix]] is given by:
:<math> \varphi(s) = \pi^{1/2} \frac{ \Gamma(s-1/2) \zeta(2s-1) }{ \Gamma(s) \zeta(2s) }. </math>

In particular, we see that if the Riemann zeta-function has a zero at <math>s_0</math>, then the determinant of the scattering matrix has a pole at <math>s_0/2</math>, and hence the Selberg zeta-function has a zero at <math>s_0/2</math>.

==References==
*{{Citation | last1=Fischer | first1=Jürgen | title=An approach to the Selberg trace formula via the Selberg zeta-function | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-15208-8 | doi=10.1007/BFb0077696 | id={{MathSciNet | id = 892317}} | year=1987 | volume=1253}}
*{{Citation | last1=Hejhal | first1=Dennis A. | title=The Selberg trace formula for PSL(2,R). Vol. I | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics, Vol. 548 | doi=10.1007/BFb0079608 | id={{MathSciNet | id = 0439755}} | year=1976 | volume=548}}
*{{Citation | last1=Hejhal | first1=Dennis A. | title=The Selberg trace formula for PSL(2,R). Vol. 2 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-12323-1 | doi=10.1007/BFb0061302 | id={{MathSciNet | id = 711197}} | year=1983 | volume=1001}}
* [[Henryk Iwaniec|Iwaniec, H.]] Spectral methods of automorphic forms, American Mathematical Society, second edition, 2002.
*{{Citation | last1=Selberg | first1=Atle | title=Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series | id={{MathSciNet | id = 0088511}} | year=1956 | journal=J. Indian Math. Soc. (N.S.) | volume=20 | pages=47–87}}
* Venkov, A. B. Spectral theory of automorphic functions. Proc. Steklov. Inst. Math, 1982.

[[Category:Zeta and L-functions]]
[[Category:Spectral theory]]

CoBoosting

2013-10-19T18:38:46Z

71.234.176.233: Add link to Named Entity Recognition in Wikipedia

In [[algebraic topology]], a branch of [[mathematics]], the '''calculus of functors''' or '''Goodwillie calculus''' is a technique for studying [[functors]] by approximating them by a sequence of simpler functors; it generalizes the [[sheafification]] of a [[presheaf]]. This sequence of approximations is formally similar to the [[Taylor series]] of a [[smooth function]], hence the term "''calculus'' of functors".

Many objects of central interest in algebraic topology can be seen as functors, which are difficult to analyze directly, so the idea is to replace them with simpler functors which are sufficiently good approximations for certain purposes.
The calculus of functors was developed by Thomas Goodwillie in a series of three papers in the 1990s and 2000s,<ref>T. Goodwillie, Calculus I: The ﬁrst derivative of pseudoisotopy theory, K-theory 4 (1990), 1-27.</ref><ref>T. Goodwillie, Calculus II: Analytic functors, K-theory 5 (1992), 295-332.</ref><ref>T. Goodwillie, Calculus III: Taylor series, Geom. Topol. 7 (2003), 645-711.</ref> and has since been expanded and applied in a number of areas.

== Examples ==
A motivational example, of central interest in [[geometric topology]], is the functor of [[embedding]]s of one [[manifold]] ''M'' into another manifold ''N,'' whose first derivative in the sense of calculus of functors is the functor of [[Immersion (mathematics)|immersions]]. As every embedding is an immersion, one obtains an inclusion of functors <math>\mathrm{Emb}(M,N) \to \mathrm{Imm}(M,N)</math> – in this case the map from a functor to an approximation is an inclusion, but in general it is simply a map.

As this example illustrates, the linear approximation of a functor (on a topological space) is its [[sheafification]], thinking of the functor as a [[presheaf]] on the space (formally, as a functor on the category of open subsets of the space), and sheaves are the linear functors.

This example was studied by Goodwillie and Michael Weiss.<ref>M. Weiss, Embeddings from the point of view of immersion theory, Part I, [[Geometry and Topology]] 3 (1999), 67-101.</ref><ref>T. Goodwillie and M. Weiss, Embeddings from the point of view of immersion theory, Part II, [[Geometry and Topology]] 3 (1999), 103-118.</ref>

== Definition ==
Here is an analogy: with the Taylor series method from calculus, you can approximate the shape of a [[smooth function]] ''f'' around a point ''x'' by using a sequence of increasingly accurate polynomial functions. In a similar way, with the calculus of functors method, you can approximate the behavior of certain kind of ''functor'' ''F'' at a particular object ''X'' by using a sequence of increasingly accurate polynomial ''functors''.

To be specific, let ''M'' be a [[smooth manifold]] and let ''O(M)'' be the category of open subspaces of ''M''—i.e. the category where the objects are the open subspaces of ''M'', and the morphisms are [[inclusion map]]s. Let ''F'' be a [[Functor#Covariance_and_contravariance|contravariant]] functor from the category ''O(M)'' to the category '''Top''' of topological spaces with continuous morphisms. This kind of functor, called a '''Top'''-valued [[presheaf]] on ''M'', is the kind of functor you can approximate using the calculus of functors method; you want to know what sort of topological space ''F(X)'' is for a particular open set ''X∈O(M)'', so you study the topology of the increasingly accurate approximations ''F0(X), F1(X), F2(X),'' and so on.

In the calculus of functors method, the sequence of approximations consists of (1) functors ''T0F, T1F, T2F'', and so on, as well as (2) [[natural transformations]] ''ηk:F → TkF'' for each integer ''k''. These natural transforms are required to be compatible, meaning that the composition <math>F \to T_{k+1}F \to T_kF</math> equals the map <math>F \to T_kF,</math> and thus form a tower <math>F \to \cdots \to T_{k+1}F \to T_kF \to \cdots T_1F \to T_0F,</math> and can be thought of as "successive approximations", just as in a Taylor series one can progressively discard higher order terms.

The approximating functors are required to be "''k''-[[Excision theorem|excisive]]" – such functors are called '''polynomial functors'''{{anchor|polynomial functor}} by analogy with [[Taylor polynomials]] – which is a simplifying condition, and roughly means that they are determined by their behavior around ''k'' points at a time, or more formally are [[Sheaf (mathematics)|sheaves]] on the [[configuration space]] of ''k'' points in the given space. The difference between the ''k''th and <math>(k-1)</math>st functors is a "homogeneous functor of degree ''k''" (by analogy with [[homogeneous polynomial]]s), which can be classified.

For the functors <math>T_kF</math> to be approximations to the original functor ''F,'' the resulting approximation maps <math>F \to T_kF</math> must be [[n-connected|''n''-connected]] for some number ''n,'' meaning that the approximating functor approximates the original functor "in dimension up to ''n''"; this may not occur. Further, if one wishes to reconstruct the original functor, the resulting approximations must be ''n''-connected for ''n'' increasing to infinity. One then calls ''F'' an '''analytic functor''', {{anchor|analytic functor}} and says that "the Taylor tower converges to the functor", in analogy with Taylor series of an analytic function.

== Branches ==
There are three branches of the calculus of functors, developed in the order:
* manifold calculus, such as embeddings,
* homotopy calculus, and
* orthogonal calculus.
Homotopy calculus has seen far wider application than the other branches.

== History ==
The notion of a sheaf and sheafification of a presheaf date to early category theory, and can be seen as the linear form of the calculus of functors. The quadratic form can be seen in the work of [[André Haefliger]] on [[Link (knot theory)|links]] of spheres in 1965, where he defined a "metastable range" in which the problem is simpler.<ref>{{citation
|title=Enlacements de sphères en codimension supérieure à 2
|first=André
|last=Haefliger
|author-link=André Haefliger}}</ref> This was identified as the quadratic approximation to the embeddings functor in Goodwillie and Weiss.

== References ==
<references/>
* {{citation
|title=Syllabus for Math 283: Calculus of Functors
|first=Brian
|last=Munson
|year=2005
|url=http://math.stanford.edu/~munson/functorcalculus/syllabus.pdf
}}

== External links ==
* [http://www.math.brown.edu/faculty/goodwillie.html Thomas Goodwillie]
* [http://www.math.wayne.edu/~klein/ John Klein]
* [http://wwwmath.uni-muenster.de/u/mweis_02/ Michael S. Weiss]

{{DEFAULTSORT:Calculus Of Functors}}
[[Category:Algebraic topology]]
[[Category:Functors]]

Schrödinger equation

2006-09-15T23:23:35Z

71.234.249.152: /* Probability currents */

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