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In [[mathematics]], the '''Ihara zeta-function''' is a [[zeta function]] associated with a finite [[Graph (mathematics)|graph]]. It closely resembles the [[Selberg zeta function|Selberg zeta-function]], and is used to relate closed paths to the spectrum of the [[adjacency matrix]]. The Ihara zeta-function was first defined by [[Yasutaka Ihara]] in the 1960s in the context of [[discrete group|discrete subgroups]] of the two-by-two [[p-adic number|p-adic]] [[special linear group]]. [[Jean-Pierre Serre]] suggested in his book ''Trees'' that Ihara's original definition can be reinterpreted graph-theoretically. It was [[Toshikazu Sunada]] who put this suggestion into practice (1985). A [[regular graph]] is a [[Ramanujan graph]] if and only if its Ihara zeta function satisfies an analogue of the [[Riemann hypothesis]].<ref>Terras (1999) p.678</ref>
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==Definition==
 
The Ihara zeta-function can be defined by a formula analogous to the Euler product for the [[Riemann zeta function]]:
 
:<math>\frac{1}{\zeta_G(u)} = \prod_{p} ({1 - u^{L(p)}})</math>
 
This product is taken over all prime walks ''p'' of the graph <math> G = (V, E) </math> - that is, closed cycles <math>p = (u_0, \cdots, u_{L(p)-1}, u_0)</math>  such that
 
:<math> (u_i, u_{(i+1)\bmod L(p)}) \in E~; \quad u_i \neq u_{(i+2) \bmod L(p)~}, </math>
 
and <math> L(p) </math> is the length of cycle ''p'', as used in the formulae above.<ref>Terras (2010) p.12</ref>
 
==Ihara's formula==
Ihara showed that for regular graphs the zeta function is a rational function. If ''G'' is ''k''-regular with [[adjacency matrix]] ''A'' then<ref>Terras (1999) p.677</ref>
 
:<math>\zeta_G(u) = \frac{(1-u^2)^{\chi(G)-1}}{\det(I - Au + (k-1)u^2I)} \  </math>
 
where χ is the [[circuit rank]].
 
The Ihara zeta-function is in fact always the reciprocal of a polynomial:
 
:<math>\zeta_G(u) = \frac{1}{\det (I-Tu)}~,</math>
 
where ''T'' is Hashimoto's edge adjacency operator. [[Hyman Bass]] gave a determinant formula involving the adjacency operator.
 
==Applications==
The Ihara zeta function plays an important role in the study of [[free group]]s, [[spectral graph theory]], and [[dynamical systems]], especially [[symbolic dynamics]], where the Ihara zeta function is an example of a [[Ruelle zeta function]].<ref name=T29>Terras (2010) p.29</ref>
 
== References ==
{{reflist}}
* {{cite journal | first=Yasutaka | last=Ihara | title=On discrete subgroups of the two by two projective linear group over <math>{\mathfrak p}</math>-adic fields | journal=J. Math. Soc. Japan | volume=18 | year=1966 | pages=219–235 | zbl=0158.27702 }}
* {{cite book | first=Toshikazu | last=Sunada | authorlink=Toshikazu Sunada |title=Curvature and Topology of Riemannian Manifolds | volume=1201 | year=1986 | pages=266–284 | doi=10.1007/BFb0075662 | chapter=L-functions in geometry and some applications | series=Lecture Notes in Mathematics | isbn=978-3-540-16770-9 | zbl=0605.58046 }}
* {{cite journal | first=H. | last=Bass | authorlink=Hyman Bass | title=The Ihara-Selberg zeta function of a tree lattice | journal=International. J. Math.| volume=3 | year=1992 | pages=717–797 | doi=10.1142/S0129167X92000357 | issue=6 | zbl=0767.11025 }}
* {{cite book | first=Harold M. | last=Stark | authorlink=Harold Stark | chapter=Multipath zeta functions of graphs | pages=601-615 | title=Emerging Applications of Number Theory | editor1-first=Dennis A. | editor1-last=Hejhal | editor1-link=Dennis Hejhal | editor2-first=Joel | editor2-last=Friedman | editor3-first=Martin C. | editor3-last=Gutzwiller | editor3-link=Martin Gutzwiller | editor4-first=Andrew M. | editor4-last=Odlyzko| editor4-link=Andrew Odlyzko | publisher=Springer | year=1999 | isbn=0-387-98824-6 | zbl=0988.11040 | series=IMA Vol. Math. Appl. | volume=109 }}
* {{cite book | first=Audrey | last=Terras | authorlink=Audrey Terras | chapter=A survey of discrete trace formulas | pages=643-681 | title=Emerging Applications of Number Theory | editor1-first=Dennis A. | editor1-last=Hejhal | editor1-link=Dennis Hejhal | editor2-first=Joel | editor2-last=Friedman | editor3-first=Martin C. | editor3-last=Gutzwiller | editor3-link=Martin Gutzwiller | editor4-first=Andrew M. | editor4-last=Odlyzko| editor4-link=Andrew Odlyzko | publisher=Springer | year=1999 | isbn=0-387-98824-6 | zbl=0982.11031| series=IMA Vol. Math. Appl. | volume=109 }}
* {{cite book | title=Zeta Functions of Graphs: A Stroll through the Garden | volume=128 | series=Cambridge Studies in Advanced Mathematics | first=Audrey | last=Terras | authorlink=Audrey Terras | publisher=[[Cambridge University Press]] | year=2010 | isbn=0-521-11367-9 | zbl=1206.05003 }}
 
 
[[Category:Zeta and L-functions]]
[[Category:Algebraic graph theory]]
 
 
{{algebra-stub}}
{{numtheory-stub}}

Latest revision as of 04:41, 13 January 2015

Although these conditions have defined characteristics, they all share a common theme which is dread and fear that is much greater than the situation calls for. Workouts helps you burn weight thereby keeping you fit and healthy. Sometimes, they will confide in their doctors or very close friends. It is normal for a child to have a certain amount of anxiety as part of their healthy development. One should make it a point to try and spot the actual cause that brings about such a condition prior to treating the problem.

WHY DO YOU IGNORE WARNING SIGNS & HINTS WHICH YOU MUST KNOW BY HEART ALREADY. And these therapy consequences are not dangerous by any means, you're just loosing fat, not loosing electrolytes. If, for example, a child lives in a home where they are exposed to math, they will grow up and most likely acquire good math skills. It lowers on consumption of serotonin within the brain and this in improves for the absorption of serotonin in your body. Herbs for Anxiety are very useful in treating acute symptoms of stress and anxiety, but you need to supplement them with other types of coping mechanisms.

There are four totally different variations of anxiety disorder treatment prescription drugs: Benzodiazepines, SSRIs, SNRIs and Tricyclic antidepressants. Dogs that are well exercised spend more of their idle time resting rather than panicking. There are definite symptoms of anxiety, but there is nothing that will show up on a brain scan. Some of the symptoms of a heart attack are chest tightness and tingling in the extremities. But, yes, hypertension is on the rise in our society, but this is largely due to diet issues.

Deep breathing is considered a far healthier way of breathing and encourages your body to improve its overall oxygen flow, which in turn cuts down on anxiety related symptoms. Nothing contributes more to anxiety like “what if” statements, when in reality we have to concentrate on “what is”. Connect with others with similar health concerns and issues. ), what can we possibly do to return to 'normal' levels of anxiety and get on with our lives. For school to be a pleasant and successful experience for your child with Autism being prepared means so much more than acquiring the material possessions that are required.

When you live with anxiety attacks, information on how to deal with them can be beneficial to your mental and physical health. He also worries about an upcoming math test, and considers getting "sick" so he can miss it. The first step to anxiety hypnosis is finding a quiet place of ease where you are comfortable and feel safe. You can't think straight and you become utterly self conscious. They are hence useful for situations where the user is aware of the possibility of panic attacks occurring, such as before public speaking.

If you adored this article and you would like to obtain more info pertaining to dealing with depression and anxiety nicely visit the site.

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