**Topics, I**

**i, Imaginary Unit**
> s.a. Euler's Equation.

@ __References__: Nahin 98 [I, history].

**Ice**
> see water.

**IceCube Detector**
> s.a. neutrino experiments;
astrophysical neutrinos.

* __Idea__: A 1-km^{3}
neutrino telescope currently (2010) under construction at the South Pole,
consisting of 5160 optical sensors deployed at depths between 1450 m and
2450 m in clear Antarctic ice distributed over 86 strings; An air shower
array covering a surface area of 1 km^{2}
above the in-ice detector will measure cosmic-ray air showers in the
energy range from 300 TeV to above 1 EeV.

@ __General references__: Karle NIMA(06)ap-conf;
Desiati ap/06-proc;
Rott NPPS(08)ap/06;
Waldenmaier NIMA(08)a0802-conf;
Karle NIMA(09)-a0812-proc [detector];
DeYoung MPLA(09)-a0906;
Halzen a0910-conf;
Hultqvist et IceCube NIMA(11)-a1003-conf [status];
Karle a1003-proc;
Kappes et IceCube AIP(10)-a1007;
news guardian(11)jan;
Gaisser a1108-proc [status and results];
> s.a. dark matter.

@ __Results__: Desiati a0812-proc [physics];
D'Agostino PhD(09)-a0910 [evidence for atmospheric-neutrino-induced cascades];
DeYoung eConf-a0910;
Karg et IceCube ASST(11)-a1011 [initial results];
Demirörs et al APP(11)-a1106 [supernova detection];
Helbing et IceCube a1107-proc [and physics beyond the standard model];
Böser et IceCube a1205-proc [neutrinos and cosmic rays];
Sullivan et IceCube NPPB(13)-a1210;
Taboada MPLA(12) [particle astrophysics, rev];
Aartsen et al PRL(13) [bound on muon neutrinos from WIMP annihilations in the Sun];
Karle et IceCube a1401-proc.

> __Online resources__:
see IceCube website;
Wikipedia page.

**Icosahedral Group**

@ __References__: Cesare & Del Duca RNC(87).

**Ideal Elements of a Physical Theory**
> see physical theories.

**Ideal Gas** > s.a. thermodynamics.

**Ideal of a Ring / Algebra**

$ __For a ring__: A
submodule of a ring *R* considered as an *R*-module.

$ __For an algebra__:
A subspace *I* of an algebra *A* is a left (right) ideal
if it is invariant under left (resp, right) multiplication by any element
of *A*, *AI* ⊂ *I* or *IA* ⊂ *I*;
In other words, for all *i* in *I* and *a* in *A*,
*ai* (resp, *ia*) belongs to *I*.

> __Online resources__:
see Wikipedia page.

**Idealizations** > see Models.

**Identical Particles** > see particle statistics.

**Identities (Mathematical Relations)**

> __In mathematics__:
see bessel functions; Bianchi Identities [for
curvature]; combinatorics; Elliptic Functions;
Gauss' Theorem; Hypergeometric Function;
integration on manifolds [Stokes' Theorem]; Schläfli
Formula; tensors [including Lovelock identity]; vector
calculus [differential and integral identities, including Green identities].

> __In physics__:
see Feynman Diagrams [shuffling identities]; Fierz
Identities; Gamma Matrices; Mandelstam Identities;
thermodynamics [fundamental identity]; Ward and
Ward-Takahashi Identities.

**Identity of Indiscernibles**
> s.a. particle statistics.

* __Idea__: A.k.a. Leibniz
principle; If two systems are qualitatively identical then they are
logically identical; It is violated by indistinguishable quantum
particles.

@ __And quantum particles__:
Castellani & Mittelstaedt FP(00) [in classical and quantum physics];
Huggett in(03)qp/02;
Ladyman & Bigaj PhSc(10)jan;
Caulton PhSc(13) [in quantum mechanics];
Acacio de Barros et al a1906 [new ontology];
Spekkens a1909 [Leibniz's
methodological principle and its significance in the work of Einstein].

> __Online resources__:
see Wikipedia page.

**i ε Term in Field Theory**

@

**Ill-Posed Problem**
> see Well-Posed Problem.

**I-Love-Q Relations**
> see neutron stars.

**Image Charge**
> see Method of Images.

**Immirzi Parameter**
> s.a. connection formulation of general
relativity and quantum gravity; Holst
Action; yang-mills gauge theory.

* __Idea__: (Also known
as Barbero-Immirzi parameter.) A parameter whose value is an ambiguity in
the connection formulation of general relativity and the quantization
procedure underlying the loop approach to quantum gravity.

* __Value__: 2003,
Dreyer proposed that the Immirizi parameter be fixed by letting the *j*
= 1 transitions of spin networks be the dominant processes contributing to
the black hole area, considering the asymptotic quasinormal modes spectrum
of a black hole (as opposed to the expected *j* = 1/2 transitions).

@ __General references__: Schücker pr(88)-a0906 [Ashtekar variables without spinors];
Immirzi CQG(97)gq/96;
Rovelli & Thiemann PRD(98)gq/97;
Krasnov CQG(98)gq/97;
Corichi & Krasnov MPLA(98);
Barros e Sá IJMPD(01)gq/00;
Samuel PRD(01);
Mena CQG(02)gq [not local];
Pérez & Rovelli PRD(06)gq/05 [physical effects];
Chou et al PRD(05)gq [meaning, scalar vs pseudo-scalar];
Fatibene et al CQG(07)-a0706 [action];
Liko CQG(12)-a1111 [conditions for physical effects in Euclidean quantum gravity];
Fatibene et al a1206
[in different dimensions and signatures];
Geiller & Noui GRG(13)-a1212 [Holst action and the covariant torsion tensor];
Perlov & Bukatin a1510
[as a solution of the simplicity constraints];
> s.a. Conformal Gravity;
models in canonical gravity [Dirac fields,
Immirzi parameter as local field]; regge calculus.

@ __And black-hole physics__: Krasnov CQG(99)gq [rotating black holes];
Rainer G&C(00)gq/99,
Garay & Mena PRD(02)gq,
Dreyer PRL(03)gq/02 [entropy];
Oppenheim PRD(04)gq/03 [quasinormal modes];
Domagała & Lewandowski CQG(04)gq [entropy];
Sadiq PLB(15)-a1410;
Zhang a1506
[quasinormal modes, 4 or more dimensions];
> s.a. black-hole entropy.

@ __Topological interpretation__: Date et al PRD(09)-a0811;
Mercuri a0903-conf;
Sengupta a0904-wd,
CQG(10)-a0911 [and wave function rescaling];
Mercuri & Randono CQG(11)-a1006 [as instanton angle];
El Naschie G&C(13);
Sengupta PRD(13)-a1304.

@ __In quantum gravity__: Immirzi NPPS(97)gq,
CQG(97)gq/96;
Rovelli & Thiemann PRD(98)gq/97;
Gambini et al PRD(99)gq/98 [Yang-Mills version];
Samuel PRD(01);
Garay & Mena PRD(02);
Mena CQG(02);
Mercuri PRD(08)-a0708 [and large gauge transformations];
Benedetti & Speziale JPCS(12)-a1111 [perturbative renormalization];
Dittrich & Ryan CQG(13)-a1209 [in discrete quantum gravity];
Charles & Livine PRD(15)-a1507 [as a cutoff].

@ __Related topics__: Açık & Ertem a0811 [effect of gup];
Broda & Szanecki PLB(10)-a1002 [derivation from the standard model];
Ellis & Mavromatos PRD(11)-a1108 [spacetime foam and supersymmetry breaking];
de Berredo-Peixoto et al JCAP(12)-a1205 [with torsion and Dirac fields, cosmology];
Panza et al PRD(14)-a1405 [and TeV-scale particle physics];
Sadiq a1510 [and the holographic principle];
Wong IJMPD(17)-a1701 [and the linking theory of shape dynamics];
Abreu et al a1808 [and Kaniadakis statistics];
> s.a. Gauge Theory of Gravity.

**Implicit Function Theorem**

**Impulsive Waves**
> see gravitational wave solutions.

**Impurities**

@ __References__: Lemeshko PRL(17)
+ Shchadilova Phy(17) [quasiparticle approach, angulons];
Liu et al a1805 [impurity particle dynamics at finite temperature].

**Incidence Algebra **
> s.a. posets.

* __Idea__: Given any locally finite
poset *P*, the incidence algebra *I*(*P*) (over \(\mathbb C\),
say) is the vector space of functions *f* : *S*(*P*) →
\(\mathbb C\), where *S*(*P*) is the set of intervals [*x*,
*y*] ≠ Ø, made into an associative algebra by the multiplication
(convolution) *fg*([*x*, *y*]):= ∑_{z
∈ [x,y]} *f*([*x*, *z*])
*g*([*z*, *y*]); The identity is δ([*x*, *y*])
= δ(*x*, *y*).

@ __References__: Sorkin MPLA(03)m.CO-proc.

> __Online resources__:
see Wikipedia page.

**Incidence Geometry**

* __Idea__: Its main areas
are projective and affine geometry and, in more recent times, the
theory of buildings and polar spaces.

@ __References__: Buekenhout ed-95 [handbook];
Ueberberg 11;
De Bruyn 16.

**Incidence Matrix**

@ __References__: Hodge 52.

**Incompleteness Theorem**
> see under Gödel's Theorem.

**Independence** > s.a. matroids.

* __Remark__: The
abstract notion has been formalized in the theory of matroids.

> __Specific notions__:
see affine structures [geometrically
independent points]; graphs [independent set];
vectors [linear independence].

**Index of a Critical Point of a Function f**

$

**Index of an Elliptic Operator D**

$

**Index of Refraction** > see refraction.

**Index of a Vector Field**

$ __Def__: If *x*
is a zero of a vector field on an *n*-dimensional manifold *M*,
the index of *v* at *x* is the degree of the map S^{n−1}
→ S^{n−1} defined by *v*
(normalized with some flat metric) on a small sphere surrounding *x*.

* __Properties__:
ind_{x}(−*v*)
= (−1)^{n}
ind_{x}(*v*).

**Index Theorem** (Atiyah-Singer)
> s.a. anomaly; fixed-point theorems.

* __Idea__: A result in
geometric analysis which relates the number of zero modes (in general
the index) of an elliptic differential operator *D* on a closed
manifold *M* to characteristic classes of the tangent bundle of *M*
and of the vector bundles on which *D* acts; "Basically a formula
that counts the number of solutions to another equation" (M Atiyah 2004,
on receiving the Abel Prize); "A cornerstone of maths, it is one of the
most fundamental results of the last 50 years" (Elmer Rees); "An index
theorem relates the difference of the numbers of solutions of two
differential equations to the topological structure of the manifold or
bundle concerned, sometimes using the heat kernels of two higher-order
differential operators as an intermediary".

* __Special cases__:
The Hirzebruch signature theorem, the Riemann-Roch theorem.

@ __General references__: Shanahan 77;
Gilkey 84; Booss & Bleecker 85;
Esposito gq/95-ln.

@ __Special manifolds__: Niemi & Semenoff pr(85) [infinite];
Peeters & Waldron JHEP(99)ht [with boundary].

@ __Generalizations__: Longo CMP(01) [quantum];
Harikumar et al JPA(07)ht/06 [*q*-deformed fuzzy sphere];
> s.a. quantum graphs.

**Indices** > see tensors.

**Indistinguishable Objects**
> s.a. Identical Particles.

* __Idea__:
Indistinguishable objects are identical objects (objects all of whose
observable properties are the same) that cannot be distinguished even in
principle.

@ __References__: Aerts et al IJTP(15)-a1410 [and human thought];
Saunders a1609-ch [and the notion of object];
Karczewski et al PRA(19)-a1812 [multipartite].

**Individuality / Individuation**

@ __In quantum mechanics__: Pešić 02;
Sant'Anna qp/04 [particles];
Jaeger FP(10) [two physical approaches];
de Ronde et al a1203
[and the Kochen-Specker theorem and the principle of superposition];
Ghirardi SHPMP(13) [and collapse];
Pylkkänen et al a1405 [in Bohm's approach];
in Kastner a1707-talk [types].

> __Different forms__:
see Identical Particles; Indistinguishability.

> __Related topics__:
see foundations of quantum mechanics [individual particles]; particle statistics [identical particles]; Quasiset Theory.

**Induced Gravity** > s.a. bianchi
models, and bianchi IX; gravity.

* __Idea__: Gravity is not
fundamental, but becomes dynamical as a result of quantum effects in the system
of heavy constituents of an underlying theory, electromagnetic or other.

* __Example__: Sakharov's theory
of gravity as a long-range Casimir force [@ NS(81)apr,
NS(90)jul28].

* __Drawbacks__: The biggest
problem is that it was shown that *G* is not calculable.

@ __General references__: Sakharov Dokl(67);
Adler PLB(80) [formula for *G*],
RMP(82);
Puthoff PRA(89) [stochastic electrodynamics];
Haisch et al phy/98-conf;
Barceló et al IJMPD(01)gq [based on general relativity analogs];
Visser MPLA(02)gq [status];
Chernitskii G&CS(02)gq-conf [from non-linear electrodynamics];
Einhorn & Jones JHEP(16)-a1511 [single scalar field].

@ __Other dynamical origin__: Dhar NPB(97) [*c* = 1 matrix model];
Laughlin IJMPA(03)gq-fs [emergent];
Wetterich PRD(04)ht/03 [from spinors];
Kan & Shiraishi PTP(04)gq/03;
Makhlin hp/04/PRL [Dirac field];
Azri CQG-a1808
[from affine connection and scalar fields];
> s.a. higgs mechanism [gravitational],
Stochastic Gravity.

@ __And cosmology__: Davidson & Gurwich PoS-gq/06 [dark matter];
Cerioni et al PLB(09)-a0906 [inflation and reheating];
> s.a. minisuperspace quantum cosmology.

**Induction, Electromagnetic**
> see Faraday's Law.

**Induction, Mathematical**

* __Analogy__: When subscribing
to a newspaper, say (i) Deliver it tomorrow; (ii) If you deliver it one day,
make sure you deliver it the following day (R Smullyan).

**Inductive Family or System**
> see sequences.

**Inductive Limit**
> see limit.

**Inequalities**

> __In Mathematics__:
see mathematical inequalities.

> __In classical physics__:
see angular momentum; black-hole geometry.

> __In quantum mechanics__:
see bell's inequalities; CHSH
Inequalities; Wigner Inequality;
states in quantum mechanics.

> __In quantum field
theory__: see Bogomolny Inequality;
effects in quantum field theory.

**Inertia** (including inertial frame, observer)
> s.a. Moment of Inertia.

**Infeld-van der Waerden Symbols**
> see Soldering Form.

**Inference**
> s.a. probability; statistics.

@ __References__: Helland book-a1206 [unified scientific basis].

**Infinite**
> s.a. Cardinality; Denumerability;
Hilbert's Hotel; non-standard analysis.

* __History__: The actual
infinite was introduced by Cantor around 1871 when studying uniqueness of
trigonometric series for cases with complicated sets of exceptional points.

* __Different infinities__:
*ω* = card Z, *ω* + 1 (notice: ≠ 1
+ *ω* = *ω*), ..., *ω* · 2
(notice: ≠ 2 · *ω* = *ω*); Surreal numbers
lie somewhere between *ω* and *ω* + 1.

@ __General references__: Cantor 15;
Zippin 62; Maor 87;
Berry & Howls PW(93)jun;
Rucker 95;
Vilenkin 95;
Aczel 01;
Barrow 03 [play];
Clegg 03;
Sergeyev 04 [arithmetic];
Barrow 05 [mathematics and physics];
Benci & Di Nasso 18 [and non-standard analysis].

@ __History__: Wallace 03 [r pw(04)apr];
Bussotti & Tapp SHPSA(09) [Spinoza's concept of infinity and Cantor's set theory];
Stillwell 10 [modern ideas and their implications];
news cosmos(17)dec [the controversy around Sergeyev's "grossone"
paper].

@ __And physics__: Donald qp/03 [many-minds and mathematical vs physical "existence"];
Laraudogoitia FP(10) [critique of argument against actual infinity];
Vidotto a1305-conf [infinities as a measure of our ignorance];
Tavakol & Gironi a1604 [use of relative or real constructed infinities in cosmology];
Perlis a1608 [taking infinity seriously].

> __Online resources__:
see Wikipedia page.

**Infinitesimal**
> s.a. Continuum; differential geometry.

* __Idea__: Numbers
that lie between zero and every positive standard number.

* __History__:
Introduced by Leibniz and Newton, they were opposed from the beginning by
Berkeley and have always very controversial; Replaced by limits, they came
back in the 1960s with non-standard analysis; A new approach was proposed
by E Nelson.

@ __References__: Bell MI(95) [and the continuum];
Kanovei et al FoS(13)-a1211 [Connes' criticisms of Robinson's infinitesimals];
Katz & Leichtnam AMM-a1304 [historical rev];
Katz & Mormann ISHPS-a1304 [as an issue in neo-Kantian philosophy of science];
Benci et al Quantum(19)-a1901 [and quantum theory].

> __Online resources__:
see Wikipedia page.

> __As a number__: see non-standard
analysis; types of numbers [extension of the reals].

**Inflation**
> s.a. phenomenology; types of inflation.

**Inflaton**
> see scalar-tensor theories.

**Influence Functional**
> see quantum systems [dissipative].

**Information Geometry / Metric**
> see types of metrics.

**Information Theory**
> s.a. information and physical theories; information
and spacetime/gravity; quantum information.

**Infraparticles**
> see Unparticles.

**Infrared Behavior of Quantum Gravity**
> see covariant quantum gravity.

**Infrared Modifications of Gravity**
> see modified general relativity.

**Infrasound**
> see acoustics.

**Ingarden Space**
> see finsler geometry.

**Inhomogeneity**
> see Homogeneity [in cosmology]; matter.

**Initial Conditions**
> s.a. cosmology and cosmological
models; Dynamics; quantum
cosmology boundary conditions; singularities.

@ __In thermodynamics__: Callender BJPS(04) ["special" initial conditions].

**Initial-Value Formulation / Problem**
> s.a. wave equations;
initial-value formulation for general relativity.

@ __References__: Finster & Grotz a1303 [for causal variational principles].

**Injective Module**
> see types of modules.

**Injectivity Radius**
> see lorentzian geometry.

**Inner Product**
> see vector.

**Instabilities**

@ __References__: in Arnold 78 [beautiful fluid example];
Sorkin ApJ(81),
ApJ(82);
Price AJP(82)apr.

@ __In statistical mechanics__: Simon & Sokal JSP(81) [balance of energy vs entropy].

> __In gravitation__:
see astrophysics; black-hole geometry
[black strings] and perturbations; fluids;
gravitational radiation.

> __In other systems__:
see classical systems [unstable]; dissipation;
solid matter; fluids [including
astrophysics]; geons; Jeans Swindle.

**Instant**
> see state of a system.

**Instantons**
> s.a. gravitational instantons.

**Insulators**
> s.a. electricity [basic laws, electric fields in matter, electric current];
Metals [transition]; solid matter.

* __Dielectric breakdown__:
The sudden decrease in the resistance of an insulator with an applied
electric field, usually accompanied by a spark.

* __Dielectric strength__:
The maximum value of the electric field before dielectric breakdown occurs.

* __Ordinary vs Mott
insulators__: In ordinary insulators every possible electron state is
filled (with two electrons of opposite spin orientation); No electric
current is then possible and the material is insulating; In a Mott
insulator only half the electronic states are occupied, but still no
electric current flows because strong electron repulsions prevent any
electron motion; Properties of Mott insulators are believed to be
important for understanding hight-*T*_{c}
superconductivity in cuprates.

* __Topological insulators__:
Materials which are non-conducting in the bulk, but with a band structure that
gives rise to conducting states along their surface; They are examples of a
topological phase and, by analogy with others, it is expected that they will
exhibit new quantization rules.

@ __Dielectric breakdown__:
Garroni et al PRS(01);
Arrayás & Trueba CP(05) [pre-breakdown streamers];
> s.a. Wikipedia page.

@ __Topological insulators__:
Kane & Mele PRL(05);
Hasan & Kane RMP(10);
Linder Phy(10)aug
[unconventional quantization rules for Landau levels in the surface states];
Prodan JPA(11)-a1010 [disordered, and non-commutative geometry];
news pw(13)apr,
Hafezi & Taylor PT(14)may [optical analog];
Kaufmann et al RVMP(16)-a1501 [invariants];
Roy et al PRB(16)-a1507 [transition between topological and trivial insulators];
Asbóth et al 16;
Schulz-Baldes a1607 [non-technical review];
news pt(18)nov;
> s.a. cohomology.

**Integrability Conditions** > see partial differential equations.

**Integrable System**
> s.a. quantum systems.

**Integral Curve of a Vector Field** > see vector field.

**Integral Domain**

$ __Def__: A
commutative ring with identity and no proper zero divisors.

**Integral Forms**

@ __References__: Catenacci et al JGP(11)-a1003 [Čech and de Rham cohomology].

**Integral Geometry** > see geometry.

**INTEGRAL Mission** > see gamma-ray astronomy.

**Integral of Motion** > see Conservation Laws.

**Integral Submanifold** > see manifolds.

**Integral Transforms**
> s.a. [integration]; fourier;
Laplace Transform.

@ __General references__: Davies 85;
Bateman 54.

@ __Other types__:
Gasaneo & Colavecchia JPA(03) [using 2-body Coulomb wave functions].

**Integration Theory**
> s.a. integration on manifolds.

**Integro-Differential Equations**
> see differential equations.

**Interaction Representation**
> see representations of quantum theory.

**Interfaces**
> s.a. Gravitating Shells; membranes;
topological defects [domain walls].

@ __References__: Avelino et al PRE(11)-a1006 [framework for dynamics].

**Interference**
> s.a. atom interferometry.

**Intergalactic Matter**
> s.a. contents of the universe; cosmic-ray propagation;
dark matter on cosmological scales; milky way
[circumgalactic medium].

@ __General references__: Scannapieco et al SA(02)oct;
Mörtsell & Goobar JCAP(03) [dust, constraints];
Simcoe AS(04)#1;
Barkana & Loeb RPP(07)ap/06 [physics and early history];
Meiksin RMP(09)-a0711 [rev];
Bower AIP(09)-a0909 [and galaxy formation];
Sun NJP(12)-a1203 [hot gas in galaxy groups];
Egan et al ApJ(14)-a1307 [simulations and observations];
Gontcharov AL(13)-a1606 [dust outside the galactic disk];
Borthakur et al ApJ(15)-a1504 [circumgalactic and interstellar medium];
Greig et al MNRAS(16)-a1509 [temperature];
Cavaliere et al ApJ(16)-a1604 [intragroup vs intracluster medium].

@ __Dynamics__: Kim et al ApJ(05)ap [dynamics and velocity field];
Evoli & Ferrara MNRAS(11)-a1101 [turbulence];
Manrique & Salvador-Solé ApJ(15)-a1502;
McQuinn ARAA(16)-a1512.

**Interior Product**

* __Idea__: A product
between a vector field *X* and a 1-form *α*,
denoted by i_{X} *α*
:= *α*(*X*).

**Intermediate Vector Bosons** > see electroweak;
particle types.

**Internal Degrees of Freedom** > see Composite Systems.

**Internal Relativity**

* __Idea__: An approach
to gravity in which the princile of equivalence is implemented not by
using an affine connectionto relate reference frames for local Minkowski
spaces as in general relativity, but by relating different local vacua of
an underlying solid-state like model.

@ __References__: Dreyer a1203 [overview].

**Interpolation Spaces**

@ __References__: Bergh & Löfström 76.

**Interpolation Techniques**
> see observational cosmology.

**Interpretation of a Theory**
> s.a. interpretations of quantum mechanics;
quantum field theory.

* __Idea__: "A
description in ordinary language of what an observer would see or
experience when the mathematical quantities used by the theory to describe
the state of the system take on any of their allowed values".

@ __References__: Curiel PhSc(09)jan
[no need for general relativity, as opposed to quantum mechanics];
Mittelstaedt FP(11)
[why treat classical physics and modern physics differently?];
De Haro & De Regt EJPS(18)-a1803 [and understanding, theories without a spacetime].

> __Specific theories__:
see formulations of electromagnetism; Unparticles.

**Intersection of Sets**
> see sets.

**Intertwiner, Intertwining Operator**
> s.a. group representations; spin networks.

* __Idea__: A
Contraction matrix for a set of representations of a group.

@ __References__: Huang et al m.QA/04 [modules for vertex operator algebras];
Bagarello JMP(09)-a0904,
JMP(10) [between different Hilbert spaces].

**Interval** > see graphs; posets.

**Invariance of a Theory** > see symmetry.

**Invariant in Dynamics** > see observables.

**Invariant Vector Field on a Lie Group, on a Manifold**
> see vector fields.

**Inverse** > see group theory;
matrix.

**Inverse Function Theorem**

**Inverse Limit** > see projective limit.

**Inverse Problems** > see formulations of classical mechanics;
higher-order lagrangians; newtonian gravity;
quantum systems; variational principles.

**Inverse Scattering** > s.a. scattering.

* __Idea__: The problem
of obtaining (the parameter values characterizing) the scattering
potential from a scattered wave.

* __And non-linear
partial differential equations__: An approach to the solution of
differential equations in which the equation appears as an integrability
condition for a pair of linear differential equations with a spectral
parameter, a stationary and an evolution equation.

@ __General theory__: Schroer AP(03)ht/01 [uniqueness in local quantum theory];
Ramm JPA(10)-a0910 [uniqueness theorem];
Melnikov a1212
[and solution of completely integrable PDEs].

@ __For Einstein's equation__:
Belinsky & Zakharov JETP(78);
Belinsky JETP(79);
Zakharov & Shabat FAA(79);
Flaschka & Newell CMP(80);
> s.a. reissner-nordström solutions.

@ __Other systems__: Ramm RPMP-a1601 [on the half-line].

**Inverse System of Spaces and Maps**
> see projective system.

**Invisibility Cloak** > see metamaterials
[for electromagnetic waves]; sound [acoustic cloak].

**Involution** > see algebra.

**Ionization / Ions** > see atomic physics.

**Irreducible Mass of a Black Hole**
> s.a. black hole geometry.

* __Idea__: The energy
contained in the black hole that cannot be extracted by Penrose
processes, i.e., degraded energy that is not stored in rotation;
Classically it can never decrease.

* __Expression__: Given by

*M*_{irr}^{2}
= *A* / (16π*G*^{2}) ,

or \({1\over2}[M^2+\sqrt{M^4-J^2}]\) for a Kerr black hole.

* __Special cases__:
For a Schwarzschild black hole it coincides with the total mass; For a
system of several black holes, it is *E*_{degr} = (∑
*M*_{irr}^{2})^{1/2}
< ∑*M*_{irr}, so, by combining
"dead" Schwarzschild black holes one can still obtain energy.

@ __Introduction of concept__: Christodoulou PRL(70).

**Irreversibility**
> see arrow of time.

**Isbell Topology on a Space of Maps**
> see topology.

**ISCO**
> stands for Innermost Stable Circular Orbit; see reissner-nordström spacetime.

**Isentropic Fluid**
> see perfect fluids.

**Ising Models**
> s.a. 2D ising models.

**Island of Stability**
> see elements.

**Island Universes**
> see history of cosmology [galaxies];
cosmological models [cosmological constant sea].

**Isolated Object / Physical System**
> s.a. systems.

* __Idea__: One which
does not interact with any other system; As pointed out by D Zeh
(1970), there can never be a truly isolated object; Is this a
conceptual difficulty for the universal validity of laws?

* __Quasi-isolated system__:
One subject to small random uncontrollable perturbations; In general
stochastically unstable; > s.a. arrow of time.

@ __References__: Cox GRG(07) [in practice].

> __Various contexts__:
see asymptotic flatness; quantum-mechanical
systems; relativistic cosmology [local effects].

**Isolated Point in a Topological Space**

* __Idea__: A point *x*
in a topological space *X* is called an isolated point of a subset
*S* of *X* if the singleton {*x*} is an open set in
the subset topology on *S*.

> __Online resources__:
see Wikipedia page.

**Isometry of a Manifold with Metric** > see differential geometry.

**Isomorphism** > s.a. graph theory
[isomorphism problem]; group theory [isomorphism problem].

* __Idea__: A mapping that
preserves all the relevant structures of objects in a category.

$ __General def__: An *f*
∈ Hom(*X*, *Y*) is an isomorphism if ∃ *g* ∈ Hom(*Y*,
*X*) such that *f* \(\circ\) *g* = id_{X}
and *g* \(\circ\) *f* = id_{X}.

$ __For sets__: A one-to-one and onto map.

$ __For linear spaces__:
A bijection *f* : *X* → *Y*, where *X*
and *Y* are linear spaces, preserving the linear structure.

**Isospin**
> s.a. modified quantum mechanics [quaternions and (iso)spin]; particle
types; QCD; standard model.

@ __References__: Fernandes & Letelier PLA(05) [motion of particle with isospin].

**Isotope**
> see nuclear physics; atomic physics [isotope effect].

**Isotopy Theory**
> see algebraic topology.

**Isotropic Coordinates**
> s.a. coordinates on a manifold
/ coordinates for schwarzschild spacetime.

$ __Def__: Coordinates in
which a metric (e.g., spatially spherically symmetric) takes the form

d*s*^{2}
= −exp{2*φ*}d*t*^{2}
+ exp{2*μ*} (d*r*^{2}
+ *r*^{2} dΩ^{2}) .

@ __References__: in Misner et al 73, ex 23.1 (p595), ex 31.7 (p840).

**Isotropic Cosmological Model / Spacetime**

> __Theoretical aspects__:
see > see bianchi models;
cosmological models in general
relativity; cosmological principle.

> __Isotropization__: see bianchi
models; cosmological models in general relativity.

> __Phenomenological
aspects__: see Anisotropy; large-scale
geometry; tests of lorentz symmetry.

**Isotropic Metric on a Manifold**

* __Idea__: An *n*-dimensional
Riemannian manifold *M* is isotropic at *p* in *M* if
there is an action of SO(*n*−1) on *M* such that *p*
is a fixed point (the only one in some neighborhood of *p*) and all
group elements act as isometries.

* __Result__: Isotropy
about every point implies homogeneity.

@ __References__: Sormani GAFA(04)math/03 [almost locally isotropic manifolds, and cosmology].

**Isotropic Modified Maxwell Theory**
> see modified electromagnetism.

**Isotropic Submanifold**
> see symplectic structures.

**Isotropy Group of a Point**

* __Idea__: The group
of spatial isometries of an asymptotically flat spacetime which leaves
a given point *p* in *M* fixed.

**Israel's Theorem**
> s.a. black-hole hair and uniqueness.

* __Idea__: The only static
and asymptotically-flat vacuum space-time possessing a regular horizon is the
Schwarzschild solution (or Reissner-Nordström in the electrovac case).

@ __References__: Israel PR(67),
CMP(68);
Herrera IJMPD(08)-a0711 [physical consequences];
Nelson PRD(10)-a1010 [in 4th-order gravity].

**It from Bit**
> see computation [the universe as a computer]; loop quantum gravity.

**Itakura-Saito Distance** > see types of metric spaces.

**Itō Calculus** > see analysis.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 22 oct 2019