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-km3 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 km2 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, AII or IAI; 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]; 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].
> Online resources: see Wikipedia page.

i ε Term in Field Theory
@ References: Sverdlov & Bombelli PRD(14)-a1306 [and continuous quantum measurement]; Witten a1307-fs [string-theory analog].

Ill-Posed Problem > see Well-Posed Problem.

Image Charge > see Method of Images.

Immersion > s.a. embedding.

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 a1701 [and the linking theory of shape dynamics]; > 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].

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
$ Def: If a is a non-degenerate (i.e., non-singular Hessian) critical point of f, its index is the number of negative terms in the quadratic expansion of f in a neighborhood of a.

Index of an Elliptic Operator D
$ Def: The difference dim(ker D) – dim(coker 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 Sn–1 → Sn–1 defined by v (normalized with some flat metric) on a small sphere surrounding x.
* Properties: indx(–v) = (–1)n indx(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].

Individuality / Individuation > s.a. foundations of quantum mechanics [individual particles]; particle statistics [identical particles].
@ 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.

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]; > 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].

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 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 [unifoed 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 14 [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].
@ 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. 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].
> 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 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 [incuding 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-Tc 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: 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]; Li et al a1501 [invariants]; Roy et al PRB(16)-a1507 [transition between topological and trivial insulators]; Asbóth et al 16; Schulz-Baldes a1607 [non-technical review]; > 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 Equations

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

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 iX 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. formulations of electromagnetism; 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?].

Intersection of Sets > see sets.

Intersection Form / Theory

Interstellar Matter

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

Mirr2 = A / (16πG2) ,

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 Edegr = (∑Mirr2)1/2 < ∑Mirr, 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 Horizon

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 = idX and g \(\circ\) f = idX.
$ For sets: A one-to-one and onto map.
$ For linear spaces: A bijection f : XY, 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

ds2 = –exp{2φ}dt2 + exp{2μ} (dr2 + r22) .

@ 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.
> 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].

Itakura-Saito Distance > see types of metric spaces.

Itō Calculus > see analysis.


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